Non-Archimedean \(q\)-integrals associated with multiple Changhee \(q\)-Bernoulli polynomials. (English) Zbl 1072.11090

Summary: Using non-Archimedean \(q\)-integration, we introduce multiple Changhee \(q\)-Bernoulli polynomials which form a \(q\)-extension of Barnes’ multiple Bernoulli polynomials. We also construct the Changhee \(q\)-zeta functions which give \(q\)-analogs of Barnes’ multiple zeta function and indicate some relationships between the Changhee \(q\)-zeta function and Daehee \(q\)-zeta function. As an immediate application of these relationships, we present a closed expression for sums of products of generalized \(q\)-Bernoulli numbers. This helps us to deal with nested sums over combinations of Barnes’ multiple zeta function, non-Archimedean binomial coefficients, and generalized \(q\)-Bernoulli polynomials.


11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11B68 Bernoulli and Euler numbers and polynomials
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)