Asymptotics of Racah polynomials with fixed parameters

Wang, X.-S.; Wong, R.

doi:10.1090/proc/13771

Asymptotics of Racah polynomials with fixed parameters
HTML articles powered by AMS MathViewer

by X.-S. Wang and R. Wong PDF

Proc. Amer. Math. Soc. 146 (2018), 1083-1096 Request permission

Abstract:

In this paper, we investigate asymptotic behaviors of Racah polynomials with fixed parameters and scaled variable as the polynomial degree tends to infinity. We start from the difference equation satisfied by the polynomials and derive an asymptotic formula in the outer region via ratio asymptotics. Next, we find the asymptotic formulas in the oscillatory region via a simple matching principle. Unlike the varying parameter case considered in a previous paper, the zeros of Racah polynomials with fixed parameters may not always be real. For this unusual case, we also provide a standard method to determine the oscillatory curve which attracts the zeros of Racah polynomials when the degree becomes large.

References

Richard Askey and James Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or $6-j$ symbols, SIAM J. Math. Anal. 10 (1979), no. 5, 1008–1016. MR 541097, DOI 10.1137/0510092
Li-Chen Chen, Mourad E. H. Ismail, and Plamen Simeonov, Asymptotics of Racah coefficients and polynomials, J. Phys. A 32 (1999), no. 3, 537–553. MR 1670247, DOI 10.1088/0305-4470/32/3/009
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
Mourad E. H. Ismail and Plamen Simeonov, Inequalities and asymptotics for a terminating ${}_4F_3$ series, Illinois J. Math. 51 (2007), no. 3, 861–881. MR 2379727
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
X.-S. Wang and R. Wong, Asymptotics of Racah polynomials with varying parameters, J. Math. Anal. Appl. 436 (2016), no. 2, 1149–1164. MR 3447001, DOI 10.1016/j.jmaa.2015.12.035
James Arthur Wilson, HYPERGEOMETRIC SERIES RECURRENCE RELATIONS AND SOME NEW ORTHOGONAL FUNCTIONS, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 2628179