## 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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