A generalized modified Bessel function and a higher level analogue of the theta transformation formula

Abstract

A new generalization of the modified Bessel function of the second kind $K_z(x)$ 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 $\cos \left(\pi z\right)M_{2z}(4\sqrt{x})-\sin \left(\pi z\right)J_{2z}(4\sqrt{x})$ and which subsumes the self-reciprocal pair involving $K_z(x)$. Its application towards finding modular-type transformations of the form $F(z,w,\alpha )=F(z,iw,\beta )$, where $\alpha \beta =1$, 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 $S{L}_2(\mathbb{Z})$. 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 $\Xi$-function and consisting of a sum of products of two confluent hypergeometric functions.

Appendix by Nico M. Temme; 32 pages, submitted for publication; in the new version, some minor typos and errors are corrected