The hypergeometric function, the confluent hypergeometric function and WKB solutions
type: Item
instance of: scholarly article
title: The hypergeometric function, the confluent hypergeometric function and WKB solutions
zbMATH Open document ID: 1496.33002
author: Takashi Aoki
author: Toshinori Takahashi
author: Mika Tanda
published in: Journal of the Mathematical Society of Japan
publication date: 2021-12-10T00:00:00Z
full work available at URL: 84528452.full
review text: Take the Gauss hypergeometric equation and instead of parameters \(a,b,c\) let us write \(a=\alpha_0+\alpha \eta, b=\beta_0+\beta\eta, c=\gamma_0+\gamma\eta\) and instead of the unknown function \(w\) let us take \(w=x^{-c/2}(1-x)^{(-1/2)(a+b-c+1)}\psi\). Then the equation is written in the form \[ (-\frac{d^2}{dx^2}+\eta^2Q)\psi=0,\quad Q=\sum_{j=0}^N \eta^{-j}Q_j(x). \] To such an equation the WKB analysis can be applied. It gives formal solutions that using the Borel summation can be transformed to analytic solutions. In the paper under review a relation between these solutions and the classical solutions of the Gauss equation is established. Also analogous results for the equation for the function \(F_{1,1}\) are obtained.
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 33C15
Mathematics Subject Classification ID: 34M40
zbMATH DE Number: 7442181
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: confluent hypergeometric differential equation
zbMATH Keywords: hypergeometric differential equation
zbMATH Keywords: Voros coefficient
zbMATH Keywords: WKB solution
reviewed by: Dmitry Artamonov
MaRDI profile type: Publication
OpenAlex ID: W3192609840
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DOI: 10.2969/JMSJ/84528452
description: scientific article; zbMATH DE number 7442181