On an operational matrix method based on generalized Bernoulli polynomials of level m

Abstract

An operational matrix method based on generalized Bernoulli polynomials of level m is introduced and analyzed in order to obtain numerical solutions of initial value problems. The most innovative component of our method comes, essentially, from the introduction of the generalized Bernoulli polynomials of level m, which generalize the classical Bernoulli polynomials. Computational results demonstrate that such operational matrix method can lead to very ill-conditioned matrix equations.

Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$

“…2, Sec. 3, p. 30]) it is possible to deduce the following formula for the integral of the product of two classical Bernoulli polynomials Using integration by parts a similar formula to (4.45) has been deduced in [16]…”

“…For a fixed m ∈ N, the generalized Bernoulli polynomials of level m are defined by means of the following generating function [13,16,[18][19][20] (2…”

“…For a fixed m ∈ N, the generalized Bernoulli polynomials of level m are defined by means of the following generating function [14,15,18,20,21,22] z (0), for all n ≥ 0. The generalized Bernoulli polynomials of level m also have been called hypergeometric Bernoulli polynomials [12].…”

“…The following results summarize some properties of the generalized Bernoulli polynomials of level m (cf. [14,15,13,18]).…”

New results on the q-generalized Bernoulli polynomials of level m

“…Ryoo [3][4][5][6][7][8] defined the q-Bernoulli polynomials using different methods and studied their properties. There are numerous recent investigations on qgeneralizations of this subject by many others author; see [9][10][11][12][13][14][15][16][17]. More recently, Mahmudov et al [18] used the q-Mittag-Le er function B [m−1,α] n,q (x, y; λ) z n [n]q!…”