, Volume 30, Issue 3, pp 399-402

Pollaczek polynomials and hypergeometric representation

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Abstract

This paper gives a solution, without the use of the three-term recurrence relation, of the problem posed in Ismail (Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005) (Problem 24.8.2, p. 658): that the hypergeometric representation of the general Pollaczek polynomials is a polynomial in cos(θ) of degree n. Chu solved in (Ramanujan J. 13(1–3): 221–225, 2007) the problem in a particular case. We use elementary properties of functions of complex variables and Pfaff’s transformation on hypergeometric 2 F 1-series.

The research is supported by NPST Program of King Saud University; project number 10-MAT1293-02.