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

## TL;DR

This paper introduces a new generalized modified Bessel function, develops its mathematical properties, and applies it to derive modular transformations and generalizations of Ramanujan's formulas, linking special functions and number theory.

## Contribution

It proposes a novel generalization of the modified Bessel function, explores its properties, and demonstrates its application in deriving advanced modular transformations and generalizations of classical formulas.

## Key findings

- Derived series and integral representations of the new function
- Established a differential-difference equation and asymptotic expansions
- Obtained 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 $SL_{2}(\mathbb{Z})$. 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.