Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-28T06:26:17.847Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  14 September 2020

Edited by
Mourad E. H. Ismail
Assisted by
Walter Van Assche
Mourad E. H. Ismail
Affiliation:
University of Central Florida
Walter Van Assche
Affiliation:
Katholieke Universiteit Leuven, Belgium
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Encyclopedia of Special Functions: The Askey-Bateman Project , pp. 357 - 384
Publisher: Cambridge University Press
Print publication year: 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdi, W. H. 1965. A basic analogue of the Bessel polynomial. Math. Nachr., 30, 209–219.Google Scholar
Ablowitz, M. J. and Ladik, J. F. 1976a. A nonlinear difference scheme and inverse scattering. Studies in Appl. Math., 55(3), 213–229.CrossRefGoogle Scholar
Ablowitz, M. J. and Ladik, J. F. 1976b. Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys., 17(6), 1011–1018.CrossRefGoogle Scholar
Abramov, S. A. 1989. Problems in computer algebra that are connected with a search for polynomial solutions of linear differential and difference equations. Moscow Univ. Comput. Math. Cybernet., 3, 63–68. Transl. from Vestn. Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 3, 56–60.Google Scholar
Abramowitz, M. and Stegun, I. A. (eds). 1965. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. Superintendent of Documents, US Government Printing Office, Washington, DC. Third printing, with corrections.Google Scholar
Ahmed, S. and Muldoon, M. E. 1983. Reciprocal power sums of differences of zeros of special functions. SIAM J. Math. Anal., 14(2), 372–382.Google Scholar
Ahmed, S., Bruschi, M., Calogero, F., Olshanetsky, M. A., and Perelomov, A. M. 1979. Properties of the zeros of the classical polynomials and of the Bessel functions. Nuovo Cimento B (11), 49(2), 173–199.Google Scholar
Ahmed, S., Laforgia, A., and Muldoon, M. E. 1982. On the spacing of the zeros of some classical orthogonal polynomials. J. London Math. Soc. (2), 25(2), 246–252.Google Scholar
Akhiezer, N. I. 1965. The Classical Moment Problem and Some Related Questions in Analysis. New York: Hafner. Translated by N. Kemmer.Google Scholar
Akhiezer, N. I. and Glazman, I. M. 1950. Theory of Linear Operators in Hilbert Space. Reprint, New York: Dover Publications, Two volumes bound as one, 1993Google Scholar
Al-Salam, W. A. and Carlitz, L. 1965. Some orthogonal q-polynomials. Math. Nachr., 30, 47–61.Google Scholar
Al-Salam, W. A. and Chihara, T. S. 1976. Convolutions of orthonormal polynomials. SIAM J. Math. Anal., 7(1), 16–28.Google Scholar
Al-Salam, W. A. and Chihara, T. S. 1987. q-Pollaczek polynomials and a conjecture of Andrews and Askey. SIAM J. Math. Anal., 18(1), 228–242.CrossRefGoogle Scholar
Al-Salam, W. A. and Ismail, M. E. H. 1983. Orthogonal polynomials associated with the Rogers–Ramanujan continued fraction. Pacific J. Math., 104(2), 269–283.CrossRefGoogle Scholar
Al-Salam, W. A. and Verma, A. 1983. q-analogs of some biorthogonal functions. Canad. Math. Bull., 26(2), 225–227.Google Scholar
Al-Salam, W. A., Allaway, W. R., and Askey, R. A. 1984. Sieved ultraspherical polynomials. Trans. Amer. Math. Soc., 284(1), 39–55.Google Scholar
Alfaro, M. P. and Vigil, L. 1988. Solution of a problem of P. Turán on zeros of orthogonal polynomials on the unit circle. J. Approx. Theory, 53(2), 195–197.Google Scholar
Alfaro, M. P., Bello Hernández, M., Montaner, J. M., and Varona, J. L. 2005. Some asymptotic properties for orthogonal polynomials with respect to varying measures. J. Approx. Theory, 135(1), 22–34.Google Scholar
Ammar, G. S. and Gragg, W. B. 1994. Schur flows for orthogonal Hessenberg matrices. Pages 27–34 of Bloch, Anthony (ed), Hamiltonian and Gradient Flows, Algorithms and Control. Fields Inst. Commun., vol. 3. Providence, RI: American Mathematical Society.Google Scholar
Andrews, G. E. 1976. The Theory of Partitions. Encyclopedia of Mathematics and its Applications, vol. 2. Reading, Mass.-London-Amsterdam: Addison-Wesley.Google Scholar
Andrews, G. E. 1981. Ramunujan’s “lost” notebook. III. The Rogers–Ramanujan continued fraction. Adv. Math., 41(2), 186–208.Google Scholar
Andrews, G. E. 1986. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, vol. 66. Washington, DC: Published for the Conference Board of the Mathematical Sciences.Google Scholar
Andrews, G. E. 1990. A page from Ramanujan’s lost notebook. Indian J. Math., 32(3), 207–216.Google Scholar
Andrews, G. E. 2005. Ramanujan’s “lost” notebook. VIII. The entire Rogers–Ramanujan function. Adv. Math., 191(2), 393–407.Google Scholar
Andrews, G. E. and Askey, R. A. 1985. Classical orthogonal polynomials. Pages 36–62 of Brezinski, C., Draux, A., Magnus, A. P., Maroni, P., and Ronveaux, A. (eds), Orthogonal Polynomials and Applications (Bar-le-Duc, 1984). Lecture Notes in Math., vol. 1171. Berlin: Springer.Google Scholar
Andrews, G. E., Askey, R. A., and Roy, R. 1999. Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge: Cambridge University Press.Google Scholar
Andrews, G. E., Berndt, B. C., Sohn, J., Yee, A. J., and Zaharescu, A. 2003. On Ramanujan’s continued fraction for (q2;q3)∞/(q;q3)∞. Trans. Amer. Math. Soc., 355(6), 2397–2411 (electronic).Google Scholar
Andrews, G. E., Berndt, B. C., Sohn, J., Yee, A. J., and Zaharescu, A. 2005. Continued fractions with three limit points. Adv. Math., 192, 231–258.Google Scholar
Annaby, M. H. and Mansour, Z. S. 2005. Basic Sturm–Liouville problems. J. Phys. A, 38(17), 3775–3797.Google Scholar
Appell, P. and Kampé de Fériet, J. 1926. Fonctions Hypergéométriques et Hypersphérique; Polynomes d’Hermite. Paris: Gauthier-Villars.Google Scholar
Area, I., Dimitrov, D. K., Godoy, E., and Ronveaux, A. 2004. Zeros of Gegenbauer and Hermite polynomials and connection coefficients. Math. Comp., 73(248), 1937–1951 (electronic).Google Scholar
Askey, R. A. 1971. Orthogonal expansions with positive coefficients. II. SIAM J. Math. Anal., 2, 340–346.Google Scholar
Askey, R. A. 1975a. A Note on the History of Series. Tech. rept. 1532. Mathematics Research Center, University of Wisconsin.Google Scholar
Askey, R. A. 1975b. Orthogonal Polynomials and Special Functions. Philadelphia, PA: Society for Industrial and Applied Mathematics.Google Scholar
Askey, R. A. 1980. Ramanujan’s extensions of the gamma and beta functions. Amer. Math. Monthly, 87(5), 346–359.CrossRefGoogle Scholar
Askey, R. A. 1983. An elementary evaluation of a beta type integral. Indian J. Pure Appl. Math., 14(7), 892–895.Google Scholar
Askey, R. A. 1985. Continuous Hahn polynomials. J. Phys. A, 18, L1017–L1019.Google Scholar
Askey, R. A. 1989a. Beta integrals and the associated orthogonal polynomials. Pages 84–121 of Alladi, K. (ed), Number Theory, Madras 1987. Lecture Notes in Math., vol. 1395. Berlin: Springer.Google Scholar
Askey, R. A. 1989b. Continuous q-Hermite polynomials when q>1. Pages 151–158 of Stanton, D. (ed), q-Series and Partitions. IMA Volumes in Mathematics and Its Applications. New York, NY: Springer.Google Scholar
Askey, R. A. 1990. Graphs as an aid to understanding special functions. Pages 3–33 of Wong, R. (ed), Asymptotic and Computational Analysis. New York, NY: Marcel Dekker.Google Scholar
Askey, R. A. and Gasper, G. 1977. Convolution structures for Laguerre polynomials. J. Analyse Math., 31, 48–68.Google Scholar
Askey, R. A. and Ismail, M. E. H. 1976. Permutation problems and special functions. Canadian J. Math., 28, 853–874.Google Scholar
Askey, R. A. and Ismail, M. E. H. 1983. A generalization of ultraspherical polynomials. Pages 55–78 of Erdős, P. (ed), Studies in Pure Mathematics. Basel: Birkhäuser.Google Scholar
Askey, R. A. and Ismail, M. E. H. 1984. Recurrence relations, continued fractions and orthogonal polynomials. Memoirs Amer. Math. Soc., 49(300), iv + 108 pp.Google Scholar
Askey, R. A. and Roy, R. 1986. More q-beta integrals. Rocky Mountain J. Math., 16(2), 365–372.CrossRefGoogle Scholar
Askey, R. A. and Wilson, J. A. 1979. A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols. SIAM J. Math. Anal., 10(5), 1008–1016.Google Scholar
Askey, R. A. and Wilson, J. A. 1982. A set of hypergeometric orthogonal polynomials. SIAM J. Math. Anal., 13(4), 651–655.Google Scholar
Askey, R. A. and Wilson, J. A. 1985. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc., 54(319), iv + 55 pp.Google Scholar
Askey, R. A. and Wimp, J. 1984. Associated Laguerre and Hermite polynomials. Proc. Roy. Soc. Edinburgh, 96A, 15–37.Google Scholar
Askey, R. A., Ismail, M. E. H., and Koornwinder, T. 1978. Weighted permutation problems and Laguerre polynomials. J. Comb. Theory Ser. A, 25(3), 277–287.Google Scholar
Askey, R. A., Rahman, M., and Suslov, S. K. 1996. On a general q-Fourier transformation with nonsymmetric kernels. J. Comp. Appl. Math., 68(1-2), 25–55.Google Scholar
Atakishiyev, N. M. and Klimyk, A. U. 2004. On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials. J. Math. Anal. Appl., 294(1), 246–257.Google Scholar
Atakishiyev, N. M. and Suslov, S. K. 1985. The Hahn and Meixner polynomials of imaginary argument and some of their applications. J. Phys. A, 18, 1583–1596.Google Scholar
Atakishiyev, N. M. and Suslov, S. K. 1992. Difference hypergeometric functions. Pages 1–35 of Gonchar, A. A. and Saff, E. B. (eds), Progress in Approximation Theory (Tampa, FL, 1990). Springer Ser. Comput. Math., vol. 19. New York: Springer.Google Scholar
Atakishiyev, N. M., Frank, A., and Wolf, K. B. 1994. A simple difference realization of the Heisenberg q-algebra. J. Math. Phys., 35(7), 3253–3260.Google Scholar
Atkinson, F. V. 1964. Discrete and Continuous Boundary Problems. Mathematics in Science and Engineering, vol. 8. New York: Academic Press.Google Scholar
Atkinson, F. V. and Everitt, W. N. 1981. Orthogonal polynomials which satisfy second order differential equations. Pages 173–181 of E. B. Christoffel (Aachen/Monschau, 1979). Basel: Birkhäuser.Google Scholar
Azor, R., Gillis, J., and Victor, J. D. 1982. Combinatorial applications of Hermite polynomials. SIAM J. Math. Anal., 13(5), 879–890.Google Scholar
Badkov, V. M. 1985. Uniform Asymptotic Representations of Orthogonal Polynomials. Sverdlovsk: Ural. Nauchn. Tsentr Akad. Nauk SSSR. Pages 41–53.Google Scholar
Badkov, V. M. 1987. Systems of orthogonal polynomials expressed in explicit form in terms of Jacobi polynomials. Math. Notes, 42, 858–863.Google Scholar
Baginski, F. E. 1991. Comparison theorems for the v-zeroes of Legendre functions Pvm(z0) when !−1<z0<1. Proc. Amer. Math. Soc., 111(2), 395–402.Google Scholar
Baik, J., Deift, P., and Johansson, K. 1999. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc., 12, 1119–1178.Google Scholar
Bailey, W. N. 1935. Generalized Hypergeometric Series. Cambridge: Cambridge University Press.Google Scholar
Bannai, E. and Ito, T. 1984. Algebraic Combinatorics I: Association Schemes. Menlo Park: Benjamin/Cummings.Google Scholar
Baratella, P. and Gatteschi, L. 1988. The bounds for the error terms of an asymptotic approximation of Jacobi polynomials. Pages 203–221 of Alfaro, M., Dehesa, J. S., Marcellán, F. J., Rubio de Francia, J. L., and Vinuesa, J. (eds), Orthogonal Polynomials and Their Applications (Segovia, Spain, 1986). Lecture Notes in Math., vol. 1329. New York: Springer.Google Scholar
Barrios, D. and López, G. 1999. Ratio asymptotics of polynomials orthogonal on arcs of the unit circle. Constr. Approx., 15, 1–31.Google Scholar
Basu, S. and Bose, N. K. 1983. Matrix Stieltjes series and network models. SIAM J. Math. Anal., 14(2), 209–222.Google Scholar
Bateman, H. 1905. A generalization of the Legendre polynomials. Proc. London Math. Soc., 3(2), 111–123.Google Scholar
Bateman, H. 1932. Partial Differential Equations. Cambridge: Cambridge University Press.Google Scholar
Bauldry, W. 1990. Estimates of asymmetric Freud polynomials on the real line. J. Approximation Theory, 63, 225–237.Google Scholar
