On the symmetries of the \(q\)-Bernoulli polynomials
Summary: \textit{B. A. Kupershmidt} [Reflection symmetries of \(q\)-Bernoulli polynomials. J. Nonlinear Math. Phys. 12, Suppl. 1, 412--422 (2005; Zbl 1362.33021)] and \textit{H. J. H. Tuenter} [Am. Math. Mon. 108, No. 3, 258--261 (2001; Zbl 0983.11008)] have introduced reflection symmetries for the \(q\)-Bernoulli numbers and the Bernoulli polynomials, 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.
symmetries of the \(q\)-Bernoulli numbers and polynomials
scientific article; zbMATH DE number 5352111
scientific article