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Asymptotics of Racah polynomials with fixed parameters

Authors: X.-S. Wang and R. Wong
Journal: Proc. Amer. Math. Soc. 146 (2018), 1083-1096
MSC (2010): Primary 41A60; Secondary 33C45
Published electronically: September 28, 2017
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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.

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  • [1] 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,
  • [2] 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,
  • [3] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884
  • [4] 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
  • [5] 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
  • [6] 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
  • [7] 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,
  • [8] 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

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Additional Information

X.-S. Wang
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70503

R. Wong
Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Keywords: Asymptotics, Racah polynomials, difference equations, matching principle
Received by editor(s): January 17, 2017
Received by editor(s) in revised form: April 11, 2017
Published electronically: September 28, 2017
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society

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