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A new generalization of the modified Bessel function of the second kind$K_{z}(x)$ is studied. Elegant series and integral representations, adifferential-difference equation and asymptotic expansions are obtained for itthereby anticipating a rich theory that it may possess. The motivation behindintroducing this generalization is to have a function which gives a new pair offunctions 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 $K_{z}(x)$. Itsapplication 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, weobtain a beautiful generalization of a famous formula of Ramanujan and Guinandequivalent to the functional equation of a non-holomorphic Eisenstein series on$SL_{2}(\mathbb{Z})$. This generalization can be considered as a higher levelanalogue of the general theta transformation formula. We then use it toevaluate an integral involving the Riemann $\Xi$-function and consisting of asum of products of two confluent hypergeometric functions.

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