Table of Contents
ISRN Applied Mathematics
Volume 2012 (2012), Article ID 475463, 14 pages
http://dx.doi.org/10.5402/2012/475463
Research Article

Generalized 𝑀 -Euler Numbers and Polynomials

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 28 October 2011; Accepted 4 December 2011

Academic Editors: A. J. Kearsley and D. Kuhl

Copyright © 2012 H. Y. Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We generalize the Euler numbers and polynomials by the generalized 𝑀 -Euler numbers 𝐸 𝑛 , 𝑀 ( π‘Ž ) and polynomials 𝐸 𝑛 , 𝑀 ( π‘₯ ∢ π‘Ž ) . For the complement theorem, 𝐸 𝑛 , 𝑀 ( π‘₯ ∢ π‘Ž ) have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of “scattering” of the zeros of the the generalized Euler polynomials 𝐸 𝑛 , 𝑀 ( π‘₯ ∢ π‘Ž ) in complex plane.