2017/06/16 by Atul Dixit, Dixit, Atul, Aashita Kesarwani +5
Chemistry · Mathematics · #11M06 #33E20 (Primary) 33C10 (Secondary) #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Molecular spectroscopy and chirality #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1706.05363
openalex publication_date 2017/06/16 · openalex created_date 2022/08/19 · openalex updated_date 2026/07/28
A new generalization of the modified Bessel function of the second kind\nKz(x) is studied. Elegant series and integral representations, a\ndifferential-difference equation and asymptotic expansions are obtained for it\nthereby anticipating a rich theory that it may possess. The motivation behind\nintroducing this generalization is to have a function which gives a new pair of\nfunctions reciprocal in the Koshliakov kernel \cos \( \π z\n\)M2z(4\√ x ) - \sin \( \π z \)J2z(4\√ x\n) and which subsumes the self-reciprocal pair involving Kz(x). Its\napplication towards finding modular-type transformations of the form F(z, w,\n\α)=F(z,iw,\β), where \α\β=1, is given. As an example, we\nobtain a beautiful generalization of a famous formula of Ramanujan and Guinand\nequivalent to the functional equation of a non-holomorphic Eisenstein series on\nSL2(\ℤ). This generalization can be considered as a higher level\nanalogue of the general theta transformation formula. We then use it to\nevaluate an integral involving the Riemann \Ξ-function and consisting of a\nsum of products of two confluent hypergeometric functions.\n