q-Partial Differential Equations for Generalized q-2D Ultraspherical Polynomials with Some Applications

Abstract

Motivated by Ismail and Zhang [Trans. Amer. Math. Soc., 2017, 369: 6779–6821], we introduce two generalized q-2D ultraspherical polynomials and deduce their corresponding q-partial differential equations. In addition, bilinear and multilinear generating functions for the generalized q-2D ultraspherical polynomials are deduced by the method of q-partial differential equations, which generalize results of Ismail and Zhang. Moreover, a new transformational identity and involving q-2D ultraspherical polynomials and linear relations for q-2D ultraspherical polynomials are obtained.

Keywords

q-partial differential equation / q-2D ultraspherical polynomial / generating function / transformational identity / linear relation / 05A30 / 33D15 / 33D45 / 05A40 / 11B65

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