Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-16T10:39:38.578Z Has data issue: false hasContentIssue false
  • English
  • Français

The Associated Ultraspherical Polynomials and their q-Analogues

Published online by Cambridge University Press:  20 November 2018

Joaquin Bustoz  and
Mourad E. H. Ismail
Joaquin Bustoz
Affiliation:
Arizona State University, Tempe, Arizona
Mourad E. H. Ismail
Affiliation:
Arizona State University, Tempe, Arizona
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that

1.1

A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence

1.2

with

1.3

Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by

1.4

with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 718 - 736
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Al-Salam, W. and Ismail, M. E. H., Orthogonal polynomials associated with Roger s- Ramanujan continued fractions, to appear, Pacific J. of Math.Google Scholar
2. Andrews, G. E., The theory of partitions, Encyclopedia of Mathematics and its Applications 2 (Addison-Wesley, Reading, 1977.Google Scholar
3. Askey, R. A., Linearization of the products of orthogonal polynomials, Problems in Analysis (Princeton University Press, Princeton, 1970), 223–238.Google Scholar
4. Askey, R. A. and Ismail, M. E. H., The Rogers q-ultraspherical polynomials, Approximation Theory III (Academic Press, 1980), 175–182.Google Scholar
5. Askey, R. A. and Ismail, M. E. H., A generalization of the ultra spherical polynomials, to appear, Studies in Analysis.Google Scholar
6. Askey, R. A. and Ismail, M. E. H., Recurrence relations, continued fractions and orthogonal polynomials, to appear.Google Scholar
7. Askey, R. A. and Wilson, J., Some basic hyper geometric orthogonal polynomials that generalize the Jacobi polynomials, to appear.Google Scholar
8. Barrucand, P. and Dickinson, D., On the associated Legendre polynomials, Orthogonal Expansions and Their Continuous Analogues (Southern Illinois University, Carbondale, 1967), 43–50.Google Scholar
9. Chihara, T. S., An introduction to orthogonal polynomials (Gordon and Breach, New York, 1978.Google Scholar
10. Cigler, J., Die Mehler- Formel fur q-Hermite-polynôme, to appear.Google Scholar
11. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, G. F., Higher transcendental functions, Vol. 1 (McGraw-Hill, New York, 1953).Google Scholar
12. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, G. F., Higher transcendental functions, Vol. 2 (McGraw-Hill, New York, 1953).Google Scholar
13. Feldheim, E., Sur les polynômes generalises de Legendre, Izv. Akad. Nauk USSR, Ser. Math. 6 (1941), 241–248.Google Scholar
14. Frank, E., A new class of continued fraction expansions for the ratios ofHeine functions, Trans. Amer. Math. Soc. 88 (1958), 288–300.Google Scholar
15. Frank, E., A new class of continued fraction expansions for the ratios of Heine functions II, Trans. Amer. Math. Soc. 95 (1960), 17–26.Google Scholar
16. Frank, E., A new class of continued fraction expansions for the ratios of Heine functions III, Trans. Amer. Math. Soc. 96 (1960), 312–320.Google Scholar
17. Hahn, W., Ûber orthogonalpolynome, die q-differenze-ngleichugen, Math. Nachr. 2 (1949), 4–34.Google Scholar
18. Lanzewizky, I. L., Uber die orthogonalitat de Fejer-Szegoschen Polynôme, C. R. (Dokl.) Acad. Sdi. USSR 3 (1941), 199–200.Google Scholar
19. Nevai, P., Orthogonal polynomials, Memoirs Amer. Math. Soc. 213 (1979).Google Scholar
20. Olver, F. W. J., Asymptotics and special functions (Academic Press, New York, 1974.Google Scholar
21. Pollaczek, F., Sur une famille de polynômes orthogonaux à quatre paramètres, C. R. Acad. Sci. Paris 230 (1950), 2254–2256.Google Scholar
22. Pollaczek, F., Sur une generalisation des polynômes de Jacobi, Mémorial des Sciences Mathématique 121 (Paris, 1956.Google Scholar
23. Shohat, J. A. and Tamarkin, J. D., The problem of moments, Mathematical Surveys, Vol. 1, Revised Edition (Amer. Math. Soc, Providence, 1950).Google Scholar
24. Slater, L. J., Generalized hyper geometric functions (Cambridge University Press, Cambridge, 1966.Google Scholar
25. Szegô, G., Ein Beitrag zur théorie der thetafunktionen, sitzungs, Pruss. Akad. Wissen., Phys. Math. Klasse (1926), 242–252.Google Scholar
26. Szegô, G., Orthogonal polynomials, Colloquium Publications 23, Fourth Edition (Amer. Math. Soc, Providence, 1975).Google Scholar
27. Toscano, L.. Polinomi associati a polinomi classici, Riv. Mat. Univ. Parma 4 (1953), 387–402.Google Scholar
28. Toscano, L.. Formule de riduzione tra funzioni e polinomi classici, Riv. Mat. Univ. Parma 6 (1955), 117–140.Google Scholar