Beckenbach, E. F., Seidel, W., and Szász, O. 1951. Recurrent determinants of Legendre and ultraspherical polynomials. Duke Math. J., 18, 1–10.Google Scholar
Bello, M. and López, G. 1998. Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle. J. Approx. Theory, 92, 216–244.Google Scholar
Berezanskii, Ju. M. 1968. Expansions in Eigenfunctions of Selfadjoint Operators. Translations of Mathematical Monographs, vol. 17. Providence, RI: American Mathematical Society. Translated from the Russian by Bolstein, R., Danskin, J. M., Rovnyak, J., and Shulman, L..Google Scholar
Berg, C. 1994. Markov’s theorem revisited. J. Approx. Theory, 78, 260–275.Google Scholar
Berg, C. 1995. Indeterminate moment problems and the theory of entire functions. J. Comput. Appl. Math., 65(1-3), 27–55. Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994).CrossRefGoogle Scholar
Berg, C. 1998. From discrete to absolutely continuous solutions of indeterminate moment problems. Arab J. Math. Sci., 4(2), 1–18.Google Scholar
Berg, C. 2004. Private communication.Google Scholar
Berg, C. and Christensen, J. P. R. 1981. Density questions in the classical theory of moments. (French summary). Ann. Inst. Fourier (Grenoble), 31(3), 99–114.Google Scholar
Berg, C. and Durán, A. J. 1995. The index of determinacy for measures and the l2-norm of orthonormal polynomials. Trans. Amer. Math. Soc., 347, 2795–2811.Google Scholar
Berg, C. and Durán, A. J. 1996. When does a discrete differential perturbation of a sequence of orthonormal polynomials belong to ℓ2? J. Funct. Anal., 136, 127–153.CrossRefGoogle Scholar
Berg, C. and Ismail, M. E. H. 1996. q-Hermite polynomials and classical orthogonal polynomials. Canad. J. Math., 48, 43–63.Google Scholar
Berg, C. and Pedersen, H. L. 1994. On the order and type of the entire functions associated with an indeterminate Hamburger moment problem. Ark. Mat., 32, 1–11.Google Scholar
Berg, C. and Pedersen, H. L. 2007. Logarithmic order and type of indeterminate moment problems. Pages 51–79 of Difference Equations, Special Functions and Orthogonal Polynomials. Hackensack, NJ: World Scientific Publisher. With an appendix by Walter Hayman.Google Scholar
Berg, C. and Szwarc, R. 2011. The smallest eigenvalue of Hankel matrices. Constr. Approx., 34(1), 107–133.Google Scholar
Berg, C. and Szwarc, R. 2014. On the order of indeterminate moment problems. Adv. Math., 250, 105–143.Google Scholar
Berg, C. and Szwarc, R. 2017. Symmetric moment problems and a conjecture of Valent. Mat.Sb. 208 (2017), no. 3, 28–53; translated in Sb. Math. 208 (2017), no. 3–4, 335–359.Google Scholar
Berg, C. and Thill, M. 1991. Rotation invariant moment problems. Acta Math., 167, 207–227.Google Scholar
Berg, C. and Valent, G. 1994. The Nevanlinna parameterization for some indeterminate Stieltjes moment problems associated with birth and death processes. Methods and Applications of Analysis, 1, 169–209.Google Scholar
Berg, C. and Valent, G. 1995. Nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes. J. Comput. Appl. Math., 57(1-2), 29–43. Proceedings of the Fourth International Symposium on Orthogonal Polynomials and Their Applications (Evian-Les-Bains, 1992).CrossRefGoogle Scholar
Berg, C., Christensen, J. P. R., and Ressel, P. 1984. Harmonic Analysis on Semigroups. Graduate Texts in Mathematics, vol. 100. New York: Springer. Theory of positive definite and related functions.Google Scholar
Berg, C., Chen, Y., and Ismail, M. E. H. 2002. Small eigenvalues of large Hankel matrices: the indeterminate case. Math. Scand., 91(1), 67–81.Google Scholar
Bergweiler, W. and Hayman, W. K. 2003. Zeros of solutions of a functional equation. Comput. Methods Funct. Theory, 3(1-2), 55–78.Google Scholar
Berndt, B. C. 2016. Integrals associated with Ramanujan and elliptic functions. Ramanujan J., 41(1), 369–389.Google Scholar
Berndt, B. C. and Sohn, J. 2002. Asymptotic formulas for two continued fractions in Ramanujan’s lost notebook. J. London, Math. Soc., 65, 271–284.Google Scholar
Biedenharn, L. and Louck, J. 1981. The Racah–Wigner Algebra in Quantum Theory. Reading: Addison-Wesley.Google Scholar
Boas, R. P. Jr. 1939. The Stieltjes moment problem for functions of bounded variation. Bull. Amer. Math. Soc., 45, 399–404.Google Scholar
Boas, R. P. Jr. 1954. Entire Functions. New York: Academic Press.Google Scholar
Bochkov, I. 2019. Polynomial birth-death processes and the second conjecture of Valent. C.R. Acad. Sci. Paris, Ser. I, 357 (2019), 247–251.Google Scholar
Bochner, S. 1929. Über Sturm–Liouvillesche polynomsysteme. Math. Zeit., 29, 730–736.CrossRefGoogle Scholar
Bochner, S. 1954. Positive zonal functions on spheres. Proc. Nat. Acad. Sci. USA, 40, 1141–1147.Google Scholar
Bonan, S., Lubinsky, D. S., and Nevai, P. 1987. Orthogonal polynomials and their derivatives. II. SIAM J. Math. Anal., 18(4), 1163–1176.Google Scholar
Bonan, S. S. and Clark, D. S. 1990. Estimates of the Hermite and the Freud polynomials. J. Approximation Theory, 63, 210–224.Google Scholar
Bonan, S. S. and Nevai, P. 1984. Orthogonal polynomials and their derivatives. I. J. Approx. Theory, 40, 134–147.Google Scholar
Bourget, J. 1866. Mémoire sur le mouvement vibratoire des membranes circulaires (June 5, 1865). Ann. Sci. Éc. Norm. Supér., III(5), 5–95.Google Scholar
Braaksma, B. L. J. and Meulenbeld, B. 1971. Jacobi polynomials as spherical harmonics. Indag. Math., 33, 191–196.Google Scholar
Bressoud, D. 1981. On partitions, orthogonal polynomials and the expansion of certain infinite products. Proc. London Math. Soc., 42, 478–500.Google Scholar
Brezinski, C. 1980. Padé-Type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, vol. 50. Basel: Birkhäuser.Google Scholar
Brown, B. M. and Ismail, M. E. H. 1995. A right inverse of the Askey–Wilson operator. Proc. Amer. Math. Soc., 123, 2071–2079.Google Scholar
Brown, B. M., Evans, W. D., and Ismail, M. E. H. 1996. The Askey–Wilson polynomials and q-Sturm–Liouville problems. Math. Proc. Cambridge Phil. Soc., 119, 1–16.Google Scholar
Bryc, W., Matysiak, W., and Szabłowski, P. J. 2005. Probabilistic aspects of Al-Salam–Chihara polynomials. Proc. Amer. Math. Soc., 133, 1127–1134.Google Scholar
Buchwalter, H. and Cassier, G. 1984a. La paramétrisation de Nevanlinna dans le problème des moments de Hamburger. Exposition. Math., 2(2), 155–178.Google Scholar
Buchwalter, H. and Cassier, G. 1984b. Mesures canoniques dans le problème classique des moments. Ann. Inst. Fourier (Grenoble), 34(2), 45–52.CrossRefGoogle Scholar
Bueno, M. I. and Marcellán, F. 2004. Darboux transformation and perturbation of linear functionals. Linear Algebra and Its Applications, 384, 215–242.Google Scholar
Bunse-Gerstner, A. and Elsner, L. 1991. Schur parameter pencils for the solution of the unitary eigenvalue problem. Linear Algebra Appl. 154/156, 741–778.Google Scholar
Burchnall, J. L. 1951. The Bessel polynomials. Canad. J. Math., 3, 62–68.Google Scholar
Burchnall, J. L. and Chaundy, T. W. 1931. Commutative ordinary differential operators. II. The identity Pn=Qm. Proc. Roy. Soc. London (A), 34, 471–485.Google Scholar
Bustoz, J. and Ismail, M. E. H. 1982. The associated classical orthogonal polynomials and their q-analogues. Canad. J. Math., 34, 718–736.CrossRefGoogle Scholar
Cachafeiro, A. and Marcellán, F. 1988. Orthogonal polynomials and jump modifications. Lecture Notes Math., 1329, 236–240.Google Scholar
Cachafeiro, A. and Marcellán, F. 1993. Modifications of Toeplitz matrices: jump functions. Rocky Mountain J. Math., 23, 521–531.Google Scholar
Calogero, F. 2001. Classical Many-Body Problems Amenable to Exact Treatments. Lecture Notes in Physics. New Series: Monographs, vol. 66. Berlin: Springer. (Solvable and/or integrable and/or linearizable…) in one-, two- and three-dimensional space.Google Scholar
Cantero, M. J., Moral, L., and Velázquez, L. 2003. Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl., 362, 29–56.CrossRefGoogle Scholar
Cantero, M. J., Moral, L., and Velázquez, L. 2007. Matrix orthogonal polynomials whose derivatives are also orthogonal. J. Approx. Theory, 146(2), 174–211.Google Scholar
Carlitz, L. 1958. On some polynomials of Tricomi. Boll. Un. Mat. Ital. (3), 13, 58–64.Google Scholar
Carlitz, L. 1959. Some formulas related to the Rogers–Ramanujan identities. Ann. Mat. (IV), 47, 243–251.Google Scholar
Cartier, P. and Foata, D. 1969. Problèmes Combinatoires de Commutation et Réarrangements. Lecture Notes in Mathematics, vol. 85. Berlin: Springer.Google Scholar
Castro, M. M. and Grünbaum, F. A. 2005. Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples. J. Nonlinear Math. Phys., 12 (suppl. 2), 63–76.Google Scholar
Castro, M. M. and Grünbaum, F. A. 2006. The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: five instructive examples. Int. Math. Res. Not., 2006, Art. ID 47602, 33.Google Scholar
Charris, J. A. and Ismail, M. E. H. 1986. On sieved orthogonal polynomials. II: Random walk polynomials. Canad. J. Math., 38(2), 397–415.Google Scholar
Charris, J. A. and Ismail, M. E. H. 1987. On sieved orthogonal polynomials. V: Sieved Pollaczek polynomials. SIAM J. Math. Anal., 18(4), 1177–1218.Google Scholar
Charris, J. A., Ismail, M. E. H., and Monsalve, S. 1994. On sieved orthogonal polynomials. X: General blocks of recurrence relations. Pacific J. Math., 163(2), 237–267.Google Scholar
Chen, Y. and Ismail, M. E. H. 1997. Ladder operators and differential equations for orthogonal polynomials. J. Phys. A, 30, 7817–7829.Google Scholar
Chen, Y. and Ismail, M. E. H. 1998. Some indeterminate moment problems and Freud-like weights. Constr. Approx., 14(3), 439–458.CrossRefGoogle Scholar
Chen, Y. and Ismail, M. E. H. 2005. Jacobi polynomials from compatibility conditions. Proc. Amer. Math. Soc., 133(2), 465–472 (electronic).Google Scholar
Chihara, T. S. 1962. Chain sequences and orthogonal polynomials. Trans. Amer. Math. Soc., 104, 1–16.Google Scholar
Chihara, T. S. 1968. On indeterminate Hamburger moment problems. Pacific J. Math., 27, 475–484.Google Scholar
Chihara, T. S. 1970. A characterization and a class of distribution functions for the Stieltjes–Wigert polynomials. Canad. Math. Bull., 13, 529–532.Google Scholar
Chihara, T. S. 1978. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach.Google Scholar
Chihara, T. S. 1982. Indeterminate symmetric moment problems. J. Math. Anal. Appl., 85(2), 331–346.Google Scholar
Chihara, T. S. 1989. Hamburger moment problems and orthogonal polynomials. Trans. Amer. Math. Soc., 315(1), 189–203.Google Scholar
Chihara, T. S. and Ismail, M. E. H. 1993. Extremal measures for a system of orthogonal polynomials. Constructive Approximation, 9, 111–119.Google Scholar
Christiansen, J. S. 2003a. The moment problem associated with the q-Laguerre polynomials. Constr. Approx., 19(1), 1–22.Google Scholar
Christiansen, J. S. 2003b. The moment problem associated with the Stieltjes–Wigert polynomials. J. Math. Anal. Appl., 277, 218–245.Google Scholar
Christiansen, J. S. 2004. Indeterminate Moment Problems within the Askey-Scheme. Ph.D. thesis, University of Copenhagen.Google Scholar
Christiansen, J. S. 2005. Indeterminate moment problems related to birth and death processes with quartic rates. J. Comp. Appl. Math., 178, 91–98.Google Scholar
Christiansen, J. S. and Ismail, M. E. H. 2006. A moment problem and a family of integral evaluations. Trans. Amer. Math. Soc., 358(9), 4071–4097.Google Scholar
Christiansen, J. S. and Koelink, E. 2006. Self-adjoint difference operators and classical solutions to the Stieltjes–Wigert moment problem. J. Approx. Theory, 140(1), 1–26.Google Scholar
Christiansen, J. S. and Koelink, E. 2008. Self-adjoint difference operators and symmetric Al-Salam–Chihara polynomials. Constr. Approx., 28(2), 199–218.Google Scholar
Ciccoli, N., Koelink, E., and Koornwinder, T. H. 1999. q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations. Methods Appl. Anal., 6(1), 109–127. Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part I.Google Scholar
