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Identities Involving Two Kinds of ***q*-Euler Polynomials and Numbers

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Abdelmejid Bayad

Département de Mathématiques

Université d'Evry Val d'Essonne

Bâtiment I.B.G.B.I., 3ème étage

23 Bd. de France

91037 Evry Cedex

France

Yoshinori Hamahata

Faculty of Engineering Science

Kansai University

3-3-35 Yamate-cho, Suita-shi

Osaka 564-8680

Japan

**Abstract:**

We introduce two kinds of *q*-Euler polynomials and numbers,
and investigate many of their interesting properties.
In particular, we establish *q*-symmetry properties of these
*q*-Euler polynomials, from which we recover the so-called
Kaneko-Momiyama identity for the ordinary Euler polynomials,
discovered recently by Wu, Sun, and Pan. Indeed,
a *q*-symmetry and *q*-recurrence formulas among sum of product
of these *q*-analogues Euler numbers and polynomials are obtained.
As an application, from these *q*-symmetry formulas
we deduce non-linear recurrence formulas for the product of the
ordinary Euler numbers and polynomials.

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Received February 15 2012;
revised version received March 31 2012.
Published in *Journal of Integer Sequences*, April 9 2012.

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