A Note on q-Bernoulli Numbers and Polynomials

Abstract

In this paper, we define a new q−analogy of the Bernoulli polynomials and the Bernoulli numbers and we deduced some important relations of them. Also, we deduced a q−analogy of the Euler-Maclaurin formulas. Finally, we present a relation between the q−gamma function and the q−Bernoulli polynomials.

A note on q-Frobenius-Euler numbers and polynomials

“…The q−Bernoulli polynomials of Hegazi and Mansour are defined by the generating function to be ∞ n=0 B n,q (x) t n [n] q ! = t e q (t) − 1 e q (xt), (see [4]). (1.5) In the special case, x = 0, B n,q (0) = B n,q are called the n−th q−Bernoulli numbers.…”

“…Kupershmidt, Hegazi and Mansour derived some interesting identities and properties related to q−Bernoulli and Euler polynomials. Recently, several authors have studied various q−extention of Bernoulli, Euler and Genocchi polynomials(see [1], [2], [4]- [14]). Let F be the set of all formal power series in variable t over C with…”

Approximation Properties of q-Bernoulli Polynomials

“…The -analogue of Euler-Maclaurin formula has been studied in [9]. The authors of [9] applied q-integral by parts to reach q-analogue of Euler-Maclaurin formula. We can not apply that approach to approximation, because it was written in terms of ( ) = , ( − [ ]).…”

A NEW APPROACH TO q-GENOCCHI NUMBERS AND POLYNOMIALS

“…This definition is motivated from Hegazi and Mansour [5]. In that work, q-Bernoulli polynomials B n (x; q) are defined by…”