Atul Dixit (Indian Institute of Technology Gandhinagar) A generalized modified Bessel function and a higher level analogue of the general 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, 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 selfreciprocal pair involving $K_{z}(x)$. Its application towards finding modulartype 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 nonholomorphic 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. 
