Segments of exponential series and regularly varying sequences

Abstract

The task of this paper is to investigate asymptotic behavior of segments of exponential series defined as $$T_{\lambda}(x):=\sum_{n<\lambda x}\dfrac{c_n}{n!}x^n,\qquad\lambdaın R^+,\quad x\toınfty,$$where $(c_n)_{nın N}$ belongs to the set of regularly varying sequences in Karamata sense of arbitrary index. Precise results are obtained.