维基百科与Wolfram MathWorld关于Beta二项分布CDF的表述是否有误？

Hey folks, let's dive into a tricky point about the Beta-binomial distribution's cumulative distribution function (CDF) as presented on Wikipedia and MathWorld.

First, here's the expression they list:

$$1- \tfrac{\mathrm{B}(\beta+n-k-1,\alpha+k+1)_3F_2(\boldsymbol{a},\boldsymbol{b};k)} {\mathrm{B}(\alpha,\beta)\mathrm{B}(n-k,k+2) (n+1)}$$

The $_3F_2(\boldsymbol{a},\boldsymbol{b};k)$ term here is a generalized hypergeometric function defined as:
$$_3F_2(1,! \alpha! +! k!+ ! 1,k! -! n! +! 1;k! +! 2,k! +! 2! -! \beta! -! n;1)!$$

The big issue here? This formulation is highly suspect. The generalized hypergeometric function $_3F_2(\cdot,\cdot,\cdot;\cdot,\cdot;1)$ included has singularities—meaning under certain parameter combinations, it's either undefined or diverges completely. That makes the entire CDF expression unreliable for practical calculations.

If you need a robust way to compute the Beta-binomial CDF, stick to the direct summation form instead. It avoids the singularity problem entirely and is straightforward to work with:
$$P(X \leq k) = \sum_{i=0}^k \binom{n}{i} \frac{\mathrm{B}(\alpha+i, \beta+n-i)}{\mathrm{B}(\alpha,\beta)}$$

内容的提问来源于stack exchange，提问作者wlad