CDF of non-central ${\chi }^2$ distribution revisited. Incomplete hypergeometric type functions approach

Abstract

The cumulative distribution function of the non-central chi-square distribution ${\chi }_{\nu }^{\prime 2}(\lambda ),\nu \in {\mathbb{R}}^{+}$ possesses an integral representation in terms of a generalized Marcum $Q$-function. Regarding some already known results, here we derive a simpler form of the cumulative distribution function for $\nu =2n\in \mathbb{N}$ degrees of freedom. Also, we express these representations in terms of an incomplete Fox-Wright function ${}_p{\Psi }_q^{(\gamma )}$ and the generalized incomplete hypergeometric functions concerning the important special cases as ${}_1{\Gamma }_1,{}_2{\Gamma }_1$ and ${}_2{\gamma }_1$. New identities are established between ${}_1{\Gamma }_1$ and ${}_2{\Gamma }_1$ as well.