Abstract
We give a new construction of the -extensions of Euler numbers and polynomials. We present new generating functions which are related to the -Euler numbers and polynomials. We also consider the generalized -Euler polynomials attached to Dirichlet's character and have the generating functions of them. We obtain distribution relations for the -Euler polynomials and have some identities involving -Euler numbers and polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate the -Euler polynomials at negative integers.
Highlights
Let C be the complex number field
We assume that q ∈ C with |q| < 1 and that the q-number is defined by x q 1 − qx / 1 − q in this paper
Many mathematicians have studied for q-Euler and q-Bernoulli polynomials and numbers see 1–18
Summary
Let C be the complex number field. We assume that q ∈ C with |q| < 1 and that the q-number is defined by x q 1 − qx / 1 − q in this paper. Many mathematicians have studied for q-Euler and q-Bernoulli polynomials and numbers see 1–18 . There are papers for the q-extensions of Euler polynomials and numbers approaching with two kinds of viewpoint among remarkable papers see 7, 10 . It is known that the Euler polynomials are defined by 2/ et 1 ext
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