Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T06:47:50.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Classical Orthogonal Polynomials as Moments

Published online by Cambridge University Press:  20 November 2018

Mourad E. H. Ismail  and
Dennis Stanton
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620, U.S.A.
Dennis Stanton
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

We show that the Meixner, Pollaczek, Meixner-Pollaczek, the continuous q-ultraspherical polynomials and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures.We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials.

Keywords

33D4533D2033C4530E05Classical orthogonal polynomialsAl-Salam-Chihara polynomialscontinuous q-ultraspherical polynomialsgenerating functionsmultilinear generating functionstransformation formulasumbral calculus
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 3 , 01 June 1997 , pp. 520 - 542
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Al-Salam, W.A. and Carlitz, L., Some orthogonal q-polynomials, Math. Nachr. 30(1965), 47–61.Google Scholar
2. Al-Salam, W.A. and Chihara, T.S., Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7(1976), 16–28.Google Scholar
3. Andrews, G.E. , On the foundations of combinatorial theory. V: Eulerian differential operators, Stud. Appl. Math. 50(1971), 345–375.Google Scholar
4. Andrews, G.E. and Askey, R.A., Classical orthogonal polynomials, . In: Polynomes Orthogonaux et Applications, (eds. C. Brezinski et ál.), Lecture Notes in Math. 1171, Springer-Verlag, Berlin, 1984. 36–63.Google Scholar
5. Askey, R.A. and Ismail, M. E. H., Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300(1984).Google Scholar
6. Askey, R.A., Rahman, M., and Suslov, S.K., On a general q-Fourier transformation with nonsymmetric kernels, J. Comput. Appl. Math. 68(1996), 25–55.Google Scholar
7. Askey, R.A. and Wilson, J.A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer.Math. Soc. 319(1985).Google Scholar
8. Berg, C. and Ismail, M. E. H., q-Hermite polynomials and the classical polynomials, Canad. J. Math. 48(1996), 43–63.Google Scholar
9. Chihara, T.S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.Google Scholar
10. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher Transcendental Functions, volume 1, McGraw-Hill, New York, 1953.Google Scholar
11. Erdelyi, A., Higher Transcendental Functions, volume 2, McGraw-Hill, New York, 1953.Google Scholar
12. Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, 1990.Google Scholar
13. Gasper, G., Positivity of the Poisson kernel for the continuous q-ultraspherical polynomials, SIAM J.Math. Anal. 14(1983), 409–420.Google Scholar
14. Gupta, D.P., Ismail, M. E. H. and Masson, D.R., Contiguous relations, basic hypergeometric functions and orthogonal polynomials. II: Associated big q-Jacobi polynomials, J. Math. Anal. Appl. 171(1993), 477–497.Google Scholar
15. Ihrig, E.C. and Ismail, M. E. H., A q-umbral calculus, J. Math. Anal. Appl. 84(1981), 178–207.Google Scholar
16. Ismail, M. E. H., Poisson kernels, in preparation.Google Scholar
17. Ismail, M. E. H., M. Rahman and Suslov, S.K., Some summation theorems and transformation formulas for q-series, Canad. J. Math. 49(1997), 543–567.Google Scholar
18. Ismail, M. E. H. and Stanton, D., On the Askey-Wilson and Rogers polynomials, Canad. J.Math. 40(1988), 1025–1045.Google Scholar
19. Joni, S.A. and Rota, G.C., Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61(1978), 93–139.Google Scholar
20. Karlin, S., Sign regularity of classical orthogonal polynomials. In: Orthogonal Expansions and Their Continuous Analogues, (ed. Haimo, D.), Southern Illinois Univ. Press, Carbondale, 1967. 55–74.Google Scholar
21. Kibble, W.F., An extension of theorem of Mehler on Hermite polynomials, Math. Proc. Cambridge Philos. Soc. 41(1945), 12–15.Google Scholar
22. Meixner, J., Unformung gewisser Reihen, deren Gleider Produkte hypergeometrischer Funktionen sind, Deutsch. Math. 6(1942), 341–349.Google Scholar
23. Nikiforov, A.F., Suslov, S.K. and Uvarov, V.B., Classical Orthogonal Polynomials of a Discrete Variable, English translation, Springer-Verlag, Berlin, 1991.Google Scholar
24. Rahman, M., A generalization of Gasper's kernel for Hahn polynomials: application to Pollaczek polynomials, Canad. J. Math. 30(1988), 133–146.Google Scholar
25. Rahman, M. and Verma, A., A q-integral representation for the Rogers q-ultraspherical polynomials and some applications, Constr. Approx. 2(1986), 1–10.Google Scholar
26. Roman, S. and Rota, G.C., The umbral calculus, Adv. inMath. 27(1978), 95–188.Google Scholar
27. Slepian, D., On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for Hermite polynomials, SIAM J. Math.Anal. 3(1972), 606–616.Google Scholar
28. Suslov, S.K., unpublished notes.Google Scholar
29. Szegö, G., Orthogonal Polynomials, Fourth Edition, Amer. Math. Soc., Providence, 1975.Google Scholar
30. Watson, G.N., A Treatise on the Theory of Bessel Functions, Second Edition, Cambridge Univ. Press, Cambridge, 1944.Google Scholar
31. Wimp, J., Some explicit Pade approximants for the function Φ’/Φand a related quadrature formula involving Bessel functions, SIAM J. Math. Anal. 16(1985), 887–896.Google Scholar