Abstract
Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (see (Araci et al. in J. Number Theory 133(10):3348-3361, 2013)). In this paper, we develop the new method of q-umbral calculus due to Roman, and we study a new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give some interesting identities on our q-Euler polynomials related to the q-Bernoulli numbers and polynomials of Hegazi and Mansour.
Highlights
Throughout this paper we will assume q to be a fixed real number between and
We introduce the q-extension of exponential function as follows:
L|p(x) denotes the action of linear functional L on the polynomial p(x), and it is well known that the vector space operations on P∗ are defined by
Summary
Throughout this paper we will assume q to be a fixed real number between and. We define the q-shifted factorials by n– ∞ ( . ) i=If x is a classical object, such as a complex number, its q-version is defined as [x]q = –qx –qWe introduce the q-extension of exponential function as follows:∞ zn eq(z) = n= [n]q! = (( – q)z : q)∞ (see [ – ]), where z ∈ C with |z|
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