I am looking for a closed form of the following infinite sum $$ \sum_{n=0}^{\infty} \frac{\left(n-\frac{1}{2}\right)!\ (r)_n }{(2n)!} w^{2n}H_{2n}(z)$$ I found a similar formula on Wolfram as $$ \sum_{n=0}^{\infty} \frac{(r)_n }{(2n)!} w^{2n}H_{2n}(z)=(w^2+1)^{-c} \ _1 F_1\left(c;\frac{1}{2};\frac{z^2 w^2}{w^2+1}\right)$$ Unfortunaly I don't know what to do with $\left(n-\frac{1}{2} \right)!$!! Is there a closed formula for this series, or an approximation.
2026-03-26 01:29:05.1774488545
Closed form for infinite sum over Hermite polynomial $H_{2n}$
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