I have been trying to understand how to obtain the integral representation of the Catalan numbers using Mellin transformations and get stuck when I obtain the inverse Mellin transformation.
2026-03-28 12:02:42.1774699362
How do I arrive at the integral representation of Catalan Numbers using Mellin Transform
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The Catalan numbers have the following integral representation (see Catalan Number Properties):
$$C_n=\int_0^4 \frac{1}{2\pi}\sqrt{\frac{4-x}{x}}\ x^n\,dx=\frac{4^n\ \Gamma\left(n+\frac{1}{2}\right)}{\sqrt{\pi}\ \Gamma(n+2)}\tag{1}$$