Differential equations for the extended 2D Bernoulli and Euler polynomials

Abstract

In this paper, we introduce the extended 2D Bernoulli polynomials by t(alpha)/(e(t) - 1)(alpha) c(xt+ytj) = Sigma(infinity)(n=0) B-n((alpha,j)) (x,y,c) t(n)/n! and the extended 2D Euler polynomials by 2(alpha)(e(t) + 1)(alpha) c(xt+ytj) = Sigma(infinity)(n=0) E-n((alpha,j)) (x,y,c) t(n)/n!, where c > 1. By using the concepts of the monomiality principle and factorization method, we obtain the differential, integro-differential and partial differential equations for these polynomials. Note that the above mentioned differential equations for the extended 2D Bernoulli polynomials reduce to the results obtained in (Bretti and Ricci in Taiwanese J. Math. 8(3): 415-428, 2004), in the special case c = e, alpha = 1. On the other hand, all the results for the second family are believed to be new, even in the case c = e, alpha = 1. Finally, we give some open problems related with the extensions of the above mentioned polynomials.