TY - JOUR

T1 - Sums of squares and products of Bessel functions

AU - Berndt, Bruce C.

AU - Dixit, Atul

AU - Kim, Sun

AU - Zaharescu, Alexandru

N1 - Funding Information:
The second author's research was partially supported by the SERB-DST grant ECR/2015/000070 . The authors sincerely thank Dmitry Vasilenko, Vice-Rector for International Relations at St. Petersburg State University of Economics, for sending them a list of the 12 publications of A.I. Popov. They also thank Arindam Roy for interesting discussions, and also for sending them a copy of [35] , Anton Lukyanenko for translating for them a section of that paper, and Jeremy Rouse for discussions on bounds for r k ( n ) .
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/11/7

Y1 - 2018/11/7

N2 - Let rk(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 Voronoï summation formula for rk(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 rk(n) and a product of two Bessel functions, and a series involving rk(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.

AB - Let rk(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 Voronoï summation formula for rk(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 rk(n) and a product of two Bessel functions, and a series involving rk(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.

KW - Analytic continuation

KW - Bessel functions

KW - Sums of squares

KW - Voronoï summation formula

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U2 - 10.1016/j.aim.2018.09.001

DO - 10.1016/j.aim.2018.09.001

M3 - Article

AN - SCOPUS:85053047863

VL - 338

SP - 305

EP - 338

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -