Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1806.01122

A note on the asymptotic expansion of the Lerch's transcendent

Xing Shi Cai·José L. López
View original source ↗Read source document ↗Export metadata ↓

Abstract

In a previous paper by Ferreira and López [Journal of Mathematical Analysis and Applications, 298(1), 2004], the authors derived an asymptotic expansion of the Lerch's transcendent $Φ(z,s,a)$ for large $\vert a\vert$, valid for $\mathrm{Re}(a)>0$, $\mathrm{Re}(s)>0$ and $z\in\mathbb{C}\setminus[1,\infty)$. In this paper we study the special case $z\ge 1$ not covered in the previous result, deriving a complete asymptotic expansion of the Lerch's transcendent $Φ(z,s,a)$ for $z > 1$ and $\mathrm{Re}(s)>0$ as $\mathrm{Re}(a)$ goes to infinity. We also show that when $a$ is a positive integer, this expansion is convergent for $\mathrm{Re}(z) \ge 1$. As a corollary, we get a full asymptotic expansion for the sum $\sum_{n=1}^{m} z^{n}/n^{s}$ for fixed $z >1 $ as $m \to \infty$. Some numerical results show the accuracy of the approximation.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xing Shi Cai, José L. López. 2018-06-04. A note on the asymptotic expansion of the Lerch's transcendent. https://doi.org/10.1080/10652469.2019.1627530

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Convolution Regularization Preserves the $L^2$-Estimate Property for $(1,n)$-Forms

In this paper, we prove that the \(L^2\)-estimate property for \((1,n)\)-forms is preserved under the standard convolution regularization. As applications, we show that any singular Hermitian metric satisfying the optimal or multiple coarse \(L^2\)-estimate property for \((n,1)\) or \((1,n)\)-forms is Griffiths semi-positive. This resolves a question posed by Deng--Ning--Wang and a question by Inayama.

math.CV↗
arXiv2026

Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains

We obtain sharp bounds for the third- and fourth-order Schippers functionals, $|σ_3(f)(0)|$ and $|σ_4(f)(0)|$, for subclasses of univalent functions associated with non-classical geometric domains. In particular, we investigate the lune-starlike class $\mathcal{S}_{\leftmoon}^*$ and the lune-convex class $\mathcal{C}_{\leftmoon}$ determined by the subordination \[ \frac{zf'(z)}{f(z)} \prec z+\sqrt{1+z^2}, \qquad 1+\frac{zf''(z)}{f'(z)} \prec z+\sqrt{1+z^2}, \] respectively, together with the bean-domain class $\mathcal{BT}_{\mathfrak{B}}$ associated with \[ \mathfrak{B}(z)=\sqrt{1+\tanh z}. \] Using Carathéodory coefficient parametrizations and extremal optimization techniques, we derive exact estimates for the higher-order Schwarzian derivatives at the origin and identify the corresponding extremal functions. In addition, geometric descriptions of the associated extremal image domains are provided to illustrate the sharpness phenomena. The obtained results further yield sharp bounds for the initial Grunsky coefficients $g_{1,1}$ and $g_{1,2}$. These findings provide a precise description of higher-order Schwarzian structures for univalent functions related to lune and bean shaped domains.

math.CV↗
arXiv2026

Cesàro operator induced by a Bergman kernel

Let $μ$ be a positive Borel measure on $[0,1)$ and $ω$ a radial weight. In this paper we consider the Cesàro-type operator $C_{μ,ω}$ induced by the reproducing kernel $B^ω$ of the weighted Bergman space $A^2_ω$, given by $$ C_{μ,ω}(f)(z)=\int_{0}^{1}f(tz)B^ω_t(z)\,dμ(t), \quad z \in \mathbb{D}, $$ for functions $f$ analytic in $\mathbb{D}$. Under the assumption that $ω$ satisfies a natural doubling property, we study the boundedness of $C_{μ,ω}$ acting on several spaces of analytic functions, including Hardy spaces $H^p$ and weighted Bergman spaces $A^p_ν$. For $0<p,q<\infty$ and a two-sided doubling weight $ν$, we completely characterize when $C_{μ,ω}: H^p \to H^q$ and $C_{μ,ω}: A^p_ν \to A^q_ν$ are bounded in terms of the interplay of tail integrals or moments of the inducing weights and the measure $μ$. Many of the results obtained are new even in the setting of standard weights or when the Bergman reproducing kernel is replaced by the Cauchy kernel. In addition, we consider $C_{μ,ω}$ acting on $H^{\infty}$, Korenblum spaces and weighted Hardy spaces.

math.CV↗