The cumulative distribution function of the non-central chi-square distribution ${\chi }_{\nu }^{\prime 2}\left(\lambda \right),\nu \in {\mathbb{R}}^{+}$ 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 ${}_p{\Psi }_q^{\left(\gamma \right)}$ and the generalized incomplete hypergeometric functions concerning the important special cases such 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.

中文翻译：

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非中心的 CDF ${\chi }^2$分布重新审视。不完全超几何类型函数方法

非中心卡方分布的累积分布函数 ${\chi }_{\nu }^{\prime 2}\left(\lambda \right),\nu \in {\mathbb{电阻}}^{+}$ 在广义 Marcum 方面具有积分表示 $问$-功能。关于一些已知的结果，这里我们推导出一个更简单的累积分布函数形式$\nu =2n\in \mathbb{N}$自由程度。此外，我们用不完整的 Fox-Wright 函数来表达这些表示${}_{磷}{\Psi }_q^{\left(\gamma \right)}$ 以及有关重要特殊情况的广义不完全超几何函数，例如 ${}_1{\Gamma }_1,{}_2{\Gamma }_1$ 和 ${}_2{\gamma }_1$. 之间建立了新的身份${}_1{\Gamma }_1$ 和 ${}_2{\Gamma }_1$ 以及。