Rogers-Ramanujan Type $q$-Exponential Operator and Stieljes-Widgert Polynomials

TL;DR

This work targets generating functions and Mehler/Rogers-type formulas for the Stieljes-Wigert polynomials by employing a Rogers-Ramanujan type $q$-exponential operator $\mathcal{R}(qD_{q})$. By combining the $q$-differential operator $D_{q}$ with the Ramanujan $q$-exponential function $\mathcal{R}_{q}(z)$, the authors obtain closed-form generating functions and a suite of bilinear and double generating identities for the bivariate polynomials $\mathrm{S}^{*}_{n}(x,y;q)$. Key results include the generating function $\sum_{n\ge0}\mathrm{S}^{*}_{n}(x,y;q)\frac{z^{n}}{(q;q)_{n}}=\frac{\mathcal{R}_{q}(zy)}{(zx;q)_{\infty}}$, a Mehler-type formula for $\mathrm{S}^{*}_{n}$, and Rogers-type double generating functions expressed through basic hypergeometric series ${}_{1}\phi_{2}$ and ${}_{0}\phi_{2}$. The operator-centric approach unifies and streamlines derivations, linking Stieljes-Wigert polynomials to Ramanujan-type $q$-exponentials and broadening the toolbox for $q$-polynomial theory.

Abstract

In this paper, we use the Rogers-Ramanujan type $q$-exponential operator $\mathcal{R}(qD_{q})$ to derive generating functions, and Mehler and Rogers formulas, for the non-normalized homogeneous Stieljes-Wigert polynomials $\mathrm{S}_{n}(x,y;q)$.

Theorems & Definitions (35)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4