Superimposing theta structure on a generalized modular relation

Show simple item record Dixit, Atul Kumar, Rahul
dc.coverage.spatial Singapore 2012-09-26T07:22:32Z 2012-09-26T07:22:32Z 2021-09
dc.identifier.citation Dixit, Atul and Kumar, Rahul, "Superimposing theta structure on a generalized modular relation", Research in the Mathematical Sciences, DOI: 10.1007/s40687-021-00277-0, vol. 8, no. 3, Sep. 2021. en_US
dc.identifier.issn 2522-0144
dc.identifier.issn 2197-9847
dc.description.abstract A generalized modular relation of the form F(z,w,?)=F(z,iw,?), where ??=1 and i=?1????, is obtained in the course of evaluating an integral involving the Riemann ?-function. This modular relation involves a surprising new generalization of the Hurwitz zeta function ?(s,a), which we denote by ?w(s,a). We show that ?w(s,a) satisfies a beautiful theory generalizing that of ?(s,a). In particular, it is shown that for 0<a<1 and w?C, ?w(s,a) can be analytically continued to Re(s)>?1 except for a simple pole at s=1. The theories of functions reciprocal in a kernel involving a combination of Bessel functions and of a new generalized modified Bessel function 1Kz,w(x), which are also essential to obtain the generalized modular relation, are developed.
dc.description.statementofresponsibility by Atul Dixit and Rahul Kumar
dc.format.extent vol. 8, no. 3
dc.language.iso en_US en_US
dc.publisher Springer Nature en_US
dc.subject Riemann zeta function en_US
dc.subject Hurwitz zeta function en_US
dc.subject Bessel functions en_US
dc.subject Theta transformation formula en_US
dc.subject Hermite�s formula en_US
dc.subject Modular relation en_US
dc.title Superimposing theta structure on a generalized modular relation en_US
dc.type Article en_US
dc.relation.journal Research in the Mathematical Sciences

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