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

TL;DR

This paper revisits the CDF of the non-central chi-square distribution, deriving simpler forms for specific degrees of freedom and expressing them through advanced hypergeometric functions, thus enhancing analytical tools for statistical distributions.

Contribution

It introduces simplified formulas for the non-central chi-square CDF at integer degrees of freedom and connects these to incomplete hypergeometric functions, establishing new identities.

Findings

Derived simpler CDF expressions for even degrees of freedom.

Expressed CDF in terms of incomplete Fox-Wright and hypergeometric functions.

Established new identities between hypergeometric functions.

Abstract

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.