[Paper Review] Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series

R. Jagannathan, K. Srinivasa Rao|ArXiv.org|Feb 27, 2006

Advanced Combinatorial MathematicsMathematics20 references94 citations

TL;DR

This paper introduces a systematic method to generalize $q$-series to $(p,q)$-series using the twin-basic number $[n]_{p,q} = (p^n - q^n)/(p - q)$, extending known $q$-identities into richer $(p,q)$-analogues. The key contribution is that $q$-results become special cases via simple parameter substitution, simplifying confluence limits and offering new algebraic and analytic tools for quantum groups and special functions.

ABSTRACT

We give a method to embed the q-series in a (p,q)-series and derive the corresponding (p,q)-extensions of the known q-identities. The (p,q)-hypergeometric series, or twin-basic hypergeometric series (diferent from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]_{p,q} = (p^n - q^n)/(p-q). This twin-basic number occurs in the theory of two-parameter quantum algebras and has been introduced independently in combinatorics. The (p,q)-identities thus derived, with doubling of the number of parameters, offer more choices for manipulations; for example, results that can be obtained via the limiting process of confluence in the usual q-series framework can be obtained by simpler substitutions. The q-results are of course special cases of the (p,q)-results corresponding to choosing p = 1. This also provides a new look for the q-identities.

Motivation & Objective

• To generalize $q$-hypergeometric identities to $(p,q)$-analogues using the twin-basic number $[n]_{p,q} = (p^n - q^n)/(p - q)$.
• To provide a unified framework for $(p,q)$-extensions of $q$-identities, such as the binomial theorem, Heine transformation, and Gauss sum.
• To demonstrate that $q$-results emerge as special cases when $p = 1$, bypassing complex limiting processes.
• To explore nontrivial $(p,q)$-generalizations of classical special functions, including Hermite polynomials and $q$-orthogonal polynomials.
• To establish a foundation for studying representations of two-parameter quantum algebras $U_{p,q}(gl(2))$ via $(p,q)$-hypergeometric series.

Proposed method

• Define the twin-basic number $[n]_{p,q} = (p^n - q^n)/(p - q)$ as the $(p,q)$-extension of the $q$-number $[n]_q = (1 - q^n)/(1 - q)$.
• Introduce the $(p,q)$-derivative $\hat{D}_{p,q}f(z) = \frac{f(pz) - f(qz)}{(p - q)z}$, which satisfies $\hat{D}_{p,q}z^n = [n]_{p,q}z^{n-1}$.
• Construct the $(p,q)$-hypergeometric series ${}_r\Phi_s$ as a generalization of the standard $q$-hypergeometric series ${}_r\phi_s$.
• Derive $(p,q)$-analogues of classical $q$-identities by embedding $q$-series into $(p,q)$-series using the twin-basic number framework.
• Utilize the ${}_1\Psi_1$ series to derive $(p,q)$-analogues of the Jacobi triple product and Euler's identity.
• Apply the method to generalize $q$-special functions, such as the continuous $(p,q)$-Hermite polynomial $\mathcal{H}_n(x|p,q)$, and explore nontrivial generalizations of $q$-orthogonal polynomials.

Experimental results

Research questions

• RQ1 How can $q$-identities be systematically extended to $(p,q)$-analogues using the twin-basic number $[n]_{p,q}$?
• RQ2 What is the role of the $(p,q)$-derivative in defining a consistent $(p,q)$-calculus and hypergeometric series?
• RQ3 Can $q$-results be recovered from $(p,q)$-results via simple parameter substitution rather than limiting processes?
• RQ4 What are the implications of the $(p,q)$-generalization for the representation theory of two-parameter quantum groups $U_{p,q}(gl(2))$?
• RQ5 How do $(p,q)$-generalizations of special functions, such as Hermite polynomials, differ from their $q$-counterparts?

Key findings

• The $(p,q)$-hypergeometric series ${}_r\Phi_s$ is defined via the twin-basic number $[n]_{p,q} = (p^n - q^n)/(p - q)$, generalizing the standard $q$-hypergeometric series.
• The $(p,q)$-binomial theorem and Heine transformation are derived as direct analogues of their $q$-counterparts, with $q$-results recovered by setting $p = 1$.
• The Gauss sum for ${}_2\phi_1$ and the Ramanujan sum for ${}_1\psi_1$ are extended to $(p,q)$-forms, with the latter yielding a $(p,q)$-analogue of the Jacobi triple product.
• The continuous $(p,q)$-Hermite polynomial $\mathcal{H}_n(x|p,q)$ is defined as $\sum_{k=0}^n \left[\begin{array}{c}n\\k\end{array}\right]_{p,q} e^{i(n-2k)\theta}$, which is not equivalent to a rescaling of the $q$-Hermite polynomial.
• The $(p,q)$-generalization leads to a two-parameter family of polynomials $H_n^{(\alpha,\beta)}(x|q)$, with $H_n^{(0,1)}(x|q)$ corresponding to the standard $q$-Hermite polynomial.
• The method allows $q$-results to be obtained from $(p,q)$-results via simple substitution ($p=1$), avoiding complex confluence limits used in traditional $q$-theory.