Deformed Homogeneous Polynomials and the Generalized q-Exponential Operator

2024/09/10 by López, Ronald Orozco
#33D15 #33D45 #Combinatorics (math.CO) #FOS: Mathematics #Primary 05A30. Secondary 11B39

paper · doi:10.48550/arxiv.2409.06878

Abstract

This paper introduces the deformed homogeneous polynomials R_n(x,y;u|q). These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials h_n(x|q), the generalized Rogers-Szegö polynomials r_n(x,y), the Stieljes-Wigert polynomials S_n(x;q), among others. Basic properties of the polynomial R_n are given, along with recurrence relations, its q-difference equation, and representations. Generating functions for the polynomials R_n(x,y;u|q) are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the q-binomial formula and the Heine transformation formula are obtained. These results are obtained via the u-deformed q-exponential operator E(yD_q|u), defined here. From this operator, we obtain for free the operators T(yD_q) the Chen, R(yD_q) of Saad, E(yD_q) of Exton, and R(yD_q) of Rogers-Ramanujan when u=1,q,√(q),q², respectively. We introduce the deformed basic hypergeometric series _rΦ_s, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.

Related

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