Journal of Integer Sequences, Vol. 15 (2012), Article 12.4.6

Identities Involving Two Kinds of q-Euler Polynomials and Numbers

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

Yoshinori Hamahata
Faculty of Engineering Science
Kansai University
3-3-35 Yamate-cho, Suita-shi
Osaka 564-8680


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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