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On the symmetries of the \(q\)-Bernoulli polynomials. (English) Zbl 1217.11022

Summary: B. A. Kupershmidt [Reflection symmetries of \(q\)-Bernoulli polynomials. J. Nonlinear Math. Phys. 12, Suppl. 1, 412–422 (2005; Zbl 1362.33021)] and 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.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
05A30 \(q\)-calculus and related topics
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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References:

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