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The homogeneous little q$q$‐Jacobi polynomials

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  • Jian Cao
  • Yue Yang
  • Sama Arjika

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.

Suggested Citation

  • Jian Cao & Yue Yang & Sama Arjika, 2025. "The homogeneous little q$q$‐Jacobi polynomials," Mathematische Nachrichten, Wiley Blackwell, vol. 298(12), pages 3791-3815, December.
    • Handle: RePEc:bla:mathna:v:298:y:2025:i:12:p:3791-3815
    DOI: 10.1002/mana.70067
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