Cohen, M. E. 1977. On Jacobi functions and multiplication theorems for integral Bessel functions. J. Math. Anal. Appl., 57, 469–475.Google Scholar
Connett, W. C. and Schwartz, A. L. 2000. Measure algebras associated with orthogonal polynomials. Pages 127–140 of Ismail, M. E. H. and Stanton, D. W. (eds), q-Series from a Contemporary Perspective (South Hadley, MA, 1998). Contemp. Math., vol. 254. American Mathematical Society.Google Scholar
Cooper, S. 2002. The Askey–Wilson operator and the 6ψ5 summation formula. South East Asian J. Math. Math. Sci., 1(1), 71–82.Google Scholar
Cycon, H. L., Froese, R. G., Kirsch, W., and Simon, B. 1987. Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Berlin: Springer.Google Scholar
Damanik, D., Pushnitski, A., and Simon, B. 2008. The analytic theory of matrix orthogonal polynomials. Surv. Approx. Theory, 4, 1–85.Google Scholar
de Boor, C. and Saff, E. B. 1986. Finite sequences of orthogonal polynomials connected by a Jacobi matrix. Linear Algebra Appl., 75, 43–55.Google Scholar
de Bruin, M. G., Saff, E. B., and Varga, R. S. 1981a. On the zeros of generalized Bessel polynomials. I. Nederl. Akad. Wetensch. Indag. Math., 43(1), 1–13.Google Scholar
de Bruin, M. G., Saff, E. B., and Varga, R. S. 1981b. On the zeros of generalized Bessel polynomials. II. Nederl. Akad. Wetensch. Indag. Math., 43(1), 14–25.Google Scholar
Deaño, A., Gil, A., and Segura, J. 2004. New inequalities from classical Sturm theorems. J. Approx. Theory, 131, 208–230.Google Scholar
DeFazio, M. V., Gupta, D. P., and Muldoon, M. E. 2007. Limit relations for the complex zeros of Laguerre and q-Laguerre polynomials. J. Math. Anal. Appl., 334(2), 977–982.Google Scholar
Deift, P. 1999. Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3. New York, NY: New York University Courant Institute of Mathematical Sciences.Google Scholar
Denisov, S. 2004. On Rakhmanov’s theorem for Jacobi matrices. Proc. Amer. Math. Soc., 132, 847–852.Google Scholar
Denisov, S. and Kupin, S. 2006. Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight. J. Approx. Theory, 139(1-2), 8–28.Google Scholar
Dette, H. and Studden, W. J. 2003. Quadrature formulas for matrix measures—a geometric approach. Linear Algebra Appl., 364, 33–64.Google Scholar
Diaconis, P. and Graham, R. L. 1985. The Radon transform on Z2k. Pacific J. Math., 118(2), 323–345.Google Scholar
Dickinson, D. J. 1954. On Lommel and Bessel polynomials. Proc. Amer. Math. Soc., 5, 946–956.Google Scholar
Dickinson, D. J., Pollack, H. O., and Wannier, G. H. 1956. On a class of polynomials orthogonal over a denumerable set. Pacific J. Math, 6, 239–247.Google Scholar
Dickson, L. E. 1939. New Course on the Theory of Equations. New York: Wiley.Google Scholar
Dilcher, K. and Stolarsky, K. 2005. Resultants and discriminants of Chebyshev and related polynomials. Trans. Amer. Math. Soc., 357, 965–981.Google Scholar
Dimitrov, D. K. 2003. Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials. J. Comput. Appl. Math., 153(1-2), 171–180. Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and Their Applications (Rome, 2001).Google Scholar
Dimitrov, D. K. and Nikolov, G. P. 2010. Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory, 162(10), 1793–1804.Google Scholar
Dimitrov, D. K. and Rafaeli, F. R. 2007. Monotonicity of zeros of Jacobi polynomials. J. Approx. Theory, 149(1), 15–29.Google Scholar
Dimitrov, D. K. and Rafaeli, F. R. 2009. Monotonicity of zeros of Laguerre polynomials. J. Comput. Appl. Math., 233(3), 699–702.Google Scholar
Dimitrov, D. K. and Rodrigues, R. O. 2002. On the behaviour of zeros of Jacobi polynomials. J. Approx. Theory, 116(2), 224–239.Google Scholar
Dominici, D. 2008. Asymptotic analysis of the Krawtchouk polynomials by the WKB method. Ramanujan J., 15(3), 303–338.Google Scholar
Donoghue, W. F. Jr. 1974. Monotone Matrix Functions and Analytic Continuation. New York: Springer. Die Grundlehren der mathematischen Wissenschaften, Band 207.CrossRefGoogle Scholar
Draux, A. 1983. Polynômes Orthogonaux Formels – Applications. Lecture Notes in Mathematics, vol. 974. Berlin: Springer.Google Scholar
Driver, K. and Duren, P. 2000. Zeros of the hypergeometric polynomials F(-n,b;2b;z). Indag. Math. (N.S.), 11(1), 43–51.Google Scholar
Driver, K. and Duren, P. 2001. Zeros of ultraspherical polynomials and the Hilbert–Klein formulas. J. Comput. Appl. Math., 135(2), 293–301.Google Scholar
Duistermaat, J. J. and Grünbaum, F. A. 1986. Differential equations in the spectral parameter. Comm. Math. Phys., 103(2), 177–240.Google Scholar
Dulucq, S. and Favreau, L. 1991. A combinatorial model for Bessel polynomials. Pages 243–249 of Brezinski, C., Gori, L., and Ronveau, A. (eds), Orthogonal Polynomials and Their Applications. Basel, Switzerland: J. C. Baltzer AG Scientific.Google Scholar
Durán, A. J. 1989. The Stieltjes moments problem for rapidly decreasing functions. Proc. Amer. Math. Soc., 107(3), 731–741.Google Scholar
Durán, A. J. 1993. Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math., 23(1), 87–104.Google Scholar
Durán, A. J. 1993. A generalization of Favard’s theorem for polynomials satisfying a recurrence relation. J. Approx. Theory, 74(1), 83–109.Google Scholar
Durán, A. J. 1995. On orthogonal polynomials with respect to a positive definite matrix of measures. Canad. J. Math., 47(1), 88–112.Google Scholar
Durán, A. J. 1996. Markov’s theorem for orthogonal matrix polynomials. Canad. J. Math., 48(6), 1180–1195.Google Scholar
Durán, A. J. 1997. Matrix inner product having a matrix symmetric second order differential operator. Rocky Mountain J. Math., 27(2), 585–600.Google Scholar
Durán, A. J. 1999. Ratio asymptotics for orthogonal matrix polynomials. J. Approx. Theory, 100(2), 304–344.Google Scholar
Durán, A. J. 2009a. Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices. J. Approx. Theory, 161(1), 88–113.Google Scholar
Durán, A. J. 2009b. A method to find weight matrices having symmetric second-order differential operators with matrix leading coefficient. Constr. Approx., 29(2), 181–205.Google Scholar
Durán, A. J. 2010. Rodrigues’ formulas for orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not., 2010(5), 824–855.Google Scholar
Durán, A. J. 2011a. A miraculously commuting family of orthogonal matrix polynomials satisfying second order differential equations. J. Approx. Theory, 163(12), 1815–1833.Google Scholar
Durán, A. J. 2011b. Rodrigues’ formulas for orthogonal matrix polynomials satisfying higher order differential equations. Experimental Mathematics, 20(1), 15–24.Google Scholar
Durán, A. J. and de la Iglesia, M. D. 2008a. Second-order differential operators having several families of orthogonal matrix polynomials as eigenfunctions. Int. Math. Res. Not., 2008, Art. ID rnn 084, 24.Google Scholar
Durán, A. J. and de la Iglesia, M. D. 2008b. Some examples of orthogonal matrix polynomials satisfying odd order differential equations. J. Approx. Theory, 150(2), 153–174.Google Scholar
Durán, A. and Daneri-Vias, E. 2001. Ratio asymptotics for orthogonal matrix polynomials with unbounded recurrence coefficients. J. Approx. Theory 110, 1–17.Google Scholar
Durán, A. J. and Defez, E. 2002. Orthogonal matrix polynomials and quadrature formulas. Linear Algebra Appl., 345, 71–84.Google Scholar
Durán, A. J. and Grünbaum, F. A. 2004. Orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not., 2004(10), 461–484.Google Scholar
Durán, A. J. and Grünbaum, F. A. 2005a. A characterization for a class of weight matrices with orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not., 2005(23), 1371–1390.Google Scholar
Durán, A. J. and Grünbaum, F. A. 2005b. Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations. J. Approx. Theory, 134(2), 267–280.Google Scholar
Durán, A. J. and Grünbaum, F. A. 2005c. Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. I. Constr. Approx., 22(2), 255–271.Google Scholar
Durán, A. J. and Grünbaum, F. A. 2007. Matrix orthogonal polynomials satisfying second-order differential equations: coping without help from group representation theory. J. Approx. Theory, 148(1), 35–48.Google Scholar
Durán, A. J. and López-Rodríguez, P. 1996. Orthogonal matrix polynomials: zeros and Blumenthal’s theorem. J. Approx. Theory, 84(1), 96–118.Google Scholar
Durán, A. J. and López-Rodríguez, P. 2007. Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. II. Constr. Approx., 26(1), 29–47.Google Scholar
Durán, A. J. and Polo, B. 2002. Gaussian quadrature formulae for matrix weights. Linear Algebra Appl., 355, 119–146.Google Scholar
Durán, A. J., López-Rodríguez, P., and Saff, E. B. 1999. Zero asymptotic behaviour for orthogonal matrix polynomials. J. Anal. Math., 78, 37–60.Google Scholar
Durand, L. 1975. Nicholson-type integrals for products of Gegenbauer functions and related topics. Pages 353–374 of Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975). New York, NY: Academic Press. Math. Res. Center, Univ. Wisconsin, Publ. No. 35.Google Scholar
Durand, L. 1978. Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal., 9(1), 76–86.Google Scholar
Elbert, Á. and Laforgia, A. 1986. Some monotonicity properties of the zeros of ultraspherical polynomials. Acta Math. Hungar., 48, 155–159.Google Scholar
Elbert, Á. and Laforgia, A. 1987. Monotonicity results on the zeros of generalized Laguerre polynomials. J. Approx. Theory, 51, 168–174.Google Scholar
Elbert, Á. and Laforgia, A. 1990. Upper bounds for the zeros of ultraspherical polynomials. J. Approx. Theory, 61, 88–97.Google Scholar
Elbert, Á. and Muldoon, M. E. 1994. On the derivative with respect to a parameter of a zero of a Sturm-Liouville function. SIAM J. Math. Anal., 25(2), 354–364.Google Scholar
Elbert, Á. and Muldoon, M. E. 1999. Inequalities and monotonicity properties for zeros of Hermite functions. Proc. Roy. Soc. Edinburgh Sect. A, 129(1), 57–75.Google Scholar
Elbert, Á. and Muldoon, M. E. 2008. Approximations for zeros of Hermite functions. Pages 117–126 of Dominici, D. and Maier, R. S. (eds), Special Functions and Orthogonal Polynomials. Contemp. Math., vol. 471. Providence, RI: American Mathematical Society.Google Scholar
Elbert, Á. and Siafarikas, P. D. 1999. Monotonicity properties of the zeros of ultraspherical polynomials. J. Approx. Theory, 97, 31–39.Google Scholar
Erdélyi, A. 1938. The Hankel transform of a product of Whittaker functions. J. London Math. Soc., 13, 146–154.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. 1953a. Higher Transcendental Functions. Vol. 1. New York: McGraw-Hill.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. 1953b. Higher Transcendental Functions. Vol. 2. New York: McGraw-Hill.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. 1955. Higher Transcendental Functions. Vol. 3. New York: McGraw-Hill.Google Scholar
Even, S. and Gillis, J. 1976. Derangements and Laguerre polynomials. Math. Proc. Camb. Phil. Soc., 79, 135–143.Google Scholar
Exton, H. 1983. q-Hypergeometric Functions and Applications. Ellis Horwood Series: Mathematics and Its Applications. Chichester: Ellis Horwood. With a foreword by Slater, L. J..Google Scholar
Favard, J. 1935. Sur les polynômes de Tchebicheff. C. R. Acad. Sci. Paris, 131, 2052–2053.Google Scholar
Faybusovich, L. and Gekhtman, M. 1999. On Schur flows. J. Phys. A, 32(25), 4671–4680.Google Scholar
Fejér, L. 1922. Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen. Math. Ann., 85, 41–48.Google Scholar
Feldheim, E. 1941. Sur les polynômes généralisés de Legendre. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR], 5, 241–248.Google Scholar
Fields, J. L. and Ismail, M. E. H. 1975. Polynomial expansions. Math. Comp., 29, 894–902.Google Scholar
Fields, J. L. and Wimp, J. 1961. Expansions of hypergeometric functions in hypergeometric functions. Math. Comp., 15, 390–395.Google Scholar
