Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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GENERALIZED EULER POWER SERIES

  • KIM, MIN-SOO (Department of Mathematics Education, Kyungnam University)
  • Received : 2020.08.02
  • Accepted : 2020.09.08
  • Published : 2020.09.30
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Abstract

This work is a continuation of our investigations for p-adic analogue of the alternating form Dirichlet L-functions $$L_E(s,{\chi})={\sum\limits_{n=1}^{\infty}}{\frac{(-1)^n{\chi}(n)}{n^s}},\;Re(s)>0$$. Let Lp,E(s, t; χ) be the p-adic Euler L-function of two variables. In this paper, for any α ∈ ℂp, |α|p ≤ 1, we give a power series expansion of Lp,E(s, t; χ) in terms of the variable t. From this, we derive a power series expansion of the generalized Euler polynomials with negative index, that is, we prove that $$E_{-n,{\chi}}(t)={\sum\limits_{m=0}^{\infty}}\(\array{-n\\m}\)E_{-(m+n),{\chi}^{t^m}},\;n{\in}{\mathbb{N}}$$, where t ∈ ℂp with |t|p < 1. Some further properties for Lp,E(s, t; χ) has also been shown.

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References

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