Generalization of the Pollaczek polynomials

Abstract

This study introduces a novel generalization of the Pollaczek polynomials in the framework of 2-orthogonality, defined via a generating function of a specific form. We investigate the key properties of these polynomials, including: • A third-order recurrence relation, • A hypergeometric representation, • Their classical character, • A solution to the generalized problem posed by Ismail (Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005, Problem 24.8.2, p. 685) without requiring a four-term recurrence relation. Furthermore, we demonstrate that the hypergeometric representation of the 2- analogue of Pollaczek polynomials yields a polynomial in cos ? of degree n.