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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 anelegant 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 bewritten in the form of an infinite series of Humbert functions. As anapplication of this result, we generalize the transformation formula for thelogarithm of the Dedekind eta function $\eta(z)$.

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