Actually the paper isn't as bad as I remember, lol. Here's a link: https://arxiv.org/pdf/1503.06211.pdf
I do take for granted that the reader knows what Ramanujan's Master Theorem is. The version I use is, if,
\(
|f(z)| \le C e^{\alpha |\Im(z)| + \rho|\Re(z)|}\,\,\alpha < \pi/2,\,\,C,\rho > 0\\
f\,\,\text{is holomorphic for}\,\,\Re(z) > 0\\
\text{Then f can be represented as}\\
\Gamma(1-z)f(z) = \sum_{n=0}^\infty f(n+1)\frac{(-1)^n}{n!(n+1-z)} + \int_1^\infty (\sum_{n=0}^\infty f(n+1)\frac{(-x)^n}{n!})x^{-z}\,dx\\
\)
Which is nothing more than a slightly tweaked version of Ramanujan's Master Theorem. I choose to write this using fractional calculus, where if,
\(
\vartheta(x) = \sum_{n=0}^\infty f(n+1) \frac{x^n}{n!}\\
f(z) = \frac{d^{z-1}}{dx^{z-1}}|_{x=0} \vartheta(x)\\
\)
I do take for granted that the reader knows what Ramanujan's Master Theorem is. The version I use is, if,
\(
|f(z)| \le C e^{\alpha |\Im(z)| + \rho|\Re(z)|}\,\,\alpha < \pi/2,\,\,C,\rho > 0\\
f\,\,\text{is holomorphic for}\,\,\Re(z) > 0\\
\text{Then f can be represented as}\\
\Gamma(1-z)f(z) = \sum_{n=0}^\infty f(n+1)\frac{(-1)^n}{n!(n+1-z)} + \int_1^\infty (\sum_{n=0}^\infty f(n+1)\frac{(-x)^n}{n!})x^{-z}\,dx\\
\)
Which is nothing more than a slightly tweaked version of Ramanujan's Master Theorem. I choose to write this using fractional calculus, where if,
\(
\vartheta(x) = \sum_{n=0}^\infty f(n+1) \frac{x^n}{n!}\\
f(z) = \frac{d^{z-1}}{dx^{z-1}}|_{x=0} \vartheta(x)\\
\)