Suppose a nonnegative function $f$ on the positive real line and $f(x)\neq\Gamma(x)$. Do we know any closed formula of the following integral $$\int_{\mathbb{R}^+} \frac{\alpha^x}{\Gamma(x)}f(x)dx,$$ where $\alpha>0$?
Thanks in advance.
2026-03-28 21:58:21.1774735101
Integration involving x at the power and the gamma function
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