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 $\varphi (s,a,\lambda )$ denote the Lerch zeta-function with a complex variable, ${\varphi }^{\ast }(s,a,\lambda )$ a slight modification of $\varphi (s,a,\lambda )$ defined by extracting the (possible) singularity of $\varphi (s,a,\lambda )$ at $s=1$, and $({\varphi }^{\ast }{)}^{(m)}(s,a,\lambda )$ for any $m\in \mathbb{Z}$ the $m$th derivative with respect to $s$ if $m\ge 0$, while if $m\le 0$ the $|m|$-th primitive defined with its initial point at $s+\infty$. The present paper aims to study asymptotic aspects of $({\varphi }^{\ast }{)}^{(m)}(s,a,\lambda )$, transformed through the Laplace-Mellin and Riemann-Liouville operators (say, ${\mathcal{LM}}_{z;\tau }^{\alpha }$ and ${\mathcal{R}\mathcal{L}}_{z;\tau }^{\alpha ,\beta }$, respectively) in terms of the variable $s$. We shall show that complete asymptotic expansions exist if $a>1$ for ${\mathcal{LM}}_{z;\tau }^{\alpha }({\varphi }^{\ast }{)}^{(m)}(s+\tau ,a,\lambda )$ and ${\mathcal{R}\mathcal{L}}_{z;\tau }^{\alpha ,\beta }({\varphi }^{\ast }{)}^{(m)}(s+\tau ,a,\lambda )$ (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).

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