Laguerre polynomial and series

Question

Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$. Consider the sum $$\sum^\infty_{j=0} \frac{1}{(b-j)}L^{m}_j(x)$$

where $b\not\in\Bbb N,x>0$ and $m\in \Bbb N$. I have found this series converges [A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 2, Special Functions (Gordon and Breach, New York, 1986), p. 703.]but I don't know how to prove that.

Here my attempt:

By Mehler's formula, for the Laguerre functions. For $|w|<1$, we have $$ \begin{aligned} & \sum_{k=0}^{\infty} \frac{\Gamma(k+1)}{\Gamma(k+\delta+1)} L_k^\delta(t) L_k^\delta(s) w^k \\ = & (1-w)^{-1}(-t s w)^{-\frac{\delta}{2}} e^{-\frac{w}{1-w}(s+t)} J_\delta\left(\frac{2(-t s w)^{\frac{1}{2}}}{1-w}\right) . \end{aligned} $$ Where $J_\delta$ is the Bessel function of order $\delta$. Multiplying by $w^{\beta},\beta>0$ and integrating with respect to $w$ betwen $0,1$, we obtain

\begin{aligned} & \sum_{k=0}^{\infty} \frac{1}{k+\beta}\frac{\Gamma(k+1)}{\Gamma(k+\delta+1)} L_k^\delta(t) L_k^\delta(s) \\ = &\int^1_0 w^\beta (1-w)^{-1}(-t s w)^{-\frac{\delta}{2}} e^{-\frac{w}{1-w}(s+t)} J_\delta\left(\frac{2(-t s w)^{\frac{1}{2}}}{1-w} \right)dw . \end{aligned}

Since $L_k^\delta(0)=\frac{\Gamma(k+\delta+1)}{\Gamma(k+1)\Gamma(\delta+1)}$ and

$\lim_{x\to 0} \big(\frac{x}{2}\big)^{-\delta}J_\delta(x)=\frac{1}{\Gamma(\delta+1)}$ the above equation becomes \begin{aligned} & \sum_{k=0}^{\infty} \frac{1}{k+\beta} L_k^\delta(t) \\ = &\int^1_0 w^\beta (1-w)^{-(1+\delta)} e^{-\frac{w}{1-w}t} dw . \end{aligned}

Thank you in advance.

Answer 1

The handbook by A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (PBM) seems hard to use, because they do not provide references to specific formulas and sometimes they do not even say for what values of the variables the formulas hold.

In your case, the formula in question is formula 5.11.1(1) on p. 703 of PBM, given there without any specifications on $\alpha$ (equal $m$ in your case) and $b$, and without any specific reference.

This formula is given (for $x>0$) as formula (26) on p. 215 of Higher transcendental functions 2 by Erdelyi (with $-a\in (-1/2,1)$ in place of your $b$) and the case $-a=-1$ is given there as formula (19) on the same page. References to these formulas are given on p. 217 of that volume. No results seem to be known for other values of $a$.