Department of

May 2017 June 2017 July 2017 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 1 2 3 1 7 8 9 10 11 12 13 4 5 6 7 8 9 10 2 3 4 5 6 7 8 14 15 16 17 18 19 20 11 12 13 14 15 16 17 9 10 11 12 13 14 15 21 22 23 24 25 26 27 18 19 20 21 22 23 24 16 17 18 19 20 21 22 28 29 30 31 25 26 27 28 29 30 23 24 25 26 27 28 29 30 31

Tuesday, June 20, 2017

**Abstract:** A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, 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 $SL_{2}(\mathbb{Z})$. 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. This is joint work with Aashita Kesarwani and Victor H. Moll, with an appendix by Nico M. Temme.