We know from this answer that for $0\leq n \leq k$, $$ \int_0^1 r^n(1-r)^{k-n}\,dr = \frac{1}{(k+1)\dbinom k n}. $$
In my case, $n$ and $k$ are integers.
But what is
$$ \int_{\varepsilon}^1 r^n(1-r)^{k-n}\,dr $$
for small $0 < \varepsilon$ ?
This question comes from the comments to the accepted answer at https://math.stackexchange.com/a/903516/66307 .
Usng a CAS, the following result was obtained $$\int_{a}^1 r^n(1-r)^{k-n}\,dr=\frac{\Gamma (n+1) \Gamma (k-n+1)}{\Gamma (k+2)}-\frac{a^{n+1} \, _2F_1(n+1,n-k;n+2;a)}{n+1}$$ You can expand the hypergeometric function as a Taylor series and then get as an approximation $$\frac{a^n \left(-\frac{\Gamma (k+2) a}{n+1}+\frac{(k-n) \Gamma (k+2) a^2}{n+2}+O\left(a^3\right)\right)+\Gamma (n+1) \Gamma (k-n+1)}{\Gamma (k+2)}$$
Added later
$$\int_{a}^1 r^n(1-r)^{k-n}\,dr=\frac{1}{(k+1)\dbinom k n}-B_a(n+1,k-n+1)$$ Expanded as a series built at $a=0$, $$B_a(n+1,k-n+1)=a^{n+1} \left(\frac{1}{n+1}+\frac{(n-k) a}{n+2}+\frac{(n-k) (n-k+1) a^2}{2 (n+3)}+O\left(a^3\right)\right)$$
The incomplete beta function. That's what “this” is. For $\epsilon=0$, see beta function.
For $k-n\in\mathbb N,~$ just expand $(1-r)^{k-n}$ using the binomial theorem.
For $k-n\not\in\mathbb N,~$ and $n\in\mathbb N,~$ let $r=1-t$, in which case your second term becomes $1-r$ $=t.~$ Now expand $(1-t)^n$.
For $\{n\}=\{k-n\}=\dfrac12$, use trigonometric substitutions, where $~\{x\}~$ denotes the fractional part of x.
In the earlier question we consider a random variable $P$ uniformly distributed on $[0,1]$ and a sequence $X_1,X_2,X_3,\ldots$ that were conditionally independent given $P$, and conditional on $P$ each $X_i$ is equal to $1$ with probability $P$ and $0$ otherwise. Then we considered the random variable $K=\min\{k\in\{1,2,3,\ldots\} : X_1+\cdots+X_k=n\}$. The probability distribution of $K$ was found to be given by $\Pr(K=k)=n/(k(k+1))$ for $k=n,n+1,n+2,\ldots.$ Finally, it was found that $\mathbb E(K)=\infty$.
The question was posed in comments, whether $\mathbb E(K)$ would be finite if $P$ had been distributed uniformly on $[\varepsilon,1]$ for some $\varepsilon>0$.
I will answer that here. That doesn't fully answer the question as posed above, but perhaps that is what is of interest.
So suppose $P$ is distributed uniformly on $[\varepsilon,1]$ and $0<\varepsilon\le 1$. Then $$ \mathbb E(K) = \mathbb E(\mathbb E(K\mid P)) \overset{(1)}{\le} \mathbb E(K\mid P=\varepsilon) \overset{(2)} = \frac n \varepsilon. $$ The equality $(2)$ is well known. I will leave the proofs of $(1)$ and $(2)$ as an exercise for now. Maybe more later$\ldots\ldots$.
