Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-18T20:12:20.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Permutation Problems and Special Functions

Published online by Cambridge University Press:  20 November 2018

Richard Askey  and
Mourad E. H. Ismail
Richard Askey
Affiliation:
The University of Wisconsin, Madison, Wisconsin
Mourad E. H. Ismail
Affiliation:
The University of Wisconsin, Madison, Wisconsin
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.

Suppose we have n events Ai, … , An and let p(Aj1 … , Ajk) be the probability that the events Ah, . . . , Ajk occur jointly. The probability P0 that none of Ai, . . . , An occur is given by Poincaré's formula, the probabilistic version of the principle of inclusion and exclusion:

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 4 , 01 August 1976 , pp. 853 - 874
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Askey, R., Orthogonal polynomials and special functions, CBMS, Vol, 21, SI AM, Philadelphia, 1975.Google Scholar
2. Askey, R. and Gasper, G., Certain rational functions whose power series have positive coefficients, Amer. Math. Monthly 79 (1972), 327–341.Google Scholar
3. Askey, R. and Gasper, G., Convolution structures for Laguerre polynomials, Journ. d'Anal. Math., to appear.Google Scholar
4. Askey, R., Ismail, M. E. H. and Rashed, T., A derangement problem, MRC Technical Report # 1522 (1975).Google Scholar
5. Boas, R. P., The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc. 4S (1939), 399–404.Google Scholar
6. Carlitz, L., Problem 71-13, A positive integral, solutions by Rousseau, C. C., Askey, R. A. and Gasper, G., SIAM Review 20 (1972), 372–378.Google Scholar
7. Debbi, D. and Gillis, J., An inequality relating to triple product integrals of Laguerre function, Israel J. Math. 18 (1974), 45–52.Google Scholar
8. Erdélyi, A., On some expressions in Laguerre polynomials, J. London Math. Soc. 13 (1938), 154–156.Google Scholar
9. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G., Higher transcendental functions, Volume 2 (McGraw-Hill, New York, 1953).Google Scholar
10. Even, S. and Gillis, J., Derangements and Laguerre polynomials, Proc. Camb. Phil. Soc. 79 (1976), 135–143,Google Scholar
11. Gillis, J., Integrals of products of Laguerre polynomials, SIAM Jour. Math. Anal. 6 (1975), 318–339.Google Scholar
12. Kaluza, Th., Elementarer Beweis einer Vermutung von K. Friedrichs und H. Lewy, Math. Zeit. 37 (1933), 689–697.Google Scholar
13. Kaplansky, I., On a generalization of the “Problème des rencontres,” Amer. Math. Monthly 1+6 (1939), 159–161.Google Scholar
14. Kaplansky, I., Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc. 50 (1944), 906–914.Google Scholar
15. Kaplansky, I. and Riordan, J., The problème des ménages, Scripta Math. 12 (1946), 113–124.Google Scholar
16. Karlin, S. and McGregor, J., The differential equations of birth and death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957), 489–546.Google Scholar
17. MacMahon, P. A., Combinatory analysis, Volume 1 (Chelsea, New York, 1960).Google Scholar
18. Mendelsohn, N. S., Symbolic solution of card matching problems, Bull. Amer. Math. 52 (1946), 918–924.Google Scholar
19. Muir, T., A treatise on the theory of determinants (Longmans, London, 1933; Reprinted: Dover, New York 1960).Google Scholar
20. Riordan, J., Three line latin rectangles, Amer. Math. Monthly 51 (1944), 450–452.Google Scholar
21. Riordan, J., An introduction to combinatorial analysis (Wiley, New York, 1958).Google Scholar
22. Roselle, D. P., Graphs, quasisymmetry and permutations with restricted positions, Duke Math. J. 41 (1974), 41–50.Google Scholar
23. Srivastava, H. M., Some expansions of generalized Whittaker functions, Proc. Camb. Phil. Soc. 61 (1965), 895–896.Google Scholar
24. Szego, G., Uber gewisse Potenzreihen mit tauter positiven Koeffizienten, Math. Zeit. 37 (1933), 674–688.Google Scholar
25. Szego, G., Orthogonal polynomials, Amer. Math. Soc. Colloquium Publications 23 (Amer. Math. Soc, Providence, 1975).Google Scholar
26. Touchard, J., Permutations discordant with two given permutations, Scripta Math. 19 (1953), 109–119.Google Scholar
27. Wyman, M. and Moser, L., On the problème des ménages, Can. J. Math. 10 (1958), 468–480.Google Scholar