Filaseta, M. and Lam, T.-Y. 2002. On the irreducibility of the generalized Laguerre polynomials. Acta Arith., 105(2), 177–182.Google Scholar
Foata, D. 1981. Some Hermite polynomial identities and their combinatorics. Adv. in Appl. Math., 2, 250–259.Google Scholar
Foata, D. and Strehl, V. 1981. Une extension multilinéaire de la formule d’Erdélyi pour les produits de fonctions hypergéométriques confluentes. C. R. Acad. Sci. Paris Sér. I Math., 293(10), 517–520.Google Scholar
Forrester, P. J. and Rogers, J. B. 1986. Electrostatics and the zeros of the classical orthogonal polynomials. SIAM J. Math. Anal., 17, 461–468.Google Scholar
Forsyth, A. R. 1918. Theory of Functions of a Complex Variable. Third edn. Vols. 1, 2. Cambridge: Cambridge University Press.Google Scholar
Foster, W. H. and Krasikov, I. 2002. Inequalities for real-root polynomials and entire functions. Adv. in Appl. Math., 29(1), 102–114.Google Scholar
Frenzen, C. L. and Wong, R. 1985. A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Canad. J. Math., 37(5), 979–1007.Google Scholar
Frenzen, C. L. and Wong, R. 1988. Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal., 19(5), 1232–1248.Google Scholar
Freud, G. 1971. Orthogonal Polynomials. New York: Pergamon Press.Google Scholar
Freud, G. 1976. On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A (1), 76, 1–6.Google Scholar
Gabardo, J.-P. 1992. A maximum entropy approach to the classical moment problem. J. Funct. Anal., 106(1), 80–94.Google Scholar
Gangolli, R. and Varadarajan, V. S. 1988. Harmonic Analysis of Spherical Functions on Real Reductive Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 101. Berlin: Springer.Google Scholar
Gantmacher, F. R. 1959. The Theory of Matrices. Vol. 1. Translated from the Russian by Hirsch, K. A.. Reprint, Providence, RI: AMS Chelsea Publishing, 1998.Google Scholar
Garrett, K., Ismail, M. E. H., and Stanton, D. 1999. Variants of the Rogers–Ramanujan identities. Adv. in Appl. Math., 23, 274–299.Google Scholar
Gasper, G. 1970. Linearization of the product of Jacobi polynomials. II. Canad. J. Math., 22, 582–593.Google Scholar
Gasper, G. 1971. Positivity and the convolution structure for Jacobi series. Ann. Math., 93, 112–118.Google Scholar
Gasper, G. 1972. Banach algebras for Jacobi series and positivity of a kernel. Ann. Math., 95, 261–280.Google Scholar
Gasper, G. 1983. A convolution structure and positivity of a generalized translation operator for the continuous q-Jacobi polynomials. Pages 44–59 of Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vols. I, II (Chicago, Ill., 1981). Wadsworth Math. Ser. Belmont, CA: Wadsworth.Google Scholar
Gasper, G. and Rahman, M. 1983. Positivity of the Poisson kernel for the continuous q-ultraspherical polynomials. SIAM J. Math. Anal., 14(2), 409–420.Google Scholar
Gasper, G. and Rahman, M. 2004. Basic Hypergeometric Series. Second edn. Cambridge: Cambridge University Press.Google Scholar
Gatteschi, L. 1949a. Approssimazione asintotica degli zeri dei polinomi ultrasferici. Univ. Roma. Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5), 8, 399–411.Google Scholar
Gatteschi, L. 1949b. Una formula asintotica per l’approssimazione degli zeri dei polinomi di Legendre. Boll. Un. Mat. Ital. (3), 4, 240–250.Google Scholar
Gatteschi, L. 1950. Sull’approssimazione asintotica degli zeri dei polinomi sferici ed ultrasferici. Boll. Un. Mat. Ital. (3), 5, 305–313.Google Scholar
Gatteschi, L. 1952. Limitazione dell’errore nella formula di Hilb e una nuova formula per la valutazione asintotica degli zeri dei polinomi di Legendre. Boll. Un. Mat. Ital. (3), 7, 272–281.Google Scholar
Gatteschi, L. 1967/1968. Una nuova rappresentazione asintotica dei polinomi di Jacobi. Univ. e Politec. Torino Rend. Sem. Mat., 27, 165–184.Google Scholar
Gatteschi, L. 1972. Sugli zeri dei polimoni ultrasferici. Pages 111–122 of Studi in onore di Fernando Giaccardi. Torino: Baccola & Gili.Google Scholar
Gatteschi, L. 1985. On the zeros of Jacobi polynomials and Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino, 149–177. International conference on special functions: Theory and computation (Turin, 1984).Google Scholar
Gatteschi, L. 1987. New inequalities for the zeros of Jacobi polynomials. SIAM J. Math. Anal., 18, 1549–1562.Google Scholar
Gatteschi, L. 1988a. Some new inequalities for the zeros of Laguerre polynomials. Pages 23–38 of Numerical Methods and Approximation Theory, III (Niš, 1987). Niš: Univ. Niš.Google Scholar
Gatteschi, L. 1988b. Uniform approximations for the zeros of Laguerre polynomials. Pages 137–148 of Numerical Mathematics, Singapore 1988. Internat. Schriftenreihe Numer. Math., vol. 86. Basel: Birkhäuser.Google Scholar
Gatteschi, L. 2002. Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math., 144(1-2), 7–27.Google Scholar
Gatteschi, L. and Pittaluga, G. 1985. An asymptotic expansion for the zeros of Jacobi polynomials. Pages 70–86 of Mathematical Analysis. Teubner-Texte Math., vol. 79. Leipzig: Teubner.Google Scholar
Gautschi, W. 1967. Computational aspects of three-term recurrence relations. SIAM Rev., 9, 24–82.Google Scholar
Gautschi, W. 2009. New conjectured inequalities for zeros of Jacobi polynomials. Numer. Algorithms, 50(3), 293–296.Google Scholar
Gautschi, W. and Giordano, C. 2008. Luigi Gatteschi’s work on asymptotics of special functions and their zeros. Numer. Algorithms, 49(1-4), 11–31.Google Scholar
Gegenbauer, L. 1874. Uber einige bestimmte Integrale, Sitz. math. natur. Klasse Akad. Wiss. Wien, 70, 433–443.Google Scholar
Gegenbauer, L. 1893. Das Additionstheorem der Functionen Cnν(x). Sitz. math. natur. Klasse Akad. Wiss. Wien, 103, 942–950.Google Scholar
Geronimo, J. S. 1992. Polynomials orthogonal on the unit circle with random recurrence coefficients. Lecture Notes in Math., 1550, 43–61.Google Scholar
Geronimo, J. S. and Van Assche, W. 1986. Orthogonal polynomials on several intervals via a polynomial mapping. Trans. Amer. Math. Soc., 308, 559–581.Google Scholar
Geronimo, J. S., Gesztesy, F., and Holden, H. 2005. Algebro-geometric solution of the Baxter–Szegő difference equation. Comm. Math. Phys., 258, 149–177.Google Scholar
Geronimus, Ya. L. 1941. On the character of the solution of the moment problem in the case of the periodic in the limit associated fraction. Bull. Acad. Sci. USSR Math., 5, 203–210.Google Scholar
Geronimus, Ya. L. 1946. On the trigonometric moment problem. Ann. of Math. (2), 47, 742–761.Google Scholar
Geronimus, Ya. L. 1961. Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval. Authorized translation from the Russian. New York: Consultants Bureau.Google Scholar
Geronimus, Ya. L. 1962. Polynomials Orthogonal on a Circle and Their Applications. Amer. Math. Soc. Transl., vol. 3. Providence, RI: American Mathematical Society.Google Scholar
Geronimus, Ya. L. 1977. Orthogonal Polynomials. Amer. Math. Soc. Transl., vol. 108. Providence, RI: American Mathematical Society. Pages 37–130.Google Scholar
Gessel, I. M. and Stanton, D. 1983. Applications of q-Lagrange inversion to basic hypergeometric series. Trans. Amer. Math. Soc., 277(1), 173–201.Google Scholar
Gessel, I. M. and Stanton, D. 1986. Another family of q-Lagrange inversion formulas. Rocky Mountain J. Math., 16(2), 373–384.Google Scholar
Gil, A., Segura, J., and Temme, N. M. 2007. Numerical Methods for Special Functions. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).Google Scholar
Gilewicz, J., Leopold, E., and Valent, G. 2005. New Nevanlinna matrices for orthogonal polynomials related to cubic birth and death processes. J. Comp. Appl. Math., 178, 235–245.Google Scholar
Gilewicz, J., Leopold, E., Ruffing, A., and Valent, G. 2006. Some cubic birth and death processes and their related orthogonal polynomials. Constr. Approx., 24(1), 71–89.Google Scholar
Gillis, J., Reznick, B., and Zeilberger, D. 1983. On elementary methods in positivity theory. SIAM J. Math. Anal., 14, 396–398.Google Scholar
Gishe, J. and Ismail, M. E. H. 2008. Resultants of Chebyshev polynomials. Z. Anal. Anwend., 27(4), 499–508.Google Scholar
Godoy, E. and Marcellán, F. 1991. An analog of the Christoffel formula for polynomial modification of a measure on the unit circle. Boll. Un. Mat. Ital., 4(7), 1–12.Google Scholar
Gohberg, I., Lancaster, P., and Rodman, L. 1982. Matrix Polynomials. Computer Science and Applied Mathematics. New York: Academic Press. [Harcourt Brace Jovanovich Publishers].Google Scholar
Goldberg, J. 1965. Polynomials orthogonal over a denumerable set. Pacific J. Math., 15, 1171–1186.Google Scholar
Golinskii, B. 1958. The analogue of Cristoffel formula for orthogonal polynomials on the unit circle and some applications. Izv. Vuz. Mat., 1(2), 33–42.Google Scholar
Golinskii, B. 1966. On certain estimates for Cristoffel kernels and moduli of orthogonal polynomials. Izv. Vuz. Mat., 1(50), 30–42.Google Scholar
Golinskii, B. 1967. On the rate of convergence of orthogonal polynomials sequence to a limit function. Ukrain. Mat. Zh., 19, 11–28.Google Scholar
Golinskii, B. and Golinskii, L. 1998. On uniform boundedness and uniform asymptotics for orthogonal polynomials on the unit circle. J. Math. Anal. Appl., 220, 528–534.Google Scholar
Golinskii, L. 2000. Operator theoretic approach to orthogonal polynomials on an arc of the unit circle. Mat. Fiz., Analiz, Geometriya, 7(1), 3–34.Google Scholar
Golinskii, L. and Khrushchev, S. 2002. Cesàro asymptotics for orthogonal polynomials on the unit circle and classes of measures. J. Approx. Theory, 115, 187–237.Google Scholar
Golinskii, L. and Zlatoš, A. 2007. Coefficients of orthogonal polynomials on the unit circle and higher-order Szegő theorems. Constr. Approx., 26(3), 361–382.Google Scholar
Golinskii, L. B. 2006. Schur flows and orthogonal polynomials on the unit circle. Mat. Sb., 197(8), 41–62.Google Scholar
Golinskii, L. B. and Nevai, P. 2001. Szegő difference equations, transfer matrices and orthogonal polynomials on the unit circle. Comm. Math. Phys., 223, 223–259.Google Scholar
Golinskii, L. B., Nevai, P., and Van Assche, W. 1995. Perturbation of orthogonal polynomials on an arc of the unit circle. J. Approx. Theory, 83, 392–422.Google Scholar
Gómez, R. and López-García, M. 2007. A family of heat functions as solutions of indeterminate moment problems. Int. J. Math. Math. Sci., Art. ID 41526, 11.Google Scholar
Gorska, K. 2016. Private communication.Google Scholar
Grenander, U. and Szegő, G. 1958. Toeplitz Forms and Their Applications. Berkeley, CA: University of California Press. Reprint, Bronx, NY: Chelsea, 1984.Google Scholar
Groenevelt, W. 2015. Orthogonality relations for Al-Salam-Carlitz polynomials of type II. J. Approx. Theory, 195, 89–108.Google Scholar
Grosjean, C. C. 1987. A property of the zeros of the Legendre polynomials. J. Approx. Theory, 50, 84–88.Google Scholar
Grosswald, E. 1978. The Bessel Polynomials. Lecture Notes in Mathematics, vol. 698. Berlin: Springer.Google Scholar
Grünbaum, F. A. 2003. Matrix valued Jacobi polynomials. Bull. Sci. Math., 127(3), 207–214.Google Scholar
Grünbaum, F. A. 2010. An urn model associated with Jacobi polynomials. Commun. Appl. Math. Comput. Sci., 5, 55–63.Google Scholar
Grünbaum, F. A. 2011. The Darboux process and a noncommutative bispectral problem: Some explorations and challenges. Pages 161–177 of Kolk, J. A. C. and van den Ban, E. P. (eds), Geometric Aspects of Analysis and Mechanics. Progress in Mathematics, vol. 292. New York, NY: Birkhäuser/Springer. In Honor of the 65th Birthday of Hans Duistermaat.Google Scholar
Grünbaum, F. A. and Tirao, J. 2007. The algebra of differential operators associated to a weight matrix. Integral Equations Operator Theory, 58(4), 449–475.Google Scholar
Grünbaum, F. A., Pacharoni, I., and Tirao, J. 2001. A matrix-valued solution to Bochner’s problem. J. Phys. A, 34(48), 10647–10656. Symmetries and integrability of difference equations (Tokyo, 2000).Google Scholar
