Expansions in series of products of eigenfunctions
type: Item
instance of: scholarly article
title: Expansions in series of products of eigenfunctions
zbMATH Open document ID: 0848.34067
DOI: 10.1007/BF03322216
published in: Results in Mathematics
publication date: 1996-10-24T00:00:00Z
review text: Consider \(m\)-fold products \(\eta = \eta_1 \dots \eta_m\) of special functions. Each of the factors satisfies a differential equation (of order \(\geq 1\) \([\geq 2\), in general]). A general method is introduced so that this product is the first component of a solution of a first order system of differential equations (Theorem 2.1). The construction of the system is explicitly given, but too lengthy to be reproduced here. If each of the functions \(\eta_j\) is a solution of an eigenvalue problem (with common eigenvalue parameter for all factors), then the system for the products can also be written as an eigenvalue problem, and the question of eigenfunction expansions for the product arises. A particular example of a product of Bessel functions \(J_{\nu_1 + {k \over 2}} (az)\) and Whittacker functions \(M_{\kappa, \nu_2 + {k \over 2}} (bz)\) is considered, where \(k \in \mathbb{Z}\). It is shown that holomorphic functions \(f\) on the Riemann surface of the logarithm satisfying the Floquet condition \(f(ze^{2 \pi i}) = e^{\nu_1 + \nu_2 + {1 \over 2}} f(z)\) can be expanded into series of \(J_{\nu_r + {k \over 2}} (az) M_{\kappa, \nu_s+ {k\over 2}} (bz)\), \((r,s) \in \{(1,2), (2,1)\}\), \(k \in \mathbb{Z}\). It is expected that this will become a standard method to obtain expansions into products of special functions.
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 33C15
zbMATH DE Number: 872710
zbMATH Keywords: special functions
zbMATH Keywords: first order system of differential equations
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: eigenfunction expansions
zbMATH Keywords: Bessel functions
zbMATH Keywords: Whittacker functions
zbMATH Keywords: Floquet condition
author: J. Saurer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322216
OpenAlex ID: W2073380721
description: scientific article; zbMATH DE number 872710