Abstract
We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted-Euler polynomials. Finally, we give a table for the solutions of the twisted-Euler polynomials.
Highlights
Introduction and notationsThroughout this paper, we use the following notations
We study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions
We find distribution relations of generalized twisted Euler numbers and polynomials
Summary
Throughout this paper, we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex numbers field, and Cp denotes the completion of algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp(p) = p−1. When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C, one normally assumes that |q| < 1. If q ∈ Cp, we normally assume that |q − 1|p < p−1/(p−1) so that qx = exp(x log q), for |x|p ≤ 1.
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