Finding the general term of a sequence defined by an integral between 0 and 1

Question

I am looking for the general term of the following sequence defined by $$ u_n = \int_0^1 F\left(-n,n+\frac{3}{2}; 2, r^2\right) \, \mathrm{d} r \, , \quad(n \ge 0) \, , $$ wherein $F$ is the hypergeometric function.

Using computer software algebra e.g. Maple, it is possible to know the terms for specified values of $n$ such as $u_0 = 1 $, $u_1 = 7/12$, $u_2=43/120$ etc.

It would be great if someone here could provide with an idea that helps in determining the expression of $u_n$.

Thanks

R

Answer 1

This is not an answer but just the result from a CAS.

$$u_n=\int_0^1\, _2F_1\left(-n,n+\frac{3}{2};2;r^2\right)\,dr=\frac{\pi \, _3\tilde{F}_2\left(-n,\frac{1}{2}-n,\frac{3}{2};2,\frac{3}{2}-n;1\right)}{4 \Gamma \left(n+\frac{3}{2}\right)}$$ where appears the regularized generalized hypergeometric function.

$n$ being an integer, the above can simplify to $$u_n=\frac{(-1)^{n+1}}{(2 n-1) (2 n+1)} \, _3F_2\left(\frac{3}{2},\frac{1}{2}-n,-n;2,\frac{3}{2}-n;1\right)$$ where appears the generalized hypergeometric function.