Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On a Neumann-type series for modified Bessel functions of the first kind
HTML articles powered by AMS MathViewer

by L. Deleaval and N. Demni PDF
Proc. Amer. Math. Soc. 146 (2018), 2149-2161 Request permission

Abstract:

In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions, while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie et al.’s formula expressing this series as a $\Phi _2$-Horn confluent hypergeometric function. In the course of the proof, we shed light on the occurrence of multiple angles in their formula through elementary symmetric functions and get a new representation of Gegenbauer polynomials.
References
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 33C45, 33C52, 33C65
  • Retrieve articles in all journals with MSC (2010): 33C45, 33C52, 33C65
Additional Information
  • L. Deleaval
  • Affiliation: Laboratoire d’Analyse et de Mathématiques appliquées , Université Paris-Est Marne-la-Vallée , 77454 Marne-la-Vallé Cedex 2, France
  • MR Author ID: 882358
  • Email: luc.deleaval@u-pem.fr
  • N. Demni
  • Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
  • MR Author ID: 816493
  • Email: nizar.demni@univ-rennes1.fr
  • Received by editor(s): March 31, 2017
  • Received by editor(s) in revised form: July 22, 2017, and July 29, 2017
  • Published electronically: December 28, 2017
  • Communicated by: Yuan Xu
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 2149-2161
  • MSC (2010): Primary 33C45, 33C52, 33C65
  • DOI: https://doi.org/10.1090/proc/13914
  • MathSciNet review: 3767365