**Question:** Is there a straightforward proof of the following relationship between the *(lower, non-regularized)* incomplete beta function $\mathcal{B}(x; a ,b)$ and the *(upper, non-regularized)* incomplete gamma function $\Gamma(s,t)$:

$$\mathcal{B}\left(\frac{\alpha}{\alpha + \mu} ; \alpha, k + 1 \right) \sim \frac{\Gamma(\alpha) \Gamma(k+1, \mu)}{\Gamma(\alpha + k + 1)} \quad \text{as } \alpha \to \infty ,$$
where $f(\alpha) \sim g(\alpha)$ as $\alpha \to \infty$ means $\displaystyle\lim\_{\alpha \to \infty} \frac{f(\alpha)}{g(\alpha)}=1$?

Notice in particular that this would be analogous to the relationship $\mathcal{B}(\alpha, k+1) = \frac{\Gamma(\alpha) \Gamma(k+1)}{\Gamma(\alpha + k + 1)}$ that one has for the *(complete)* beta function and the *(complete)* gamma function. I believe I was able to show that the above asymptotic relationship would be equivalent to showing the convergence of the negative binomial CDF to the Poisson CDF.

**Background+definitions:** The negative binomial distribution with "number of successes" $\alpha > 0$, $\alpha \in \mathbb{R}$, expectation/mean $\mu > 0$, $\mu \in \mathbb{R}$, and "probability of success" $\frac{\alpha}{\alpha + \mu}$ has the following CDF for $k \ge 0$, $ k \in \mathbb{N}$:
$$\frac{\mathcal{B}\left(\frac{\alpha}{\alpha + \mu} ; \alpha, k + 1 \right)}{\mathcal{B}(\alpha, k+1)}, \quad \text{where} \quad \mathcal{B}(x; a, b) := \int\_{0}^x t^{a-1} (1-t)^{b-1} \mathrm{d}t . $$
The Poisson distribution with expectation/mean $\mu > 0$, $\mu \in \mathbb{R}$ has the following CDF for $k \ge 0$, $k \in \mathbb{N}$:
$$\frac{\Gamma(k+1, \mu)}{\Gamma(k+1)}, \quad \text{where} \quad \Gamma(s,t) := \int\_{t}^{\infty} e^{-u} u^{s-1} \mathrm{d}u . $$

**Note:** One can also show the convergence in distribution using either the PMF or MGF, and that it is easier than using the CDF. However, I want to understand the proof specifically using the CDFs. (As a sanity check that the formulas given for the CDFs are at least plausible.)

Here are some related, but distinct, questions on this website: [(1)](https://stats.stackexchange.com/questions/236728/negative-binomial-mgf-converges-to-poisson-mgf) [(2)](https://stats.stackexchange.com/questions/203972/why-is-the-negative-binomial-typically-preferred-over-the-poisson) [(3)](https://stats.stackexchange.com/questions/178101/the-relationship-between-the-negative-binomial-distribution-and-poisson-distribu) [(4)](https://stats.stackexchange.com/questions/496472/proof-of-poisson-distribution-as-a-limiting-case-of-the-negative-binomial-distri) [(5)](https://stats.stackexchange.com/questions/118220/poisson-as-a-limiting-case-of-negative-binomial) [(6)](https://stats.stackexchange.com/questions/221828/relationship-between-poisson-binomial-negative-binomial-distributions-and-norm) [(7)](https://stats.stackexchange.com/questions/468331/incomplete-beta-function-increasing-in-alpha)

As a *community wiki* answer I typed up an outline of what I tried so far, which either does not work or which I was not able to complete. Either way the calculations involved are not straightforward.