Evaluation of generalized Laguerre function integrals using orthogonality relations

Question

(NB - I am not asking to be spoon-fed with complete solutions, just pointing out any useful transformations, or giving general pointers would suffice.)

The orthogonality relation for generalized Laguerre functions is given as the following:

$$ \int_0^{\infty} x^\alpha \ e^{-x} \ L_n^\alpha (x) \ L_m^\alpha (x) \ dx = \frac{\Gamma\left( \alpha + n +1 \right)}{n!} \delta_{mn}$$

If we have to evaluate integrals in which we have $x^{(\alpha + k)}$ in the integrand instead of $x^{\alpha}$ as it occurs in the orthogonality relation, i.e. one is dealing with:

$$I = \int_0^{\infty} x^{(\alpha + k)} \ e^{-x} \ L_n^\alpha (x) \ L_m^\alpha (x) \ dx \ \ \ \ \ ,$$

where $\alpha$ is a positive half integer.

How do we evaluate these type of integrals, in situations where:

• $k$ is a positive integer.
• $k$ is a negative integer.
• $k$ is a positive half-integer.

Answer 1

Sorry, I can only provide the opposite, that is, the final answer without a clean derivation. The result is

$$\int_0^\infty x^{\alpha + k} L_m^{(\alpha)} L_n^{(\alpha)} e^{-x} dx = \\ \frac {(-1)^n \,(m + \alpha)!} {n!} \sum_{i = 0}^m \frac {(-1)^i \,(i + k)! \,(i + k + \alpha)!} {i! \,(i + \alpha)! \,(m - i)! \,(i + k - n)!}, \\ m, n, 2k, \alpha - \frac 1 2 \in \mathbb N^0,$$

which has a closed form in terms of the regularized hypergeometric function ${_3\hspace{-1px}{\widetilde F}\hspace{-2px}_2}$. For $-k \in \mathbb N$, the answer is the same, but the $i$th term $a_i$ is understood as $\lim_{\epsilon \to 0} a_{i + \epsilon}$.