Grünbaum, F. A., Pacharoni, I., and Tirao, J. 2002. Matrix valued spherical functions associated to the complex projective plane. J. Funct. Anal., 188(2), 350–441.Google Scholar
Grünbaum, F. A., Pacharoni, I., and Tirao, J. 2003. Matrix valued orthogonal polynomials of the Jacobi type. Indag. Math. (N.S.), 14(3-4), 353–366.Google Scholar
Grünbaum, F. A., Pacharoni, I., and Tirao, J. 2004. An invitation to matrix-valued spherical functions: Linearization of products in the case of complex projective space P2(ℂ). Pages 147–160 of Modern Signal Processing. Math. Sci. Res. Inst. Publ., vol. 46. Cambridge: Cambridge University Press.Google Scholar
Grünbaum, F. A., Pacharoni, I., and Tirao, J. 2005. Matrix valued orthogonal polynomials of Jacobi type: The role of group representation theory. Ann. Inst. Fourier (Grenoble), 55(6), 2051–2068.Google Scholar
Grünbaum, F. A., Pacharoni, I., and Tirao, J. 2011. Two stochastic models of a random walk in the U(n)-spherical duals of U(n+1). Ann. Mat. Pura Appl., 1–27.Google Scholar
Gupta, D. P. and Muldoon, M. E. 2007. Inequalities for the smallest zeros of Laguerre polynomials and their q-analogues. JIPAM. J. Inequal. Pure Appl. Math., 8(1), Article 24, 7 pp. (electronic).Google Scholar
Habsieger, L. 2001a. Integer zeros of q-Krawtchouk polynomials in classical combinatorics. Adv. in Appl. Math., 27(2-3), 427–437. Special issue in honor of Dominique Foata’s 65th birthday (Philadelphia, PA, 2000).Google Scholar
Habsieger, L. 2001b. Integral zeroes of Krawtchouk polynomials. Pages 151–165 of Codes and Association Schemes (Piscataway, NJ, 1999). DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 56. Providence, RI: American Mathematical Society.Google Scholar
Habsieger, L. and Stanton, D. 1993. More zeros of Krawtchouk polynomials. Graphs Combin., 9(2), 163–172.Google Scholar
Hayman, W. K. 2005. On the zeros of a q-Bessel function. Pages 205–216 of Complex Analysis and Dynamical Systems II. Contemp. Math., vol. 382. Providence, RI: American Mathematical Society.Google Scholar
Hayman, W. K. and Ortiz, E. L. 1975/76. An upper bound for the largest zero of Hermite’s function with applications to subharmonic functions. Proc. Roy. Soc. Edinburgh Sect. A, 75.Google Scholar
Heine, E. 1961. Handbuch der Kugelfunctionen. Theorie und Anwendungen. Band I, II. Zweite umgearbeitete und vermehrte Auflage. Thesaurus Mathematicae, No. 1. Würzburg, 1961: Physica.Google Scholar
Helgason, S. 1978. Differential Geometry, Lie Groups, and Symmetric Spaces. Corrected reprint, Graduate Studies in Mathematics, vol. 34. Providence, RI: American Mathematical Society, 2001.Google Scholar
Hille, E. 1993. Über die Nullstellen der Hermiteschen Polynome. Jahresber. Deutsch Math.-Verein., 44, 162–165.Google Scholar
Hong, Y. 1986. On the nonexistence of nontrivial perfect e-codes and tight 2e-designs in Hamming schemes H(n,q) with e≥3 and q≥3. Graphs Combin., 2(2), 145–164.Google Scholar
Horn, R. A. and Johnson, C. R. 1992. Matrix Analysis. Cambridge: Cambridge University Press.Google Scholar
Ismail, M. E. H. 1981. The basic Bessel functions and polynomials. SIAM J. Math. Anal., 12, 454–468.Google Scholar
Ismail, M. E. H. 1982. The zeros of basic Bessel functions, the functions Jv+ax(x), and associated orthogonal polynomials. J. Math. Anal. Appl., 86(1), 1–19.Google Scholar
Ismail, M. E. H. 1985a. On sieved orthogonal polynomials I: Symmetric Pollaczek polynomials. SIAM J. Math. Anal., 16, 1093–1113.Google Scholar
Ismail, M. E. H. 1985b. A queueing model and a set of orthogonal polynomials. J. Math. Anal. Appl., 108, 575–594.Google Scholar
Ismail, M. E. H. 1986a. Asymptotics of the Askey–Wilson polynomials and q-Jacobi polynomials. SIAM J. Math. Anal., 17, 1475–1482.Google Scholar
Ismail, M. E. H. 1986b. On sieved orthogonal polynomials II: Orthogonality on several intervals. Trans. Amer. Math. Soc., 294, 89–111.Google Scholar
Ismail, M. E. H. 1987. The variation of zeros of certain orthogonal polynomials. Adv. in Appl. Math., 8(1), 111–118.Google Scholar
Ismail, M. E. H. 1993. Ladder operators for q-1-Hermite polynomials. Math. Rep. Royal Soc. Canada, 15, 261–266.Google Scholar
Ismail, M. E. H. 1995. The Askey–Wilson operator and summation theorems. Pages 171–178 of Ismail, M., Nashed, M. Z., Zayed, A., and Ghaleb, A. (eds), Mathematical Analysis, Wavelets and Signal Processing. Contemporary Mathematics, vol. 190. Providence, RI: American Mathematical Society.Google Scholar
Ismail, M. E. H. 1998. Discriminants and functions of the second kind of orthogonal polynomials. Results Math., 34, 132–149.Google Scholar
Ismail, M. E. H. 2000a. An electrostatic model for zeros of general orthogonal polynomials. Pacific J. Math., 193, 355–369.Google Scholar
Ismail, M. E. H. 2000b. More on electronic models for zeros of orthogonal polynomials. Numer. Funct. Anal. and Optimiz., 21(1-2), 191–204.Google Scholar
Ismail, M. E. H. 2003. Difference equations and quantized discriminants for q-orthogonal polynomials. Adv. in Appl. Math., 30(3), 562–589.Google Scholar
Ismail, M. E. H. 2005a. Asymptotics of q-orthogonal polynomials and a q-Airy function. Internat. Math. Res. Notices, 2005(18), 1063–1088.Google Scholar
Ismail, M. E. H. 2005b. Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge: Cambridge University Press.Google Scholar
Ismail, M. E. H. 2005c. Determinants with orthogonal polynomial entries. J. Comp. Appl. Anal., 178, 255–266.Google Scholar
Ismail, M. E. H. and Jing, N. 2001. q-discriminants and vertex operators. Adv. in Appl. Math., 27, 482–492.Google Scholar
Ismail, M. E. H. and Kelker, D. 1976. The Bessel polynomial and the student t-distribution. SIAM J. Math. Anal., 7(1), 82–91.Google Scholar
Ismail, M. E. H. and Li, X. 1992a. Bounds for extreme zeros of orthogonal polynomials. Proc. Amer. Math. Soc., 115, 131–140.Google Scholar
Ismail, M. E. H. and Li, X. 1992b. On sieved orthogonal polynomials. IX. Orthogonality on the unit circle. Pacific J. Math., 153, 289–297.Google Scholar
Ismail, M. E. H. and Masson, D. R. 1991. Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math., 21(1), 359–375.Google Scholar
Ismail, M. E. H. and Masson, D. R. 1994. q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals. Trans. Amer. Math. Soc., 346, 63–116.Google Scholar
Ismail, M. E. H. and Muldoon, M. 1991. A discrete approach to monotonicity of zeros of orthogonal polynomials. Trans. Amer. Math. Soc., 323, 65–78.Google Scholar
Ismail, M. E. H. and Muldoon, M. E. 1995. Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal., 2(1), 1–21.Google Scholar
Ismail, M. E. H. and Mulla, F. S. 1987. On the generalized Chebyshev polynomials. SIAM J. Math. Anal., 18(1), 243–258.Google Scholar
Ismail, M. E. H. and Rahman, M. 1991. Associated Askey–Wilson polynomials. Trans. Amer. Math. Soc., 328, 201–237.Google Scholar
Ismail, M. E. H. and Rahman, M. 1998. The q-Laguerre polynomials and related moment problems. J. Math. Anal. Appl., 218(1), 155–174.Google Scholar
Ismail, M. E. H. and Ruedemann, R. 1992. Relation between polynomials orthogonal on the unit circle with respect to different weights. J. Approximation Theory, 71, 39–60.Google Scholar
Ismail, M. E. H. and Simeonov, P. 1998. Strong asymptotics for Krawtchouk polynomials. J. Comput. Appl. Math., 100, 121–144.Google Scholar
Ismail, M. E. H. and Simeonov, P. 2015. Complex Hermite polynomials: Their combinatorics and integral operators. Proc. Amer. Math. Soc., 143, 1397–1410.Google Scholar
Ismail, M. E. H. and Stanton, D. 1988. On the Askey–Wilson and Rogers polynomials. Canad. J. Math., 40, 1025–1045.Google Scholar
Ismail, M. E. H. and Stanton, D. 1997. Classical orthogonal polynomials as moments. Canad. J. Math., 49, 520–542.Google Scholar
Ismail, M. E. H. and Stanton, D. 2002. q-Integral and moment representations for q-orthogonal polynomials. Canad. J. Math., 54, 709–735.Google Scholar
Ismail, M. E. H. and Stanton, D. 2003a. Applications of q-Taylor theorems. J. Comp. Appl. Math., 153(1-2), 259–272.Google Scholar
Ismail, M. E. H. and Stanton, D. 2003b. q-Taylor theorems, polynomial expansions and interpolation of entire functions. J. Approx. Theory, 123, 125–146.Google Scholar
Ismail, M. E. H. and Stanton, D. 2003c. Tribasic integrals and identities of Rogers–Ramanujan type. Trans. Amer. Math. Soc., 355, 4061–4091.Google Scholar
Ismail, M. E. H. and Stanton, D. 2006. Ramanujan’s continued fractions via orthogonal polynomials. Adv. Math., 203, 170–193.Google Scholar
Ismail, M. E. H. and Tamhankar, M. V. 1979. A combinatorial approach to some positivity problems. SIAM J. Math. Anal., 10, 478–485.Google Scholar
Ismail, M. E. H. and Valent, G. 1998. On a family of orthogonal polynomials related to elliptic functions. Illinois J. Math., 42(2), 294–312.Google Scholar
Ismail, M. E. H. and Wilson, J. 1982. Asymptotic and generating relations for the q-Jacobi and the 4ϕ3 polynomials. J. Approx. Theory, 36, 43–54.Google Scholar
Ismail, M. E. H. and Wimp, J. 1998. On differential equations for orthogonal polynomials. Methods Appl. Anal., 5, 439–452.Google Scholar
Ismail, M. E. H. and Witte, N. 2001. Discriminants and functional equations for polynomials orthogonal on the unit circle. J. Approx. Theory, 110, 200–228.Google Scholar
Ismail, M. E. H. and Zhang, C. 2007. Zeros of entire functions and a problem of Ramanujan. Adv. Math., 209, 363–380.Google Scholar
Ismail, M. E. H. and Zhang, R. 1988. On the Hellmann–Feynman theorem and the variation of zeros of special functions. Adv. in Appl. Math., 9, 439–446.Google Scholar
Ismail, M. E. H. and Zhang, R. 1994. Diagonalization of certain integral operators. Adv. Math., 109, 1–33.Google Scholar
Ismail, M. E. H. and Zhang, R. 2005. New proofs of some q-series results. Pages 285–299 of Ismail, M. E. H. and Koelink, E. H. (eds), Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman. Developments in Mathematics, vol. 13. New York: Springer.Google Scholar
Ismail, M. E. H., Stanton, D., and Viennot, G. 1987. The combinatorics of the q-Hermite polynomials and the Askey–Wilson integral. European J. Combinatorics, 8, 379–392.Google Scholar
Ismail, M. E. H., Letessier, J., and Valent, G. 1988. Linear birth and death models and associated Laguerre and Meixner polynomials. J. Approx. Theory, 55, 337–348.Google Scholar
Ismail, M. E. H., Valent, G., and Yoon, G. J. 2001. Some orthogonal polynomials related to elliptic functions. J. Approx. Theory, 112(2), 251–278.Google Scholar
Ismail, M. E. H., Nikolova, I., and Simeonov, P. 2004. Difference equations and discriminants for discrete orthogonal polynomials. Ramanujan J., 8, 475–502.Google Scholar
Ismail, M. E. H. and Zeng, J. 2010. Addition theorems via continued fractions. Trans. Amer. Math. Soc., 362(2), 957–983.Google Scholar
Jackson, F. H. 1903. On generalized functions of Legendre and Bessel. Trans. Royal Soc. Edinburgh, 41, 1–28.Google Scholar
Jackson, F. H. 1903–1904. The application of basic numbers to Bessel’s and Legendre’s functions. Proc. London Math. Soc. (2), 2, 192–220.Google Scholar
Jackson, F. H. 1904–1905. The application of basic numbers to Bessel’s and Legendre’s functions, II. Proc. London Math. Soc. (2), 3, 1–20.Google Scholar
Jensen, J. L. W. V. 1913. Recherches sur la théorie des équations. Acta Math., 36(1), 181–195.Google Scholar
Jones, W. B. and Thron, W. 1980. Continued Fractions: Analytic Theory and Applications. Reading, MA: Addison-Wesley.Google Scholar
Jordan, C. 1965. Calculus of Finite Differences. New York: Chelsea.Google Scholar
Kadell, K. W. J. 2005. The little q-Jacobi functions of complex order. Pages 301–338 of Ismail, M. E. H. and Koelink, E. H. (eds), Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman. Developments in Mathematics, vol. 13. New York: Springer.Google Scholar
Kalnins, E. G. and Miller, W. 1988. q-series and orthogonal polynomials associated with Barnes’ first lemma. SIAM J. Math. Anal., 19, 1216–1231.Google Scholar
