The homogeneous little q$q$‐Jacobi polynomials

Jian Cao, Yue Yang and Sama Arjika

Mathematische Nachrichten, 2025, vol. 298, issue 12, 3791-3815

Abstract: Motivated by the q$q$‐operational equation for Rogers–Szegö polynomials [Sci. China Math. 66(2023), no. 6, 1199–1216], it is natural to ask whether some general q$q$‐polynomials exist, which are solutions of certain q$q$‐operational equations, q$q$‐difference equations, and q$q$‐partial differential equations. In this paper, based on the importance of little q$q$‐Jacobi polynomials, we define two homogeneous little q$q$‐Jacobi polynomials and search their corresponding q$q$‐operational equations, q$q$‐difference equations, and q$q$‐partial differential equations by the technique of noncommutative q$q$‐binomial theorem and recurrence relations. In addition, we deduce some generating functions for homogeneous little q$q$‐Jacobi polynomials by methods of q$q$‐operational equation, q$q$‐difference equation, and q$q$‐partial differential equation. Moreover, we consider recurrence relations for homogeneous little q$q$‐Jacobi polynomials.

Date: 2025
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