A Generalization of q-Binomial Theorem
Abstract
By using Liu's -partial differential equations theory, we prove that if an analytic function in several variables satisfies a system of -partial differential equations, if and only if it can be expanded in terms of homogeneous -Al-Salam-Carlitz polynomials. As an application, we proved that for and , \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 -binomial theorem or so-called Cauchy theorem.
Cite
@article{arxiv.2204.11625,
title = {A Generalization of q-Binomial Theorem},
author = {Qi Bao},
journal= {arXiv preprint arXiv:2204.11625},
year = {2022}
}
Comments
The error in this manuscript is that the right side of equation (3.1) is not suitable for formula (1.1). This causes formula (3.1) to be incorrect. Therefore, theorem 1.2 is also incorrect. However, the second part of this manuscript about the theorem of q-partial differential equation theory is still the correct conclusion