Kaplansky, I. 1944. Symbolic solution of certain problems in permutations. Bull. Amer. Math. Soc., 50, 906–914.Google Scholar
Karlin, S. and McGregor, J. 1957a. The classification of birth and death processes. Trans. Amer. Math. Soc., 86, 366–400.Google Scholar
Karlin, S. and McGregor, J. 1957b. The differential equations of birth and death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc., 85, 489–546.Google Scholar
Karlin, S. and McGregor, J. 1958. Many server queuing processes with Poisson input and exponential service time. Pacific J. Math., 8, 87–118.Google Scholar
Karlin, S. and McGregor, J. 1959. Random walks. Illinois J. Math., 3, 66–81.Google Scholar
Karlin, S. and Szegő, G. 1960/1961. On certain determinants whose elements are orthogonal polynomials. J. Anal. Math., 8, 1–157.Google Scholar
Karp, D. 2001. Holomorphic spaces related to orthogonal polynomials and analytic continuation of functions. Pages 169–187 of Saitoh, S., Hayashi, N., and Yamamoto, M. (eds), Analytic Extension Formulas and Their Applications (Fukuoka, 1999/Kyoto, 2000). Int. Soc. Anal. Appl. Comput., vol. 9. Dordrecht, The Netherlands: Kluwer Academic Publisher.Google Scholar
Khruschev, S. 2001. Schur’s algorithm, orthogonal polynomials, and convergence of Wall’s continued fractions in L2(T). J. Approx. Theory, 108, 161–248.Google Scholar
Khruschev, S. 2002. Classification theorems for general orthogonal polynomials on the unit circle. J. Approx. Theory, 116, 268–342.Google Scholar
Khruschev, S. 2003. Turán measures. J. Approx. Theory, 122, 112–120.Google Scholar
Kibble, W. F. 1945. An extension of theorem of Mehler on Hermite polynomials. Proc. Cambridge Philos. Soc., 41, 12–15.Google Scholar
Killip, R. and Nenciu, I. 2005. CMV: The unitary analogue of Jacobi matrices. Preprint arXiv:math.SG/0508113.Google Scholar
Koekoek, R. and Swarttouw, R. 1998. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogues. Reports of the Faculty of Technical Mathematics and Informatics 98-17. Delft University of Technology, Delft.Google Scholar
Koekoek, R., Lesky, P. A., Swarttouw, R. F. 2010. Hypergeometric Orthogonal Polynomials and their q-Analogues. Springer Monographs in Mathematics, Springer-Verlag: Berlin Heidelberg.Google Scholar
Koelink, E. 1997. Addition formulas for q-special functions. Pages 109–129 of Ismail, M. E. H., Masson, D. R., and Rahman, M. (eds), Special Functions, q-Series and Related Topics (Toronto, ON, 1995). Fields Inst. Commun., vol. 14. Providence, RI: American Mathematical Society.Google Scholar
Koelink, H. T. 1996. On Jacobi and continuous Hahn polynomials. Proc. Amer. Math. Soc., 124(3), 887–898.Google Scholar
Koelink, H. T. and Van der Jeugt, J. 1998. Convolutions for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Anal., 29(3), 794–822.Google Scholar
Koornwinder, T. H. 1978. Positivity proofs for linearization and connection coefficients for orthogonal polynomials satisfying an addition formula. J. London Math. Soc., 18(2), 101–114.Google Scholar
Koornwinder, T. H. 1981. Clebsch–Gordan coefficients for SU(2) and Hahn polynomials. Nieuw Arch. Wisk. (3), 29(2), 140–155.Google Scholar
Koornwinder, T. H. 1982. Krawtchouk polynomials, a unification of two different group theoretic interpretations. SIAM J. Math. Anal., 13(6), 1011–1023.Google Scholar
Koornwinder, T. H. 1984a. Jacobi functions and analysis on noncompact semisimple Lie groups. Pages 1–85 of Askey, R. A., Koornwinder, T. H., and Schempp, W. (eds), Special Functions: Group Theoretical Aspects and Applications. Dordrecht: Reidel.Google Scholar
Koornwinder, T. H. 1984b. Orthogonal polynomials with weight function (1-x)α(1+x)β+Mδ(x+1)+Nδ(x-1). Canad. Math Bull., 27, 205–214.Google Scholar
Koornwinder, T. H. 1990. Orthogonal polynomials in connection with quantum groups. Pages 257–292 of Nevai, P. (ed), Orthogonal Polynomials; Theory and Practice. Nato ASI Series C: Mathematical and Physcal Science, vol. 294. Dordrecht: Kluwer Academic Publishers.Google Scholar
Koornwinder, T. H. 1993. Askey–Wilson polynomials as zonal spherical functions on the SU(2) quantum group. SIAM J. Math. Anal., 24(3), 795–813.Google Scholar
Koornwinder, T. H. 2004. On q-1-Al-Salam–Chihara Polynomials. Informal note.Google Scholar
Koornwinder, T. H. 2005. A second addition formula for continuous q-ultraspherical polynomials. Pages 339–360 of Ismail, M. E. H. and Koelink, E. (eds), Theory and Applications of Special Functions. Developments in Mathematics, vol. 13. New York: Springer.Google Scholar
Koornwinder, T. H. 2006. Lowering and raising operators for some special orthogonal polynomials. Pages 227–238 of Jack, Hall–Littlewood and Macdonald Polynomials. Contemp. Math., vol. 417. Providence, RI: American Mathematical Society.Google Scholar
Krall, H. L. and Frink, O. 1949. A new class of orthogonal polynomials. Trans. Amer. Math. Soc., 65, 100–115.Google Scholar
Krasikov, I. 2003. Bounds for zeros of the Laguerre polynomials. J. Approx. Theory, 121, 287–291.Google Scholar
Krasikov, I. 2007. Inequalities for orthonormal Laguerre polynomials. J. Approx. Theory, 144(1), 1–26.Google Scholar
Krasikov, I. and Litsyn, S. 1996. On integral zeros of Krawtchouk polynomials. J. Combin. Theory Ser. A, 74(1), 71–99.Google Scholar
Krein, M. 1949. Infinite J-matrices and a matrix-moment problem. Doklady Akad. Nauk SSSR (N.S.), 69, 125–128.Google Scholar
Krein, M. G. 1949. Fundamental aspects of the representation theory of Hermitian operators with deficiency index (m,m). Ukrain. Math. Zh., 1, 3–66. Amer. Math. Soc. Transl. (2) 97 (1970), 75–143.Google Scholar
Krein, M. G. and Nudel’man, A. A. 1977. The Markov Moment Problem and Extremal Problems. Translations of Mathematical Monographs, vol. 50. Providence, RI: American Mathematical Society.Google Scholar
Kuznetsov, A. 2016. Constructing Measures with Identical Moments. arXiv:1607.08003.Google Scholar
Kuznetsov, A. 2017. Solving the mystery integral. In Li, X. and Nashed, M. Z. (eds), Frontiers in Orthogonal Polynomials and q-Series. World Scientific.Google Scholar
Kwon, K. H. 2002. Orthogonal Polynomials I. Lecture Notes. KAIST, Seoul.Google Scholar
Labelle, J. and Yeh, Y. N. 1989. The combinatorics of Laguerre, Charlier, and Hermite polynomials. Stud. Appl. Math., 80(1), 25–36.Google Scholar
Laforgia, A. 1981. A monotonic property for the zeros of ultraspherical polynomials. Proc. Amer. Math. Soc., 83(4), 757–758.Google Scholar
Laforgia, A. and Muldoon, M. E. 1986. Some consequences of the Sturm comparison theorem. Amer. Math. Monthly, 93, 89–94.Google Scholar
Landau, H. J. 1987. Maximum entropy and the moment problem. Bull. Amer. Math. Soc. (N.S.), 16(1), 47–77.Google Scholar
Lanzewizky, I. L. 1941. Über die orthogonalität der Fejer–Szegöschen polynome. C. R. Dokl. Acad. Sci. URSS (N.S.), 31, 199–200.Google Scholar
Laplace, P. S. 1782. Théorie des attractions des sphéroides et de la figure des planètes. Mémoires de Mathématique et de Physique tirés des registres de l’Académie royale des Sciences, Paris, 113–196.Google Scholar
Legendre, A. M. 1785. Recherches sur l’attraction des sphéorides homogènes. Mémoires de Mathématique et de Physique presentés à l’Académie royale des Sciences par divers savans, Paris, 10, 411–434.Google Scholar
Legendre, A. M. 1789. Suite des recherches sur la figure des planètes. Mémoires de Mathématique et de Physique tirés des registres de l’Académie royale des Sciences, Paris, 372–454.Google Scholar
Leipnik, R. 1981. The lognormal distribution and strong nonuniqueness of the moment problem. Teor. Veroyatnost. i Primenen., 26(4), 863–865.Google Scholar
Lew, J. S. and Quarles, D. A. Jr. 1983. Nonnegative solutions of a nonlinear recurrence. J. Approx. Theory, 38(4), 357–379.Google Scholar
Li, X. and Wong, R. 2000. A uniform asymptotic expansion for Krawtchouk polynomials. J. Approx. Theory, 106(1), 155–184.Google Scholar
Lorch, L. 1977. Elementary comparison techniques for certain classes of Sturm–Liouville equations. Pages 125–133 of Berg, G., Essén, M., and Pleijel, A. (eds), Differential Equations (Proc. Conf. Uppsala, 1977). Stockholm: Almqvist and Wiksell.Google Scholar
Lorentzen, L. and Waadeland, H. 1992. Continued Fractions with Applications. Amsterdam: North-Holland.Google Scholar
Louck, J. D. 1981. Extension of the Kibble–Slepian formula for Hermite polynomials using Boson operator methods. Adv. in Appl. Math., 2, 239–249.Google Scholar
Lubinsky, D. S. 1994. Zeros of orthogonal and biorthogonal polynomials: Some old, some new. Pages 3–15 of Nonlinear Numerical Methods and Rational Approximation, II (Wilrijk, 1993). Math. Appl., vol. 296. Dordrecht: Kluwer Academic Publisher.Google Scholar
Makai, E. 1952. On monotonicity property of certain Sturm–Liouville functions. Acta Math. Acad. Sci. Hungar., 3, 15–25.Google Scholar
Marcellán, F. and Maroni, P. 1992. Sur l’adjonction d’une masse de Dirac á une forme régulière et semi-classique. Ann. Mat. Pura Appl., 162, 1–22.Google Scholar
Marden, M. 1966. Geometry of Polynomials. Second edn. Mathematical Surveys, No. 3. Providence, RI: American Mathematical Society.Google Scholar
Maroni, P. 1987. Prolégomènes à l’étude des polynômes orthogonaux semi-classiques. Ann. Mat. Pura Appl. (4), 149, 165–184.Google Scholar
Máté, A. and Nevai, P. G. 1982. Remarks on E. A. Rakhmanov’s paper: “The asymptotic behavior of the ratio of orthogonal polynomials” [Mat. Sb. (N.S.) 103(145) (1977), no. 2, 237–252; MR 56 #3556]. J. Approx. Theory, 36(1), 64–72.Google Scholar
Máté, A., Nevai, P., and Totik, V. 1985. Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle. Constr. Approx., 1, 63–69.Google Scholar
Máté, A., Nevai, P., and Totik, V. 1987a. Extensions of Szegő’s theory of orthogonal polynomials, II. Constr. Approx., 3(1), 51–72.Google Scholar
Máté, A., Nevai, P., and Totik, V. 1987b. Strong and weak convergence of orthogonal polynomials. Amer. J. Math., 109, 239–281.Google Scholar
Máté, A., Nevai, P., and Totik, V. 1991. Szegő’s extremum problem on the unit circle. Ann. Math., 134, 433–453.Google Scholar
Mazel, D. S., Geronimo, J. S., and Hayes, M. H. 1990. On the geometric sequences of reflection coefficients. IEEE Trans. Acoust. Speech Signal Process., 38, 1810–1812.Google Scholar
Mehta, M. L. 2004. Random Matrices. Third edn. San Diego: Elsevier.Google Scholar
Meixner, J. 1934. Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. London Math. Soc., 9, 6–13.Google Scholar
Meixner, J. 1942. Unformung gewisser Reihen. deren Gleider Produkte hypergeometischer Funktionen sind. Deutsch. Math., 6, 341–349.Google Scholar
Mhaskar, H. 1990. Bounds for certain Freud polynomials. J. Approx. Theory, 63, 238–254.Google Scholar
Mhaskar, H. and Saff, E. 1990. On the distribution of zeros of polynomials orthogonal on the unit circle. J. Approx. Theory, 63, 30–38.Google Scholar
Mikaelyan, L. 1978. The analogue of Cristoffel formula for orthogonal polynomials on the unit circle. Dokl. Akad. Nauk Arm. SSR, 67(5), 257–263.Google Scholar
Milne-Thomson, L. M. 1933. The Calculus of Finite Differences. New York: Macmillan.Google Scholar
Moak, D. 1981. The q-analogue of the Laguerre polynomials. J. Math. Anal. Appl., 81(1), 20–47.Google Scholar
Mukaihira, A. and Nakamura, Y. 2000. Integrable discretization of the modified KdV equation and applications. Inverse Problems, 16, 413–424.Google Scholar
Mukaihira, A. and Nakamura, Y. 2002. Schur flow for orthogonal polynomials on the unit circle and its integrable discretization. J. Comput. Appl. Math., 139, 75–94.Google Scholar
Muldoon, M. E. 1993. Properties of zeros of orthogonal polynomials and related functions. J. Comput. Appl. Math., 48(1-2), 167–186.Google Scholar
Muldoon, M. E. 2008. Continuous ranking of zeros of special functions. J. Math. Anal. Appl., 343, 436–445.Google Scholar
Naimark, M. A. 1947. Extremal spectral functions of a symmetric operator. Izv. Akad. Nauk SSSR ser matem, 11, 327–344.Google Scholar
