Asymptotic expansions for the Laplace-Mellin and Riemann-Liouville transforms of Lerch zeta-functions

Abstract

For a complex variable $s$ and real parameters $a$ and $\lambda$ with $a>0$, let $\phi(s,a,\lambda)$ denote the Lerch zeta-function with a complex variable, $\phi^{\ast}(s,a,\lambda)$ a slight modification of $\phi(s,a,\lambda)$ defined by extracting the (possible) singularity of $\phi(s,a,\lambda)$ at $s=1$, and $(\phi^{\ast})^{(m)}(s,a,\lambda)$ for any $m\in\mathbb{Z}$ the $m$th derivative with respect to $s$ if $m\geq0$, while if $m\leq0$ the $|m|$-th primitive defined with its initial point at $s+\infty$. The present paper aims to study asymptotic aspects of $(\phi^{\ast})^{(m)}(s,a,\lambda)$, transformed through the Laplace-Mellin and Riemann-Liouville operators (say, $\mathcal{LM}_{z;\tau}^{\alpha}$ and $\mathcal{RL}_{z;\tau}^{\alpha,\beta}$, respectively) in terms of the variable $s$. We shall show that complete asymptotic expansions exist if $a>1$ for $\mathcal{LM}_{z;\tau}^{\alpha}(\phi^{\ast})^{(m)}(s+\tau,a,\lambda)$ and $\mathcal{RL}_{z;\tau}^{\alpha,\beta}(\phi^{\ast})^{(m)}(s+\tau,a,\lambda)$ (Theorems~1--4), as well as for their iterated variants (Theorems~5--10), when the `pivotal' parameter $z$ (of the transforms) tends to both $0$ and $\infty$ through appropriate sectors. Most of our results include any vertical ray in their region of validity; this allows us to deduce complete asymptotic expansions along vertical lines $(s,z)=(\sigma,it)$ as $t\to\pm\infty$ (Corollaries~2.1,~4.1,~6.1 and~8.1).