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New approach to the complete sum of products of the twisted \((h,q)\)-Bernoulli numbers and polynomials. (English) Zbl 1163.11015

In this paper the higher-order twisted \((h,q)\)-Bernoulli polynomials and numbers are defined, and a new approach to the complete sums of products of twisted \((h,q)\)-Bernoulli polynomials and numbers is used. The \(p\)-adic \(q\)-Volkenborn integral is used to evaluate summations of the form: \[ B^{(h,v)}_{m,w}(y_1+y_2+ \cdots +y_v,q)=\sum_{l_1, l_2, \cdots, l_v \geq 0 ; l_1+l_2+ \cdots + l_v = m }{m \choose l_1,l_2, \cdots, l_v} \prod^v_{j=1}B^{(h)}_{l_j,w}(y_j,q), \] where \(B^{(h)}_{m,w}(y_j,q)\) is the twisted \((h,q)\)-Bernoulli polynomials. Several new identities involving \((h,q)\)-Bernoulli polynomials and numbers are also obtained.

MSC:

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