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 Luo1 , Bai-Ni Guo2 , Feng Qi2 and Lokenath Debnath4

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
 4Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539, USA

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.