REMARKS ON TAUBERIAN THEOREMS FOR EXP-LOG FUNCTIONS

Author:

Karl-Heinz INDLEKOFER

2010 Mathematics Subject Classification:

40E05, 11N45, 05A15, 60C05.

Keywords:

arithmetical semigroups, generating functions for algebraic and combinatorial structures, Tauberian theorems for power series.

Abstract:

In this paper, we consider functions F, holomorphic in the open unit disk D:=\{  y \in \mathbb C : |y| <1 \}, of the form  F(y)=\sum_{n=0}^{\infty}f(n)y^n  = \exp \left(\sum_{m=1}^{\infty}\frac{\lambda_f(m)}{m}y^m  \right)=\frac{H(y)}{(1-y)^{\delta}}, \delta>0,  where \lambda_f(m)={\rm O}(1), m \in \mathbb N. If H(y)={\rm O}(1) for y \in D and \lim_{y \rightarrow 1{-}}H(y)=A exists and is different from zero, then f(n) \sim \frac{A n^{\delta -1}}{\Gamma(\delta)} as n \rightarrow \infty.

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Indl-2013

Vol. 8 (16), 2013