Abstract
We generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . We observe an interesting phenomenon of “scattering” of the zeros of the generalized -Euler polynomials in complex plane.
Highlights
Many mathematicians have studied in the area of the Euler numbers and polynomials see 1–15
We investigate some properties which are related to the generalized q, w -Euler numbers En,q,w a and polynomials En,q,w x : a
We find generating functions of the generalized q, w Euler numbers En,q,w a and polynomials En,q,w x : a
Summary
Many mathematicians have studied in the area of the Euler numbers and polynomials see 1–15. The Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. In 14 , we introduced that Euler equation En x 0 has symmetrical roots for x 1/2 see 14. It is the aim of this paper to observe an interesting phenomenon of “scattering” of the zeros of the generalized q, w -Euler polynomials En,q,w x : a in complex plane. When one talks of qextension, 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 assume 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 xq 1 − qx 1−q
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