Is there a standard integral that can be used to evaluate $\int_{0}^{\infty} e^{-t}t^{a-1}\left[ L_{n}^{a+1} (t)\right]^{2}dt$ In order to estimate the normalized wave function of a particular potential, the condition for normalizing the wave function requires that  $$N^2\int_{0}^{\infty} e^{-t}t^{a-1}\left[ L_{n}^{a+1} (t)\right]^{2}dt=1$$ with $a+1>0$. the tables of integral, series and product by Gradshteyn and Ryzhik provide me with
$$\int_{0}^{\infty} e^{-t}t^{a}\left[ L_{n}^{a} (t)\right]^{2}dt= \frac{\Gamma(a+n+1)}{n!}$$
which look similar but not the same as my integral. Is there a standard integral that can evaluate my integral?
 A: Mathematica gives a closed form answer for definite $n$, which is summarized by the formula
$$\int_{0}^{\infty} e^{-t}t^{a-1}\left[ L_{n}^{a+1} (t)\right]^{2}dt=\frac{\Gamma (a)}{n!}(a+2 n+2)  
\prod _{p=3}^{n+1} (a+p),\;\;a>0,$$
for $n\geq 2$, while the integral equals $\Gamma(a)$ for $n=0$ and $(4+a)\Gamma(a)$ for $n=1$.

This is a special case of a result due to Rassias and Srivastava [source]:


Take $\lambda=a-1$, $m=n$, $\alpha=\beta=a+1$, and use that
$$\sum _{k=0}^n \binom{-2}{n-k}^2\binom{a+k-1}{k} =\sum _{k=0}^n (n-k+1)^2\binom{a+k-1}{k} =\frac{(a+2 n+2) \Gamma (a+n+2)}{n!\,\Gamma (a+3)}.$$

Further generalizations, involving the product of more than two Laguerre polynomials, are in Some integrals of the products of Laguerre polynomials (2001).
A: Here one can try to use the recurrence in $(j,k)$ for the more general, "bilinearized" integrals
$$I_{j,k,m}:=\int_0^\infty e^{-t}t^{a-1+m} L^{a+1}_j(t)L^{a+1}_k(t)\,dt,$$
based in a straightforward manner on the recurrence
$$L^{a+1}_{n+1}(t)=\frac{(2n+2+a-t)L^{a+1}_n(t)-(n+a+1)L^{a+1}_{n-1}(t)}{n+1}$$
in $n$ for the generalized Laguerre polynomials, with $L^{a+1}_0(t)=1$ and $L^{a+1}_1(t)=2+a-t$.
If one can now come up with a right conjecture on the expression for the $I_{j,k,m}$'s, then we could try to prove the conjecture by induction on $j$ and $k$. Perhaps, one can try to "bilinearize" the expression in Carlo Beenakker's answer to obtain the conjecture for the expression for the $I_{j,k,m}$'s.