Nenciu, I. 2005. Lax pairs for the Ablowitz–Ladik system via orthogonal polynomials on the unit circle. IMRN, 11, 647–686.Google Scholar
Nevai, P. 1983. Orthogonal polynomials associated with exp(-x4). Canadian Math. Soc. Conference Proceedings, 263–285.Google Scholar
Nevai, P. 1991. Weakly convergent sequences of functions and orthogonal polynomials. J. Approx. Theory, 65, 322–340.Google Scholar
Nevai, P. and Totik, V. 1989. Orthogonal polynomials and their zeros. Acta Sci. Math. (Szeged), 53, 99–104.Google Scholar
Nikolov, G. and Uluchev, R. 2004. Inequalities for real-root polynomials. Proof of a conjecture of Foster and Krasikov. Pages 201–216 of Approximation Theory: A Volume Dedicated to Borislav Bojanov. Prof. M. Drinov Academic Publ. House, Sofia.Google Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F., and Clark, C. W. (eds). 2010. NIST Handbook of Mathematical Functions. Cambridge: Cambridge University Press.Google Scholar
Ortiz, E. L. and Rivlin, T. J. 1983. Another look at the Chebyshev polynomials. Amer. Math. Monthly, 90(1), 3–10.Google Scholar
Pacharoni, I. 2009. Matrix spherical functions and orthogonal polynomials: An instructive example. Rev. Un. Mat. Argentina, 50(1), 1–15.Google Scholar
Pacharoni, I. and Román, P. 2008. A sequence of matrix valued orthogonal polynomials associated to spherical functions. Constr. Approx., 28(2), 127–147.Google Scholar
Pacharoni, I. and Tirao, J. 2007a. Matrix valued orthogonal polynomials arising from the complex projective space. Constr. Approx., 25(2), 177–192.Google Scholar
Pacharoni, I. and Tirao, J. 2007b. Three term recursion relation for spherical functions associated to the complex hyperbolic plane. J. Lie Theory, 17(4), 791–828.Google Scholar
Pacharoni, I. and Tirao, J. 2013. One step spherical functions of the pair (SU(n+1),U(n)). In Huckleberry, A., Penkov, I., and Zuckerman, G. (eds), Lie Groups: Structures, Actions and Representations. Progress in Mathematics, vol. 306. Basel: Birkhäuser. Also available at http://arxiv.org/abs/1209.4500.Google Scholar
Pacharoni, I., Tirao, J., and Zurrián, I. 2014. Spherical functions associated to the 3-dimensional sphere. Ann. Mat. Pura Appl., 193, no. 6, 1727–1778.Google Scholar
Parlett, B. N. 1980. The Symmetric Eigenvalue Problem. Corrected reprint, Classics in Applied Mathematics, vol. 20. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1998.Google Scholar
Pedersen, H. L. 1995. Stieltjes moment problems and the Friedrichs extension of a positive definite operator. J. Approx. Theory, 83(3), 289–307.Google Scholar
Pedersen, H. L. 1997. La paramétrisation de Nevanlinna et le problème des moments de Stieltjes indéterminé. Exposition. Math., 15(3), 273–278.Google Scholar
Pedersen, H. L. 1998. On Krein’s theorem for indeterminacy of the classical moment problem. J. Approx. Theory, 95(1), 90–100.Google Scholar
Pedersen, H. L. 2009. Logarithmic order and type of indeterminate moment problems. II. J. Comput. Appl. Math., 233(3), 808–814.Google Scholar
Peherstorfer, F. and Steinbauer, R. 1995. Characterization of general orthogonal polynomials with respect to a functional. J. Comp. Appl. Math., 65, 339–355.Google Scholar
Peherstorfer, F. and Steinbauer, R. 1999. Mass-points of orthogonality measures on the unit circle. East J. Approx., 5, 279–308.Google Scholar
Periwal, V. and Shevitz, D. 1990. Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett., 64, 1326–1329.Google Scholar
Pollaczek, F. 1949. Sur une généralisation des polynômes de Legendre. C. R. Acad. Sci. Paris, 228, 1363–1365.Google Scholar
Pollaczek, F. 1956. Sur une Généralisation des Polynômes de Jacobi. Memorial des Sciences Mathematique, vol. 131. Paris: Gauthier-Villars.Google Scholar
Pólya, G. and Szegő, G. 1976. Problems and Theorems in Analysis. II. Reprint, Classics in Mathematics. Berlin: Springer. Theory of functions, zeros, polynomials, determinants, number theory, geometry. Translated from the German by C. E. Billigheimer. 1998.Google Scholar
Qiu, W.-Y. and Wong, R. 2004. Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comp. Meth. Func. Theory, 4(1), 189–226.Google Scholar
Rahman, M. 1981. The linearization of the product of continuous q-Jacobi polynomials. Canad. J. Math., 33(4), 961–987.Google Scholar
Rahman, M. 1984. A simple evaluation of Askey and Wilson’s q-beta integral. Proc. Amer. Math. Soc., 92(3), 413–417.Google Scholar
Rahman, M. 1988. A generalization of Gasper’s kernel for Hahn polynomials. Canad. J. Math., 30(133-146), 373–381.Google Scholar
Rahman, M. and Tariq, Q. 1997. Poisson kernels for associated q-ultrasperical polynomials. Methods and Applications of Analysis, 4, 77–90.Google Scholar
Rahman, M. and Verma, A. 1986a. Product and addition formulas for the continuous q-ultraspherical polynomials. SIAM J. Math. Anal., 17(6), 1461–1474.Google Scholar
Rahman, M. and Verma, A. 1986b. A q-integral representation of Rogers’ q-ultraspherical polynomials and some applications. Constructive Approximation, 2, 1–10.Google Scholar
Rainville, E. D. 1960. Special Functions. New York: Macmillan.Google Scholar
Rakhmanov, E. A. 1977. On the asymptotics of the ratio of orthogonal polynomials. Mat. Sb., 103, 237–252. English translation in Math. USSR Sb. 32 (1977), 199–213.Google Scholar
Rakhmanov, E. A. 1980. Steklov’s conjecture in the theory of orthogonal polynomials. Math. USSR Sb., 36, 549–575.Google Scholar
Rakhmanov, E. A. 1982a. Estimates of the growth of orthogonal polynomials whose weight is bounded away from zero. Math. USSR Sb., 42, 237–263.Google Scholar
Rakhmanov, E. A. 1982b. On the asymptotics of the ratio of orthogonal polynomials. II. Mat. Sb., 118, 104–117. English translation in Math. USSR Sb. 47 (1983), 105–117.Google Scholar
Riesz, M. 1923. Sur le problème des moments et le théorème de Parseval correspondant. Acta Litt. Ac. Sci. Szeged, 1, 209–225.Google Scholar
Rogers, L. J. 1894. Second memoir on the expansion of certain infinite products. Proc. London Math. Soc., 25, 318–343.Google Scholar
Rogers, L. J. 1895. Third memoir on the expansion of certain infinite products. Proc. London Math. Soc., 26, 15–32.Google Scholar
Román, P. and Tirao, J. 2006. Spherical functions, the complex hyperbolic plane and the hypergeometric operator. Internat. J. Math., 17(10), 1151–1173.Google Scholar
Román, P. and Tirao, J. 2012. The spherical transform of any K-type in a locally compact group. J. Lie Theory, 22, 361–395.Google Scholar
Roman, S. and Rota, G.-C. 1978. The umbral calculus. Adv. Math., 27, 95–188.Google Scholar
Romanov, R. 2017. Order problem for canonical systems and a conjecture of Valent. Trans. Amer. Math. Soc., 369 (2017), no. 2, 1061–1078.Google Scholar
Routh, E. 1884. On some properties of certain solutions of a differential equation of the second order. Proc. London Math. Soc., 16, 245–261.Google Scholar
Rui, B. and Wong, R. 1994. Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal., 1(3), 294–313.Google Scholar
Rui, B. and Wong, R. 1996. Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math., 96(3), 307–338.Google Scholar
Saff, E. B. and Totik, V. 1992. What parts of a measure’s support attract zeros of the corresponding orthogonal polynomials. Proc. Amer. Math. Soc., 114, 185–190.Google Scholar
Saff, E. B. and Varga, R. S. 1977. On the zeros and poles of Padé approximants to ez. II. Pages 195–213 of Saff, E. B. and Varga, R. S. (eds), Padé and Rational Approximations: Theory and Applications. New York: Academic Press.Google Scholar
Sarmanov, I. O. 1968. A generalized symmetric gamma-correlation. Dokl. Akad. Nauk SSSR, 179, 1279–1281.Google Scholar
Sarmanov, O. V. and Bratoeva, Z. N. 1967. Probabilistic properties of bilinear expansions of Hermite polynomials. Theor. Probability Appl., 12, 470–481.Google Scholar
Schur, I. 1929. Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, I. Sitzungsber. Preuss. Akad. Wissensch. Phys.-Math. Kl., 23, 125–136.Google Scholar
Schur, I. 1931. Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome. J. Reine Angew. Math., 165, 52–58.Google Scholar
Schwartz, H. M. 1940. A class of continued fractions. Duke J. Math., 6, 48–65.Google Scholar
Segura, J. 2003. On the zeros and turning points of special functions. J. Comput. Appl. Math., 153, 433–440.Google Scholar
Sharapudinov, I. I. 1988. Asymptotic properties of Krawtchouk polynomials. Mat. Zametki, 44(5), 682–693, 703.Google Scholar
Shohat, J. A. 1936. The relation of the classical orthogonal polynomials to the polynomials of Appell. Amer. J. Math., 58, 453–464.Google Scholar
Shohat, J. A. 1938. Sur les polynômes orthogonèaux généraliséès. C. R. Acad. Sci., 207, 556–558.Google Scholar
Shohat, J. A. 1939. A differential equation for orthogonal polynomials. Duke Math. J., 5, 401–417.Google Scholar
Shohat, J. A. and Tamarkin, J. D. 1950. The Problem of Moments. revised edn. Providence, RI: American Mathematical Society.Google Scholar
Siegel, C. L. 1929. Über einige Anwendungen Diophantischer Approximationery. Abh. der Preuss. Akad. der Wissenschaften. Phys-math. Kl. Nr. 1.Google Scholar
Simon, B. 2004a. Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. Vol. 54, part 1. Providence, RI: American Mathematical Society Colloquium Publications.Google Scholar
Simon, B. 2004b. Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory. Vol. 54, part 2. Providence, RI: American Mathematical Society Colloquium Publications.Google Scholar
Simon, B. 2005a. Fine structure of the zeros of orthogonal polynomials. II. OPUC with competing exponential decay. J. Approx. Theory, 135(1), 125–139.Google Scholar
Simon, B. 2005b. Sturm oscillation and comparison theorems. Pages 29–43 of Sturm–Liouville Theory. Basel: Birkhäuser.Google Scholar
Simon, B. 2006. Fine structure of the zeros of orthogonal polynomials. I. A tale of two pictures. Electron. Trans. Numer. Anal., 25, 328–368 (electronic).Google Scholar
Simon, B. 2007a. CMV matrices: Five years after. J. Comput. Appl. Math., 208(1), 120–154.Google Scholar
Simon, B. 2007b. Zeros of OPUC and the long time asymptotics of Schur and related flows. Inverse Probl. Imaging, 1(1), 189–215.Google Scholar
Simon, B. and Totik, V. 2005. Limits of zeros of orthogonal polynomials on the circle. Math. Nachr., 278(12-13), 1615–1620.Google Scholar
Simon, B. and Zlatoš, A. 2005. Higher-order Szegő theorems with two singular points. J. Approx. Theory, 134(1), 114–129.Google Scholar
Sinap, A. 1995. Gaussian quadrature for matrix valued functions on the real line. J. Comput. Appl. Math., 65(1-3), 369–385. Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994).Google Scholar
Sinap, A. and Van Assche, W. 1994. Polynomial interpolation and Gaussian quadrature for matrix-valued functions. Linear Algebra Appl., 207, 71–114.Google Scholar
Slater, L. J. 1966. Generalized Hypergeometric Functions. Cambridge: Cambridge University Press.Google Scholar
Slepian, D. 1972. On the symmetrized Kronecker power of a matrix and extensions of Mehler’s formula for Hermite polynomials. SIAM J. Math. Anal., 3, 606–616.Google Scholar
Srivastava, H. M. and Singhal, J. P. 1973. New generating functions for Jacobi and related polynomials. J. Math. Anal. Appl., 41, 748–752.Google Scholar
Stanton, D. 1984. Orthogonal polynomials and Chevalley groups. Pages 87–128 of Askey, R. A., Koornwinder, T. H., and Schempp, W. (eds), Special Functions: Group Theoretical Aspects and Applications. NATO Sci. Ser. II Math. Phys. Chem. Dordrecht: D. Reidel.Google Scholar
Stieltjes, T.-J. 1885a. Sur les polynômes de Jacobi. C.R. Acad. Sci. Paris, 100, 620–622. Reprinted in Œuvres Complètes, vol. 1, pp. 442–444.Google Scholar
Stieltjes, T.-J. 1885b. Sur quelques théorèmes d’algèbre. C. R. Acad. Sci. Paris, 100, 439–440. Reprinted in Œuvres Complètes, vol. 1, pp. 440–441.Google Scholar
Stieltjes, T.-J. 1887. Sur les racines de l’équation Xn=0. Acta Math., 9(1), 385–400.Google Scholar
