It's been several days that I'm confronted to this integral, without much success in its resolution.
To give you more details, in my case:
- $n$ is an integer $>1$
- $m=n-2$
- $p,q \in \{n-1, n\}$
- $x \in [1/2, 1]$
- $\theta \in (-\infty, 0]$
The integral becomes:
$$\int_{0}^{x}u^{p-1}(1-u)^{q-1}e^{-\theta u}L_n^{(n-2)}(\theta u)\mathrm du$$
I am aware of a close form for $x=1$ (from $\textit{Table of integrals, series and products}$, Gradshteyn and Ryzhik, eq (7.415)), but I can't find any for an arbitrary $x$.
I have tried several approaches, such as transforming the Laguerre polynomial into hypergeometric function, or developing $(1-u)^{q-1}$ with the Newton formula to recombine later, but non of them were successful.
Could someone help me with finding wether or not there is a close form formula for this integral, and give me some insight in the case it does?
Thank you in advance.