A Generalization of q-Binomial Theorem

TL;DR

This paper generalizes the q-binomial theorem by linking solutions of q-partial differential equations to expansions in homogeneous (q,c)-Al-Salam-Carlitz polynomials, providing a broader mathematical framework.

Contribution

It introduces a new characterization of functions satisfying q-partial differential equations via (q,c)-Al-Salam-Carlitz polynomials, extending the classical q-binomial theorem.

Findings

Established a new expansion criterion for functions using q-partial differential equations.

Derived a generalized q-binomial theorem as an application.

Connected differential equations with polynomial expansions in q-series.

Abstract

By using Liu's $q$-partial differential equations theory, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, if and only if it can be expanded in terms of homogeneous $(q,c)$-Al-Salam-Carlitz polynomials. As an application, we proved that for $c\neq0$ and $\max \{|cq|,|x|\}<1$, \begin{align*} \sum_{n=0}^{\infty} \frac{ (a;q)_n }{(cq;q)_n}x^n=(ax/c;q)_{\infty} \sum_{n=0}^{\infty} \frac{x^n}{(cq;q)_n}, \end{align*} which is a generalization of famous $q$-binomial theorem or so-called Cauchy theorem.