Stieltjes, T.-J. 1894. Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8, 9, 1–122, 1–47.Google Scholar
Stieltjes, T.-J. 1993. Œuvres Complètes/Collected papers. Vols. I, II. Berlin: Springer. Reprint of the 1914–1918 edition.Google Scholar
Stone, M. H. 1932. Linear Transformations in Hilbert Space. Reprint, American Mathematical Society Colloquium Publications, vol. 15. Providence, RI: American Mathematical Society, 1990.Google Scholar
Szász, O. 1950. On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3), 5, 125–127.Google Scholar
Szász, O. 1951. On the relative extrema of the Hermite orthogonal functions. J. Indian Math. Soc. (N.S.), 15.Google Scholar
Szegő, G. 1915. Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion. Math. Ann., 76(4), 490–503.Google Scholar
Szegő, G. 1920. Beiträge zur Theorie der Toeplitzschen Formen. Math. Z., 6(3-4), 167–202.Google Scholar
Szegő, G. 1921. Beiträge zur Theorie der Toeplitzschen Formen. II. Math. Z., 9(3-4), 167–190.Google Scholar
Szegő, G. 1926. Beiträge zur Theorie der Thetafunktionen. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl., XIX, 242–252. Reprinted in Collected Papers, (R. A. Askey, ed.), vol. I, Boston: Birkhauser, 1982.Google Scholar
Szegő, G. 1936. On some Hermitian forms associated with two given curves of the complex plane. Trans. Amer. Math. Soc., 40(3), 450–461.Google Scholar
Szegő, G. 1950a. On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc., 1, 731–737.Google Scholar
Szegő, G. 1950b. On the relative extrema of Legendre polynomials. Boll. Un. Mat. Ital. (3), 5.Google Scholar
Szegő, G. [1939] 1975. Orthogonal Polynomials. Fourth edn. Providence, RI: American Mathematical Society. American Mathematical Society, Colloquium Publications, vol. XXIII.Google Scholar
Szwarc, R. 1992. Connection coefficients of orthogonal polynomials. Canad. Math. Bull., 35(4), 548–556.Google Scholar
Szwarc, R. 2005. Orthogonal polynomials and Banach algebras. Pages 103–139 of Inzell Lectures on Orthogonal Polynomials. Adv. Theory Spec. Funct. Orthogonal Polynomials, vol. 2. Hauppauge, NY: Nova Sci. Publ.Google Scholar
Tirao, J. A. 1977. Spherical functions. Rev. Un. Mat. Argentina, 28(2), 75–98.Google Scholar
Tirao, J. A. 2003. The matrix-valued hypergeometric equation. Proc. Natl. Acad. Sci. USA, 100(14), 8138–8141 (electronic).Google Scholar
Tirao, J. A. 2011. The algebra of differential operators associated to a weight matrix: a first example. In Milies, C. P. (ed), Groups, Algebras and Applications. Contemporary Mathematics, vol. 537. Providence, RI: American Mathematical Society. XVIII Latin American Algebra Colloquium, August 3–8, 2009, in São Paulo, Brazil.Google Scholar
Toda, M. 1989. Theory of Nonlinear Lattices. Second edn. Springer Series in Solid-State Sciences, vol. 20. Berlin: Springer.Google Scholar
Todd, J. 1950. On the relative extrema of the Laguerre orthogonal functions. Boll. Un. Mat. Ital. (3), 5, 122–125.Google Scholar
Tricomi, F. G. 1947. Sugli zeri delle funzioni di cui si conosce una rappresentazione asintotica. Ann. Mat. Pura Appl. (4), 26, 283–300.Google Scholar
Tricomi, F. G. 1954. Funzioni Ipergeometriche Confluenti. Roma: Edizioni Cremonese.Google Scholar
Tricomi, F. G. 1957. Integral Equations. Reprint, New York: Dover Publications, 1985.Google Scholar
Turán, P. 1980. On some open problems of approximation theory. J. Approx. Theory, 29(1), 23–85. P. Turán memorial volume. Translated from the Hungarian by P. Szüsz.Google Scholar
Tyan, S. and Thomas, J. B. 1975. Characterization of a class of bivariate distribution functions. J. Multivariate Anal., 5, 227–235.Google Scholar
Tyan, S. G., Derin, H., and Thomas, J. B. 1976. Two necessary conditions on the representation of bivariate distributions by polynomials. Ann. Statist., 4(1), 216–222.Google Scholar
Underhill, C. 1972. On the Zeros of Generalized Bessel Polynomials. Internal note. University of Salford.Google Scholar
Uvarov, V. B. 1959. On the connection between polynomials, orthogonal with different weights. Dokl. Acad. Nauk SSSR, 126, 33–36.Google Scholar
Uvarov, V. B. 1969. The connection between systems of polynomials that are orthogonal with respect to different distribution functions. Ž. Vyčisl. Mat. i Mat. Fiz., 9, 1253–1262.Google Scholar
Valent, G. 1994. Asymptotic analysis of some associated orthogonal polynomials connected with elliptic functions. SIAM J. Math. Anal., 25(2), 749–775.Google Scholar
Valent, G. 1995. Associated Stieltjes-Carlitz polynomials and a generalization of Heun’s differential equation. J. Comput. Appl. Math., 57(1-2), 293–307. Proceedings of the Fourth International Symposium on Orthogonal Polynomials and Their Applications (Evian-Les-Bains, 1992).Google Scholar
Valent, G. 1996a. Co-recursivity and Karlin–McGregor duality for indeterminate moment problems. Constr. Approx., 12(4), 531–553.Google Scholar
Valent, G. 1996b. Exact solutions of some quadratic and quartic birth and death processes and related orthogonal polynomials. J. Comput. Appl. Math., 67(1), 103–127.Google Scholar
Valent, G. 1999. Indeterminate moment problems and a conjecture on the growth of the entire functions in the Nevanlinna parametrization. Pages 227–237 of Applications and Computation of Orthogonal Polynomials (Oberwolfach, 1998). Internat. Ser. Numer. Math., vol. 131. Basel: Birkhäuser.Google Scholar
Valent, G. and Van Assche, W. 1995. The impact of Stieltjes’ work on continued fractions and orthogonal polynomials: Additional material. J. Comput. Appl. Math., 65(1-3), 419–447. Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994).Google Scholar
Van Assche, W. 1993. The impact of Stieltjes’ work on continued fractions and orthogonal polynomials. Pages 5–37 of Dijk, Van, G. (ed), Thomas Jan Stieltjes, Œuvres Complètes/Collected papers. Vol. I. Berlin: Springer.Google Scholar
Van Assche, W. 2007. Rakhmanov’s theorem for orthogonal matrix polynomials on the unit circle. J. Approx. Theory, 146(2), 227–242.Google Scholar
Van Assche, W. and Magnus, A. P. 1989. Sieved orthogonal polynomials and discrete measures with jumps dense in an interval. Proc. Amer. Math. Soc., 106(1), 163–173.Google Scholar
Van der Jeugt, J. 1997. Coupling coefficients for Lie algebra representations and addition formulas for special functions. J. Math. Phys., 38(5), 2728–2740.Google Scholar
Van der Jeugt, J. and Jagannathan, R. 1998. Realizations of su(1,1) and Uq(su(1,1)) and generating functions for orthogonal polynomials. J. Math. Phys., 39(9), 5062–5078.Google Scholar
Van Deun, J. 2007. Electrostatics and ghost poles in near best fixed pole rational interpolation. Electron. Trans. Numer. Anal., 26, 439–452.Google Scholar
van Eijndhoven, S. J. L. and Meyers, J. L. H. 1990. New orthogonality relations for the Hermite polynomials and related Hilbert spaces. J. Math. Anal. Appl., 146, 89–98.Google Scholar
Verblunsky, S. 1935. On positive harmonic functions: A contribution to the algebra of Fourier series. Proc. London Math. Soc., 38, 125–157.Google Scholar
Verblunsky, S. 1936. On positive harmonic functions (second paper). Proc. London Math. Soc., 40, 290–320.Google Scholar
Verma, A. 1972. Some transformations of series with arbitrary terms. Ist. Lombardo Accad. Sci. Lett. Rend. A, 106, 342–353.Google Scholar
Viennot, G. 1983. Une théorie combinatoire de polynômes orthogonaux generaux. Université de Québec à Montréal. Lecture notes.Google Scholar
Vilenkin, N. Ja. 1968. Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs, vol. 22. Providence, RI: American Mathematical Society. Translated from the Russian by Singh, V. N..Google Scholar
Vilenkin, N. Ja. and Klimyk, A. U. 1991–1993. Representation of Lie Groups and Special Functions, vols. 1–3. Mathematics and Its Applications (Soviet Series), vols. 72, 74, 75. Dordrecht, The Netherlands: Kluwer Academic Publishers Group.Google Scholar
Vinet, L. and Zhedanov, A. 2004. A characterization of classical and semiclassical orthogonal polynomials from their dual polynomials. J. Comput. Appl. Math., 172(1), 41–48.Google Scholar
Volkmer, H. 2008. Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math., 213(2), 488–500.Google Scholar
Wall, H. S. 1948. Analytic Theory of Continued Fractions. New York, NY: D. Van Nostrand.Google Scholar
Wall, H. S. and Wetzel, M. 1944. Quadratic forms and convergence regions for continued fractions. Duke Math. J., 11, 89–102.Google Scholar
Wallisser, R. 2000. On Lambert’s proof of the irrationality of π. Pages 521–530 of Algebraic Number Theory and Diophantine Analysis (Graz, 1998). Berlin: de Gruyter.Google Scholar
Watkins, D. S. 1993. Some perspectives on the eigenvalue problem. SIAM Rev., 35(3), 430–471.Google Scholar
Watson, G. N. 1944. A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press.Google Scholar
Wendroff, B. 1961. On orthogonal polynomials. Proc. Amer. Math. Soc., 12, 554–555.Google Scholar
Whittaker, E. T. and Watson, G. N. 1927. A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Fourth edn. Reprint, Cambridge Mathematical Library, Cambridge: Cambridge University Press, 1996.Google Scholar
Widder, D. V. 1941. The Laplace Transform. Princeton Mathematical Series, vol. 6. Princeton, NJ: Princeton University Press.Google Scholar
Widom, H. 1967. Polynomials associated with measures in the complex plane. J. Math. Mech., 16, 997–1013.Google Scholar
Wigert, S. 1923. Sur les polynomes orthogonaux et l’approximation des fonctions continues. Ark. Mat. Astronom. Fys., 17(18), 15 pages.Google Scholar
Wilson, J. A. 1980. Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal., 11(4), 690–701.Google Scholar
Wilson, J. A. 1982. Hypergeometric Series Recurrence Relations and Properties of Some Orthogonal Polynomials. (Preprint).Google Scholar
Wilson, J. A. 1991. Asymptotics for the 4F3 polynomials, J. Approx. Theory, 66, no. 1, 58–71.Google Scholar
Wilson, M. W. 1970. Nonnegative expansions of polynomials. Proc. Amer. Math. Soc., 24, 100–102.Google Scholar
Wimp, J. 1985. Some explicit Padé approximants for the function Φ′/Φ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal., 16(4), 887–895.Google Scholar
Wimp, J. 1987. Explicit formulas for the associated Jacobi polynomials and some applications. Canad. J. Math., 39(4), 983–1000.Google Scholar
Wintner, A. 1929. Spektraltheorie der unendlichen Matrizen. Leipzig: S. Hirzel.Google Scholar
Wong, R. and Zhang, J.-M. 1994a. Asymptotic monotonicity of the relative extrema of Jacobi polynomials. Canad. J. Math., 46(6), 1318–1337.Google Scholar
Wong, R. and Zhang, J.-M. 1994b. On the relative extrema of the Jacobi polynomials Pn(0,-1)(x). SIAM J. Math. Anal., 25(2), 776–811.Google Scholar
Zhani, D. 1984. Problème des Moments Matriciels sur la Droite: Construction d’une Famille de Solutions et Questions D’unicité. Publications du Département de Mathématiques. Nouvelle Série. D [Publications of the Department of Mathematics. New Series. D], vol. 84. Lyon: Université Claude-Bernard Département de Mathématiques. Dissertation, Université de Lyon I, Lyon, 1983.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Edited by Mourad E. H. Ismail, University of Central Florida
  • Assisted by Walter Van Assche, Katholieke Universiteit Leuven, Belgium
  • Book: Encyclopedia of Special Functions: The Askey-Bateman Project
  • Online publication: 14 September 2020
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Edited by Mourad E. H. Ismail, University of Central Florida
  • Assisted by Walter Van Assche, Katholieke Universiteit Leuven, Belgium
  • Book: Encyclopedia of Special Functions: The Askey-Bateman Project
  • Online publication: 14 September 2020
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Edited by Mourad E. H. Ismail, University of Central Florida
  • Assisted by Walter Van Assche, Katholieke Universiteit Leuven, Belgium
  • Book: Encyclopedia of Special Functions: The Askey-Bateman Project
  • Online publication: 14 September 2020
Available formats
×