This came up in the computation of an ensemble average in quantum mechanics. According to Mathematica, we have the curious identity \begin{equation} \sum_{n=0}^\infty \exp(-bn)L_n(2a) = \frac{\exp\left(b+\frac{2a}{1-e^b}\right)}{e^b-1} \end{equation} for real $a,b$ and $b \neq 0$. How would one prove this? I have been unable to find a reference for this identity. I would conjecture that this is somehow related to the generating function just by its form, but I am not sure.
2026-03-25 09:49:31.1774432171
Series identity of Laguerre polynomials
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The generating polynomial of Laguerre functions is $$\sum_{n=0}^{\infty}t^n L_n(x) = \dfrac{\exp\left(-\dfrac{tx}{1-t}\right)}{1-t}$$