Dixit, AtulAtulDixitKesarwani, AashitaAashitaKesarwaniMoll, Victor H.Victor H.Moll2025-08-302025-08-302018-03-0110.1016/j.jmaa.2017.10.0502-s2.0-85032367617https://d8.irins.org/handle/IITG2025/22318A new generalization of the modified Bessel function of the second kind K<inf>z</inf>(x) is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel cos⁡(πz)M<inf>2z</inf>(4x)−sin⁡(πz)J<inf>2z</inf>(4x) and which subsumes the self-reciprocal pair involving K<inf>z</inf>(x). Its application towards finding modular-type transformations of the form F(z,w,α)=F(z,iw,β), where αβ=1, is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on SL<inf>2</inf>(Z). This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann Ξ-function and consisting of a sum of products of two confluent hypergeometric functions.falseAsymptotic expansion | Basset's formula | Bessel functions | Ramanujan–Guinand formula | Riemann Ξ-function | Theta transformation formulaArticlehttps://arxiv.org/pdf/1706.0536310960813385-4181 March 20188arJournal8WOS:000418310900023