Generalized Neumann and Kapteyn expansions
type: Item
instance of: scholarly article
title: Generalized Neumann and Kapteyn expansions
zbMATH Open document ID: 0863.33003
DOI: 10.1155/S1048953395000384
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 1996-02-26T00:00:00Z
full work available at URL: https://eudml.org/doc/47644
review text: The author studies the function \[ \begin{aligned} _nX_\nu \bigl((a); (b); z\bigr) = & \left( {z \over 2} \right)^\nu \Gamma \left[ \begin{matrix} a_1+ \textstyle {1\over 2} \nu, \dots, a_n + \textstyle {1\over 2} \nu \\ b_1 + \textstyle {1\over 2} \nu, \dots, b_n+ \textstyle {1\over 2} \nu,1 + \nu \end{matrix} \right] \\ & _nF_{n+1} \left.\left[\begin{matrix} a_1 + \textstyle {1\over 2} \nu, \dots, a_n + \textstyle {1\over 2} \nu \\ b_1+ \textstyle {1\over 2} \nu, \dots, b_n + \textstyle {1\over 2} \nu, \;1+ \nu \end{matrix} \right|- \textstyle {1\over 4} z^2 \right]; \end{aligned} \] it reduces to \(J_\nu(z)\) for \(n=0\). A generating function and some recurrence formulas are found. Moreover, an expansion of Neumann type for \(\left({z\over 2}\right)^\nu\) is established, as well as the expansion \[ \left( {kz \over 2} \right)^{\mu-\nu} {_gX_\nu} \bigl((a); (b); kz\bigr) = \sum^\infty_{n=0} A(n)_qX_{\mu + 2n} \bigl((a'); (b'); z\bigr); \] the primed parameters are free; the coefficient \(A(n)\) is a gamma fraction multiplied by a \(_{g+q+2} F_{g+q+1} [k^2]\). Finally, an expansion of Kapteyn type for \(\left({z\over 2} \right)^\nu\) is established.
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 847759
zbMATH Keywords: Neumann series
zbMATH Keywords: Kapteyn series
author: H. Exton
reviewed by: Per W. Karlsson
MaRDI profile type: Publication
OpenAlex ID: W1972806307
Recommended article: Q5531782
Recommended article: Q5577739
Recommended article: Further results on generalized Kapteyn-type expansions
Recommended article: Generalizations of the Neumann system
Recommended article: Q4735403
Recommended article: Convergence of generalized Kapteyn expansion
Recommended article: Q3592915
description: scientific article; zbMATH DE number 847759