[Paper Review] Asymptotic formulae of two divergent bilateral basic hypergeometric series

Hironori Mori, Takeshi Morita|arXiv (Cornell University)|Nov 15, 2016

Mathematical functions and polynomials9 references3 citations

TL;DR

This paper derives new asymptotic formulae for two divergent bilateral basic hypergeometric series, ${}_1\psi_1(0;b;q,x)$ and ${}_1\psi_0(a;--;q,x)$, using the $q$-Borel-Laplace transformation to address connection problems in $q$-difference equations. The key contribution is establishing well-defined $q\to 1-0$ limits that recover classical confluent hypergeometric-type functions, providing a bridge between $q$-special functions and their classical counterparts.

ABSTRACT

We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1ψ_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection formulae of $q$-difference equation for ${}_1ψ_1 ( a; b; q, z )$. Moreover, we show that our formulae have the $q o 1 - 0$ limit.

Motivation & Objective

• To derive asymptotic formulae for two divergent bilateral basic hypergeometric series, ${}_1\psi_1(0;b;q,x)$ and ${}_1\psi_0(a;--;q,x)$, using $q$-Borel-Laplace resummation.
• To address the connection problem for $q$-difference equations by relating divergent bilateral series to convergent unilateral series.
• To establish the $q\to 1-0$ limit of the derived formulae, recovering known classical special functions.
• To provide a foundation for constructing full connection formulae in $q$-difference equation theory.

Proposed method

• The $q$-Borel-Laplace transformation is applied to resum divergent formal power series solutions of $q$-difference equations.
• The $q$-Borel transformation of the first kind is defined as $\mathcal{B}_q^+ f(\xi) = \sum_{n\geq 0} a_n q^{n(n-1)/2} \xi^n$ for a formal series $f(x) = \sum_{n\geq 0} a_n x^n$.
• The $q$-Laplace transformation of the first kind is used as $\mathcal{L}_{q,\lambda}^+ \varphi(x) = \sum_{n\in\mathbb{Z}} \frac{\varphi(\lambda q^n)}{\theta_q(\lambda q^n / x)}$, with $\theta_q$ being the $q$-theta function.
• The method constructs resummation maps from divergent bilateral series to convergent unilateral series via analytic continuation and $q$-periodic coefficients.
• The $q$-Borel-Laplace framework allows handling formal solutions with divergent power series components in $q$-difference equations.
• Rescaling $x \mapsto (1-q)x$ or $x \mapsto x/(1-q)$ enables the extraction of classical limits as $q \to 1-0$.

Experimental results

Research questions

• RQ1 How can divergent bilateral basic hypergeometric series be resummed using $q$-Borel-Laplace transformations to yield meaningful asymptotic formulae?
• RQ2 What is the connection between divergent bilateral series and convergent unilateral series in the context of $q$-difference equations?
• RQ3 What classical special functions emerge in the $q\to 1-0$ limit of the derived $q$-asymptotic formulae?
• RQ4 Can the $q$-Borel-Laplace method provide a systematic approach to constructing full connection formulae for $q$-difference equations?
• RQ5 How do the $q$-resummation results relate to known classical confluent hypergeometric functions?

Key findings

• The asymptotic formula for ${}_1\psi_1(0;b;q,x)$ is expressed via $q$-Borel-Laplace transformation as $\widetilde{\psi}_\lambda^\mathrm{A}(x) = \frac{(q;q)_\infty}{(q^\beta;q)_\infty} \frac{\theta_q(\lambda)}{\theta_q(q^{1-\beta}\lambda)} \frac{\theta_q\left(\frac{q^{1-\beta}}{(1-q)}\frac{\lambda}{x}\right)}{\theta_q\left(\frac{1}{(1-q)}\frac{\lambda}{x}\right)} \sum_{n\geq 0} \frac{(1-q)^n}{(q;q)_n} x^n$, with a well-defined $q\to 1-0$ limit.
• The $q\to 1-0$ limit of $\widetilde{\psi}_\lambda^\mathrm{A}((1-q)x)$ yields $\Gamma(\beta) x^{1-\beta} e^x$, recovering a classical confluent hypergeometric-type function.
• The asymptotic formula for ${}_1\psi_0(a;--;q,x)$ is given by $\widetilde{\psi}_\lambda^\mathrm{B}(x) = \frac{(q;q)_\infty}{(q^{1-\alpha};q)_\infty} \frac{\theta_q(q^\alpha\lambda)}{\theta_q(\lambda)} \frac{\theta_q\left(\frac{q^{\alpha+1}}{(1-q)}\frac{x}{\lambda}\right)}{\theta_q\left(\frac{q}{(1-q)}\frac{x}{\lambda}\right)} \sum_{n\geq 0} \frac{(1-q)^n}{(q;q)_n} \left(\frac{1}{q^\alpha x}\right)^n$, with a consistent $q$-resummation.
• The $q\to 1-0$ limit of $\widetilde{\psi}_\lambda^\mathrm{B}(x/(1-q))$ yields $\Gamma(1-\alpha) x^{-\alpha} e^{1/x}$, matching a known classical solution of the confluent hypergeometric equation.
• The $q$-Borel-Laplace transformation successfully resolves the connection problem for divergent bilateral series by mapping them to convergent unilateral series with $q$-periodic coefficients.
• The classical limits confirm consistency with known special functions, validating the $q$-resummation framework as a bridge to classical analysis.