International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 59, Pages 3769-3776
doi:10.1155/S0161171203112070

Generalizations of Bernoulli numbers and polynomials

Qiu-Ming Luo,1 Bai-Ni Guo,2 Feng Qi,2 and Lokenath Debnath3

1Department of Broadcast-Television Teaching, Jiaozuo University, Henan, Jiaozuo City 454002, China
2Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Henan, Jiaozuo City 454000, China
3Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA

Received 3 December 2001

Copyright © 2003 Hindawi Publishing Corporation. 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

The concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x), and the generalized Bernoulli numbers Bn(a,b) are generalized to the one Bn(x;a,b,c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between Bn, Bn(x), Bn(a,b), and Bn(x;a,b,c) are established.