<s>[INST] Equality involving Appell hypergeometric function After some algebra, Wolfram online integrator gave me the following:
$$\tag{1} \int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t = c\ \cdot t^{3/2}\ \operatorname{F}_1 \left( \frac{3}{2}; -\frac{1}{2}, 2-N; \frac{5}{2}; \frac{t}{2}, \frac{t}{1-a}\right)+C$$
where:


*

*$\operatorname{F}_1$ is the Appell hypergeometric function of two variables defined by the expansion:
$$\operatorname{F}_1(\alpha; \beta, \beta^\prime ; \gamma; x,y):= \sum_{m,n=0}^\infty \frac{(\alpha)_{m+n}\ (\beta)_m\ (\beta^\prime)_n}{m!\ n!\ (\gamma)_{m+n}}\ x^m\ y^n$$
($(a_k):=\frac{\Gamma (a+k)}{\Gamma(a)}$ is the Pochhammer symbol), which converges in the region $|x|,|y|<1$;

*$N\geq 3$ an integer, $a\in ]0,1[$ and $t\in [0,1-a]$ (so the RHside makes sense);

*$c=c(a,N)$ is a known constant and $C$ is an arbitrary constant (coming from indefinite integration).


My question is:

How can I get (1) without using any software?

I suppose a series expansion of the integrand and term by term integration should be used, but I could not figure out how to do explicit computations.
Neither I succeeded in simplifying the derivative of the LHside to get the integrand...
Any hint will be appreciated.
 [/INST] I found also the following term-by-term integration solution.
One has:
$$(1-a-t)^{N-2}\ \sqrt{2t-t^2} = \sqrt{2}\ (1-a)^{N-2}\ \sqrt{t}\ \left( 1-\frac{t}{1-a}\right)^{N-2}\ \sqrt{1-\frac{t}{2}}$$
and by the binomial theorem:
$$\begin{split}
\left( 1-\frac{t}{1-a}\right)^{N-2} &= \sum_{k=0}^{N-2} (-1)^k\ \binom{N-2}{k}\ \left(\frac{t}{1-a}\right)^k\\
&= \sum_{k=0}^{N-2} \frac{(2-N)_k}{k!}\ \left(\frac{t}{1-a}\right)^k\\
&= \sum_{k=0}^\infty \frac{(2-N)_k}{k!}\ \left(\frac{t}{1-a}\right)^k\\
\sqrt{1-\frac{t}{2}} &= \sum_{k=0}^\infty (-1)^k\ \binom{1/2}{k}\ \left( \frac{t}{2}\right)^k\\
&= \sum_{k=0}^\infty \frac{(-1/2)_k}{k!}\ \left( \frac{t}{2}\right)^k\; ;
\end{split}$$
now, since both series in the rightmost sides converge absolutely, Mertens theorem applies and one can take the Cauchy product:
$$\begin{split} \left( 1-\frac{t}{1-a}\right)^{N-2}\ \sqrt{1-\frac{t}{2}} &= \left( \sum_{k=0}^\infty \frac{(2-N)_k}{k!}\ \left(\frac{t}{1-a}\right)^k\right)*\left( \sum_{k=0}^\infty \frac{(-1/2)_k}{k!}\ \left( \frac{t}{2}\right)^k\right)\\
&= \sum_{k=0}^\infty \sum_{h=0}^k \frac{(2-N)_h}{h!}\ \left(\frac{t}{1-a}\right)^h\ \frac{(-1/2)_{k-h}}{(k-h)!}\ \left( \frac{t}{2}\right)^{k-h}\\
&= \sum_{k=0}^\infty \sum_{h=0}^k \frac{(2-N)_h\ (-1/2)_{k-h}}{h!\ (k-h)!}\ \frac{1}{2^h\ (1-a)^{k-h}}\ t^k
\end{split}$$
with the rigthmost side converging absolutely and uniformly on compact subsets.
Therefore one gets:
$$(1-a-t)^{N-2}\ \sqrt{2t-t^2} = \sqrt{2}\ (1-a)^{N-2}\ \sum_{k=0}^\infty \sum_{h=0}^k \frac{(2-N)_h\ (-1/2)_{k-h}}{h!\ (k-h)!}\ \frac{1}{2^h\ (1-a)^{k-h}}\ t^{k+1/2}$$
and a rearrangement of the RHside yields:
$$(1-a-t)^{N-2}\ \sqrt{2t-t^2} = \sqrt{2}\ (1-a)^{N-2}\ \sum_{m,n=0}^\infty \frac{(2-N)_m\ (-1/2)_n}{m!\ n!}\ \frac{1}{2^m\ (1-a)^n}\ t^{m+n+1/2}\; ;$$
thus term-by-term integration shows that:
$$\begin{split}
\int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t &= \sqrt{2}\ (1-a)^{N-2}\ \sum_{m,n=0}^\infty \frac{(2-N)_m\ (-1/2)_n}{m!\ n!}\ \frac{1}{2^m\ (1-a)^n}\ \int t^{m+n+1/2}\ \text{d} t\\
&= \sqrt{2}\ (1-a)^{N-2}\ \sum_{m,n=0}^\infty \frac{(2-N)_m\ (-1/2)_n}{m!\ n!\ (3/2 +m+n)}\ \frac{1}{2^m\ (1-a)^n}\ t^{m+n+3/2}\ + C\\
&= \frac{2\sqrt{2}}{3}\ (1-a)^{N-2}\ \sum_{m,n=0}^\infty \frac{(3/2)_{m+n}\ (2-N)_m\ (-1/2)_n}{m!\ n!\ (5/2)_{m+n}}\ \frac{1}{2^m\ (1-a)^n}\ t^{m+n+3/2}\\
&\phantom{=} +C\\
&= \frac{2\sqrt{2}}{3}\ (1-a)^{N-2}\ t^{3/2}\ F_1\left( \frac{3}{2}; -\frac{1}{2}, 2-N; \frac{5}{2}; \frac{t}{2}, \frac{t}{1-a}\right) +C
\end{split}$$
because:
$$\frac{3}{2}\ \frac{1}{m+n+3/2} = \frac{(3/2)_{m+n}}{(5/2)_{m+n}}\; .$$
</s>