[Paper Review] On the $q$-partial differential equations and $q$-series

Zhi-Guo Liu|arXiv (Cornell University)|May 6, 2018

Mathematical functions and polynomials27 references19 citations

TL;DR

This paper establishes a general expansion theorem for analytic functions in several complex variables that satisfy a system of $q$-partial differential equations, showing they can be expressed as series in Rogers-Szegő polynomials. The key contribution is a general $q$-transformation formula that unifies and extends known results, including Watson’s $q$-analog of Whipple’s theorem and Carlitz’s $q$-Mehler formula, via the theory of $q$-exponential operators and $q$-series identities.

ABSTRACT

Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the Rogers-Szegő polynomials. This expansion theorem allows us to develop a general method for proving $q$-identities. A general $q$-transformation formula is derived, which implies Watson's $q$-analog of Whipple's theorem as a special case. A multilinear generating function for the Rogers-Szegő polynomials is given. The theory of $q$-exponential operator is revisited.

Motivation & Objective

• To develop a general method for proving $q$-identities using $q$-partial differential equations.
• To establish an expansion theorem for analytic functions in several variables satisfying $q$-PDEs in terms of Rogers-Szegő polynomials.
• To derive a general $q$-transformation formula that encompasses known results such as Watson’s $q$-analog of Whipple’s theorem.
• To revisit and extend the theory of the $q$-exponential operator in the context of $q$-series and orthogonal polynomials.
• To construct multilinear generating functions for Rogers-Szegő and Stieltjes-Wigert polynomials.

Proposed method

• Using the theory of functions of several complex variables to analyze solutions of $q$-partial differential equations.
• Applying the $q$-derivative operator $\mathcal{D}_{q,x}$ and higher-order $q$-derivative formulas to derive identities involving $q$-shifted factorials.
• Employing the Rogers-Szegő polynomials $h_n(x,y|q)$ and their $q$-derivative properties to construct generating functions.
• Introducing the $q$-exponential operator $T(b\mathcal{D}_{q,a})$ to generate transformation formulas for $q$-series.
• Deriving a multilinear generating function for Rogers-Szegő polynomials via $q$-exponential operator identities.
• Extending classical $q$-series transformations, such as Sears’ ${}_3\phi_2$ formula, to include parameters and $q$-shifted factorials.

Experimental results

Research questions

• RQ1 Can analytic functions satisfying a system of $q$-partial differential equations be expanded in terms of Rogers-Szegő polynomials?
• RQ2 What is the general form of a $q$-transformation formula that unifies known $q$-series identities?
• RQ3 How can the $q$-exponential operator be used to derive new generating functions and transformation formulas?
• RQ4 What is the multilinear generating function for Rogers-Szegő polynomials, and how does it relate to $q$-series identities?
• RQ5 How do the $q$-partial differential equations for Rogers-Szegő and Stieltjes-Wigert polynomials lead to new $q$-Mehler formulas?

Key findings

• Any analytic function in several variables satisfying a system of $q$-partial differential equations admits an expansion in terms of Rogers-Szegő polynomials.
• A general $q$-transformation formula is derived, which reduces to Watson’s $q$-analog of Whipple’s theorem when specialized.
• A multilinear generating function for Rogers-Szegő polynomials is explicitly constructed using $q$-exponential operators.
• The theory of the $q$-exponential operator is revisited and extended, leading to new identities and transformation formulas.
• Carlitz’s extension of the $q$-Mehler formula for Rogers-Szegő polynomials is generalized via the $q$-exponential operator framework.
• An extension of Rogers’s summation formula and the Andrews-Askey integral is obtained through the proposed method.