重新访问非中心$χ^2$分布的CDF。不完全超几何型函数方法
CDF of non-central $χ^2$ distribution revisited. Incomplete hypergeometric type functions approach
摘要：
非中心卡方分布$χ_ν'^2(λ),\, ν\in\mathbb{R}^+$的累积分布函数具有广义Marcum $Q$函数的积分表示。关于一些已知的结果，在这里，我们推导出一个更简单的形式的累积分布函数的$ν= 2n \in\mathbb{N}$自由度。此外，我们表示的不完全Fox-Wright函数${}_pΨ_q^{(γ)}$和广义不完全超几何函数的重要特殊情况下，如${}_1Γ_1,\, {}_2Γ_1$和${}_2γ_1$。在${}_1Γ_1$和${}_2Γ_1$之间也建立了新的身份。
英文摘要：
The cumulative distribution function of the non-central chi-square distribution $χ_ν'^2(λ),\, ν\in\mathbb{R}^+$ 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 $ν= 2n \in\mathbb{N}$ degrees of freedom. Also, we express these representations in terms of an incomplete Fox-Wright function ${}_pΨ_q^{(γ)}$ and the generalized incomplete hypergeometric functions concerning the important special cases as ${}_1Γ_1,\, {}_2Γ_1$ and ${}_2γ_1$. New identities are established between ${}_1Γ_1$ and ${}_2Γ_1$ as well.
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