Integral of generalized Laguerre polynomials

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 $\beta$?

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