ON THE SUM AND MAXIMAL TERM OF TAYLOR-DIRICHLET TYPE SERIES

Abstract

The article deals with Taylor-Dirichlet type series of the form $F(x)=\sum\nolimits_{k=0}^{+\infty}a_ke^{x\lambda_k+\tau(x)\beta_k},$ where $(\lambda_k)$ and $(\beta_k)$ are some sequence of non-negative numbers, and
$\tau(x)$ is a non-negative non-decreasing function, $a_k\geq 0$ $(k\geq 0)$. The class of such functions we denote $\mathcal{TD}(\Lambda,\beta,\tau)$. The main statement of the paper is Theorem 2: Let a sequence {$(\lambda_n+\beta_n)$} be increasing, a sequence $\beta=(\beta_n)$ be non-decreasing and a positive function $\tau$ be such that $\tau(x+h)-\tau(x)\ge h$ $(x > 0, h>0)$. If the condition $\sum_{k=0}^{\infty}{(\lambda_{k+1}+\beta_{k+1}-\lambda_k-\beta_k)^{-1}}<+\infty$ is fulfilled, then for every function $F\in\mathcal{TD}(\Lambda,\beta,\tau)$ the asymptotic relation
$
F(x)=(1+o(1))\mu(x,F)
$
holds as $x\to +\infty$ outside some set $E\subset [0,+\infty)$ of finite Lebesgue measure ($\int\limits_{E}dx<+\infty$), where $\mu(x,F)=\max\{|a_k|e^{\tau(x)\beta_k+x\lambda_k}\colon k\geq 0\}$.
Theorem 2 was proved earlier (1998) under the conditions of strict increasing of the sequences $(\lambda_n)$ and $(\beta_n)$.