[Paper Review] On the $(p,q)$-Gamma and the $(p,q)$-Beta functions

P. Njionou Sadjang|arXiv (Cornell University)|Jun 22, 2015

Advanced Mathematical Identities5 references18 citations

TL;DR

This paper introduces the $(p,q)$-Gamma and $(p,q)$-Beta functions as generalizations of the classical Gamma and Beta functions using $(p,q)$-calculus. It establishes fundamental properties, including a $(p,q)$-version of the Legendre multiplication formula and Gauss' multiplication formula, extending known $q$-special function results to the broader $(p,q)$-framework with applications in quantum calculus and special functions.

ABSTRACT

We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.

Motivation & Objective

• To generalize the classical Gamma and Beta functions using $(p,q)$-calculus for applications in quantum groups and special functions.
• To extend known $q$-special function results to the more general $(p,q)$-framework.
• To establish fundamental identities and functional equations for the new $(p,q)$-Gamma and $(p,q)$-Beta functions.
• To derive $(p,q)$-analogues of classical multiplication formulas, including Legendre and Gauss-type identities.
• To unify and generalize previous work on $q$-Gamma and $q$-Beta functions within a broader $(p,q)$-calculus setting.

Proposed method

• Define the $(p,q)$-Gamma function using infinite products involving $(p,q)$-shifted factorials: $\Gamma_{p,q}(x) = \frac{(p\ominus q)_{p,q}^\infty}{(p^x\ominus q^x)_{p,q}^\infty}(p-q)^{1-x}$.
• Introduce the $(p,q)$-Beta function via the relation $B_{p,q}(x,y) = \frac{\Gamma_{p,q}(x)\Gamma_{p,q}(y)}{\Gamma_{p,q}(x+y)}$, generalizing the classical Beta-Gamma relation.
• Utilize $(p,q)$-factorials, $(p,q)$-binomial coefficients, and $(p,q)$-powers to derive functional identities and recurrence relations.
• Establish the $(p,q)$-Legendre multiplication formula: $\Gamma_{p,q}(2x)\Gamma_{p^2,q^2}(\frac{1}{2}) = (p+q)^{2x-1}\Gamma_{p^2,q^2}(x)\Gamma_{p^2,q^2}(x+\frac{1}{2})$.
• Prove the $(p,q)$-Gauss multiplication formula: $\Gamma_{p,q}(nx)\prod_{k=1}^{n-1}\Gamma_{p^n,q^n}(\frac{k}{n}) = [n]_{p,q}^{nx-1}\prod_{k=0}^{n-1}\Gamma_{p^n,q^n}(x+\frac{k}{n})$.
• Use properties of infinite products and $(p,q)$-shifted factorials to verify identities and derive recurrence relations for the $(p,q)$-Beta function.

Experimental results

Research questions

• RQ1 How can the classical Gamma and Beta functions be generalized within the $(p,q)$-calculus framework?
• RQ2 What are the fundamental functional equations and recurrence relations satisfied by the $(p,q)$-Gamma and $(p,q)$-Beta functions?
• RQ3 Can classical multiplication formulas like Legendre’s and Gauss’s be extended to the $(p,q)$-setting?
• RQ4 How do the $(p,q)$-Gamma and $(p,q)$-Beta functions relate to known $q$-special functions as $p \to 1$?
• RQ5 What are the structural properties of the $(p,q)$-shifted factorials and their role in defining these generalized functions?

Key findings

• The $(p,q)$-Gamma function is defined via an infinite product involving $(p,q)$-shifted factorials, generalizing the classical and $q$-Gamma functions.
• The $(p,q)$-Legendre multiplication formula is established as $\Gamma_{p,q}(2x)\Gamma_{p^2,q^2}(\frac{1}{2}) = (p+q)^{2x-1}\Gamma_{p^2,q^2}(x)\Gamma_{p^2,q^2}(x+\frac{1}{2})$.
• The $(p,q)$-Gauss multiplication formula is derived: $\Gamma_{p,q}(nx)\prod_{k=1}^{n-1}\Gamma_{p^n,q^n}(\frac{k}{n}) = [n]_{p,q}^{nx-1}\prod_{k=0}^{n-1}\Gamma_{p^n,q^n}(x+\frac{k}{n})$.
• The $(p,q)$-Beta function satisfies recurrence relations: $B_{p,q}(x,y+1) = \frac{[y]_{p,q}}{[x+y]_{p,q}}B_{p,q}(x,y)$ and $B_{p,q}(x+n,y) = \frac{(p^x\ominus q^x)^n_{p,q}}{(p^{x+y}\ominus q^{x+y})^n_{p,q}}B_{p,q}(x,y)$.
• As $p \to 1$, the $(p,q)$-Gamma and $(p,q)$-Beta functions reduce to their $q$-analogues, confirming consistency with prior work.
• The functional identities are proven using algebraic manipulation of infinite products and $(p,q)$-factorial identities, establishing the framework’s consistency.