×

Asymptotic approximations between the hahn-type polynomials and Hermite, Laguerre and charlier polynomials. (English) Zbl 1168.33309

Summary: It has been shown by C. Ferreira et al. [Adv. Appl. Math. 31, No. 1, 61–85 (2003; Zbl 1029.33004)] and J. L. López and N. M. Temme [Methods Appl. Anal. 6, No. 2, 131–146 (1999; Zbl 0958.33004); J. Comput. Appl. Math. 133, No. 1–2, 623–633 (2001; Zbl 0990.33010)] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic expansions.
In this paper, we continue with that investigation and establish asymptotic connections between the fourth level and the two lower levels: we derive twelve asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Hermite, Charlier and Laguerre polynomials. From these expansions, several limits between polynomials are derived. Some numerical experiments give an idea about the accuracy of the approximations and, in particular, about the accuracy in the approximation of the zeros of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of the zeros of the Hermite, Charlier and Laguerre polynomials.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A10 Approximation by polynomials
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
26C10 Real polynomials: location of zeros
33F05 Numerical approximation and evaluation of special functions
65D20 Computation of special functions and constants, construction of tables
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ferreira, C., López, J.L., Mainar, E.: Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials. Adv. Appl. Math. 31, 61–85 (2003) · Zbl 1029.33004 · doi:10.1016/S0196-8858(02)00552-3
[2] Frenzen, C.L., Wong, R.: Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal. 19, 1232–1248 (1988) · Zbl 0654.33004 · doi:10.1137/0519087
[3] Godoy, E., Ronveaux, A., Zarzo, A., Area, I.: On the limit relations between classical continuous and discrete orthogonal polynomials. J. Comput. Appl. Math. 91, 97–105 (1998) · Zbl 0934.33013 · doi:10.1016/S0377-0427(98)00026-0
[4] Godoy, E., Ronveaux, A., Zarzo, A., Area, I.: Transverse limits in the Askey tableau. J. Comput. Appl. Math. 99, 327–335 (1998) · Zbl 0933.65147 · doi:10.1016/S0377-0427(98)00155-1
[5] Koekoek, R., Swarttouw, R.F.: Askey scheme or hypergeometric orthogonal polynomials, http://aw.twi.tudelft.nl/koekoek/askey (1999)
[6] López, J.L., Temme, N.M.: Approximations of orthogonal polynomials in terms of Hermite polynomials. Methods Appl. Anal. 6, 131–146 (1999) · Zbl 0958.33004
[7] López, J.L., Temme, N.M.: Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel and Buchholz polynomials. J. Math. Anal. Appl. 239, 457–477 (1999) · Zbl 0979.33004 · doi:10.1006/jmaa.1999.6584
[8] López, J.L., Temme, N.M.: The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133, 623–633 (2001) · Zbl 0990.33010 · doi:10.1016/S0377-0427(00)00683-X
[9] Rui, B., Wong, R.: Uniform asymptotic expansions of Charlier polynomials. Methods Appl. Anal. 1, 294–313 (1994) · Zbl 0846.41025
[10] Wolfram, S.: Wolfram research web page, http://functions.wolfram.com (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.