On the Symmetries of the q -Bernoulli Polynomials

Abstract

Kupershmidt and Tuenter have introduced reflection symmetries for the $q$-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of the $q$-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for the $q$-Bernoulli polynomials, we can obtain some interesting relationships between $q$-Bernoulli numbers and polynomials.