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

The cumulative distribution function of the non-central chi-square distribution $\chi_\nu'^2(\lambda),\, \nu\in\mathbb{R}^+$ 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.