Kupershmidt and Tuenter have introduced reflection symmetries for the -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 -Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for the -Bernoulli polynomials, we can obtain some interesting relationships between -Bernoulli numbers and polynomials.
"On the Symmetries of the -Bernoulli Polynomials." Abstr. Appl. Anal. 2008 1 - 7, 2008. https://doi.org/10.1155/2008/914367