The generalized modified Bessel function $K_{z,w}(x)$ at $z=1/2$ and Humbert functions

TL;DR

This paper explores a generalization of the modified Bessel function, expressing a specific case as a series of Humbert functions and applying it to extend a transformation formula for the Dedekind eta function.

Contribution

It demonstrates that the generalized Bessel function at a specific parameter can be expressed via Humbert functions and applies this to generalize a classical eta function transformation.

Findings

$K_{1/2,w}(x)$ can be written as an infinite series of Humbert functions.

The transformation formula for $ ext{log} \, ext{eta}(z)$ is generalized.

Provides new insights into the structure of generalized Bessel functions.

Abstract

Recently Dixit, Kesarwani, and Moll introduced a generalization $K_{z,w}(x)$ of the modified Bessel function $K_{z}(x)$ and showed that it satisfies an elegant theory similar to $K_{z}(x)$. In this paper, we show that while $K_{\frac{1}{2}}(x)$ is an elementary function, $K_{\frac{1}{2},w}(x)$ can be written in the form of an infinite series of Humbert functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta function $\eta(z)$.