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

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.