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.