English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 $\phi(s,a,\lambda)$ definedby 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 $m$thderivative with respect to $s$ if $m\geq0$, while if $m\leq0$ the $|m|$-thprimitive defined with its initial point at $s+\infty$. The present paper aimsto study asymptotic aspects of $(\phi^{\ast})^{(m)}(s,a,\lambda)$, transformedthrough 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 completeasymptotic 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), whenthe `pivotal' parameter $z$ (of the transforms) tends to both $0$ and $\infty$through appropriate sectors. Most of our results include any vertical ray intheir region of validity; this allows us to deduce complete asymptoticexpansions along vertical lines $(s,z)=(\sigma,it)$ as $t\to\pm\infty$(Corollaries~2.1,~4.1,~6.1 and~8.1).

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