Question

Let $P_{k}^{\alpha}(q)$ be generalized Laguerre polynomials defined as

$$P_{k}^{\alpha}(q)=\sum_{r=0}^k \tbinom{k}{r}(-1)^r\frac{\Gamma(k+\alpha+1)}{\Gamma(k+\alpha-r+1)}q^{k-r},$$

where $k$ is an integer, and $\alpha$ is real.

Now the question becomes how to evalueate the following integral for given real $\alpha,\beta$ and integer $k$,

$$J^k(\alpha,\beta)=\int_{0}^{\infty} q^{\beta}[P_{k}^{\alpha}(q)]^2e^{-q}dq.$$

Things become easy when $\beta-\alpha$ is an integer, I would like to ask is there a closed form for general $\alpha$ and $\beta$?

Answer 1

There is an expression in terms of the generalized hypergeometric function 3F2. The best references I know in this direction are:

1) J.S. Dehesa et al, "Information Theory of D-Dimensional Hydrogenic Systems: Application to Circular and Rydberg States", International Journal of Quantum Chemistry, Vol 110, 1529–1548 (2010).

2) V. F. Tarasov, "Exact numerical values of diagonal matrix elements 〈rk〉 nl , as n≤8 and −7≤k≤4, and the symmetry of Appell’s function F2(1,1)", Int J Mod Phys B (2004) 18:3177–3184.