Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$. Assume $0<\beta<1$. Is there a closed formula for this sum $$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$
where $b>0$ and $m\in \Bbb N$
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$\begingroup$ the series will not converge if $m\neq 0$. $\endgroup$– Carlo BeenakkerCommented Aug 18, 2023 at 10:57
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$\begingroup$ why this series not converge $\endgroup$– zoran VicovicCommented Aug 18, 2023 at 11:04
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1$\begingroup$ the terms decay as $j^{\beta-1}j^{m/2-1/4}$ $\endgroup$– Carlo BeenakkerCommented Aug 18, 2023 at 11:16
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$\begingroup$ Thank you @CarloBeenakker $\endgroup$– zoran VicovicCommented Aug 18, 2023 at 18:01
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