In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^{s-1}f\left(x\right)dx$$ in closed form computing the inverse mellin transform using the Residue theorem gets us formulae (usually asymptotic, but sometimes exact) for the behavior of $f\left(x\right)$ as $x$ decreases to $0$ and/or as $x$ increases to $\infty$. Ex:
$$\int_{0}^{\infty}x^{s-1}\sum_{n=0}^{\infty}e^{-2^{n}x}dx=\frac{\Gamma\left(s\right)}{1-2^{-s}}$$ implies: $$\sum_{n=0}^{\infty}e^{-2^{n}x}=\frac{1}{2}-\frac{\gamma+\ln x}{\ln2}+\frac{1}{\ln2}\sum_{k\in\mathbb{Z}^{\times}}\Gamma\left(\frac{2k\pi i}{\ln2}\right)x^{-\frac{2k\pi i}{\ln2}}-\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}}{2^{n}-1}\frac{x^{n}}{n!}$$ an (emotionally satisfying!) identity which holds for all x>0.
But, now consider the p-adic analogue: $$\mathscr{M}_{p}\left\{ f\right\} \left(s\right)=\int_{\mathbb{Z}_{p}}\left|\mathfrak{y}\right|_{p}^{s-1}f\left(\mathfrak{y}\right)d\mathfrak{y}$$ where $d\mathfrak{y}$ is the haar probability measure on $\mathbb{Z}_{p}$ and $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}$ is sufficiently nice. Is there a similar interpretation for the inverse mellin transform of $\mathscr{M}_{p}\left\{ f\right\} $ in terms of the asymptotics of $f$? Answers or references would be appreciated.
Addendum: I'm asking this because I have specific p-adic mellin transforms whose analytic continuations I've computed, and I would like to know what I can do with them, and how—if at all—I can use them to make conclusions about the asymptotics or value distributions of the functions I transformed.
