Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T12:41:52.655Z Has data issue: false hasContentIssue false
  • English
  • Français

Associated Continuous Hahn Polynomials

Published online by Cambridge University Press:  20 November 2018

Dharma P. Gupta ,
Mourad E. H. Ismail  and
David R. Masson
Dharma P. Gupta
Affiliation:
Department of Mathematics, York University, North York, Ontario M3J 1P3
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida, USA 33620-5700
David R. Masson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1
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.

Explicit solutions to the recurrence relation for associated continuous Hahn polynomials are derived using 3F2 contiguous relations. These solutions are used to obtain a new continued fraction and the associated absolutely continuous measure. An exceptional case is shown to yield entry 33 in Chapter 12 of Ramanujan's second notebook.

Keywords

33A65Continued fractionscontiguous relationshypergeometric functionsorthogonal polynomialsreal and complex weight functions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1263 - 1280
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Askey, R., Continuous Hahn polynomials, J. Phys. A: Math. Gen. 18(1985), L1017-L1019.Google Scholar
2. Askey, R. and Wilson, J., A set of hypergeometricpolynomials, SIAM J. Math. Anal. 13(1982), 651–655.Google Scholar
3. Atakishiyev, N.M. and Suslov, S.K., The Hahn and Meixner polynomials of an imaginary argument and some of their applications, J. Phys. A: Math. Gen. 18(1985), 1583–1596.Google Scholar
4. Bailey, W.N., Generalized hypergeometric series. Cambridge University Press, Cambridge, 1935, reprinted by Hafner, New York, 1972.Google Scholar
5. Bailey, W.N., Contiguous hypergeometric functions of the type 3F2O), Proc. Glasgow Math. Assoc. 2(1954), 62–65.Google Scholar
6. Berndt, B.C., Lamphere, R.L. and Wilson, B.M., Chapter 12 of Ramanujans second notebook: continued fractions, Rocky Mountain J. Math. 15(1985), 235–310.Google Scholar
7. Gasper, G. and Rahman, M., Basic hypergeometric series. Cambridge University Press, Cambridge, 1990.Google Scholar
8. Gautschi, W., Computational aspects of three term recurrence relations, SIAM Review 9(1967), 24–82.Google Scholar
9. Ismail, M.E.H., Letessier, J., Valent, G. and Wimp, J., Two families of associated Wilson polynomials, Can. J. Math. 42(1990), 659–695.Google Scholar
10. Ismail, M.E.H. and Libis, C.A., Contiguous relations, basic hypergeometric functions and orthogonal polynomials, I, J. Math. Anal. Appl. 141(1989), 349–372.Google Scholar
11. Ismail, M.E.H. and Rahman, M., Associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. (1991), to appear.Google Scholar
12. Masson, D.R., Difference equations, continued fractions, Jacobi matrices and orthogonal polynomials. In: Nonlinear Numerical Methods and Rational Approximation, 1987, A. Cuyt, éd., D. Reidel, Dordrecht, 1988,239–257.Google Scholar
13. Masson, D.R., Some continued fractions of Ramanujan and Meixner-Pollaczek polynomials, Canad. Math. Bull. 32(1989), 177–181.Google Scholar
14. Masson, D.R., Associated Wilson polynomials, Constructive Approximation 7(1991), 521–534.Google Scholar
15. Masson, D.R. and Repka, J., Spectral theory of Jacobi matrices in l2(Z ) and su(1, 1) Lie algebra, SIAM J. Math. Anal., 22(1991), 1131–1146.Google Scholar
16. Szegö, G., Orthogonal polynomials. Fourth edition, Amer. Math. Soc, Providence, Rhode Island, 1975.Google Scholar
17. Wilson, J.A., Three term contiguous relations and some new orthogonal polynomials, In: Padé and Rational Approximation, eds. E. Saff and R. Varga, Academic Press, New York, 1977,227–232.Google Scholar
18. Wilson, J.A., Hypergeometric series, recurrence relations and some new orthogonal functions. Doctoral Dissertation, University of Wisconsin-Madison, 1978.Google Scholar
19. Wimp, J., The associated Jacobi polynomials, Can. J. Math. 39(1987), 983–1000.Google Scholar
20. Zhang, L.C., Ramanujan's continued fractions for products of gamma functions, to appear.Google Scholar