Universität zu Köln

Berndt, Bruce C., Dixit, Atul, Kim, Sun and Zaharescu, Alexandru (2018). Sums of squares and products of Bessel functions. Adv. Math., 338. S. 305 - 339. SAN DIEGO: ACADEMIC PRESS INC ELSEVIER SCIENCE. ISSN 1090-2082

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Abstract

Let r(k) (n) denote the number of representations of the positive integer n as the sum of k squares. We rigorously prove for the first time a Voronoi summation formula for r(k)(n), k >= 2, proved incorrectly by A.I. Popov and later rediscovered by A.P. Guinand, but without proof and without conditions on the functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of r(k) (n) and a product of two Bessel functions, and a series involving r(k) (n) and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G.H. Hardy, and of A.L. Dixon and W.L. Ferrar, as well as of a classical result of A.I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar. (C) 2018 Elsevier Inc. All rights reserved.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Berndt, Bruce C. UNSPECIFIED UNSPECIFIED UNSPECIFIED Dixit, Atul UNSPECIFIED UNSPECIFIED UNSPECIFIED Kim, Sun UNSPECIFIED UNSPECIFIED UNSPECIFIED Zaharescu, Alexandru UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-165912
DOI:	10.1016/j.aim.2018.09.001
Journal or Publication Title:	Adv. Math.
Volume:	338
Page Range:	S. 305 - 339
Date:	2018
Publisher:	ACADEMIC PRESS INC ELSEVIER SCIENCE
Place of Publication:	SAN DIEGO
ISSN:	1090-2082
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language IDENTITIES INVOLVING COEFFICIENTS; VALUES Multiple languages Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/16591