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

A new generalization of the modified Bessel function of the second kind Kz(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)M2z(4x)−sin⁡(πz)J2z(4x) and which subsumes the self-reciprocal pair involving Kz(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 SL2(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.