Abstract
Kupershmidt and Tuenter have introduced reflection symmetries for theq-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 theq-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for theq-Bernoulli polynomials, we can obtain some interesting relationships betweenq-Bernoulli numbers and polynomials.
Highlights
I cannot obtain the extended formulae of Theorems 2.1 and 2.3 related to the Carlitz’s q-Bernoulli numbers and polynomials
We suggest the following two questions
We note that Carlitz’s q-Bernoulli numbers can be written by βn,q x n q dμq
Summary
Kupershmidt and Tuenter have introduced reflection symmetries for the q-Bernoulli numbers and the Bernoulli polynomials in 2005 , 2001 , respectively. 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. 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.
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