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Identities of symmetry for higher-order \(q\)-Bernoulli polynomials. (English) Zbl 1318.11030

Summary: Recently, the higher-order Carlitz’s \(q\)-Bernoulli polynomials are represented as \(q\)-Volkenborn integral on \(\mathbb Z_p\) by the second author [Russ. J. Math. Phys. 9, No. 3, 288–299 (2002; Zbl 1092.11045)]. A question was asked by him in [Abstr. Appl. Anal. 2008, Article ID 914367, 7 p. (2008; Zbl 1217.11022)] as to finding the extended formulae of symmetries for Bernoulli polynomials which are related to Carlitz \(q\)-Bernoulli polynomials. In this paper, we give some new identities of symmetry for the higher-order Carlitz \(q\)-Bernoulli polynomials which are derived from multivariate \(q\)-Volkenborn integrals on \(\mathbb Z_p\). We note that they are a partial answer to that question.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)