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

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 $\eta(z)$.