New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials

In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach to the complete sums of products of twisted (h, q)-Bernoulli polynomials and numbers. p-adic q-Volkenborn integral is used to evaluate summations of the following form: $${array}{*{20}{c}} {\,Bm,w(h,v)({y_1} + {y_2} + ... + {yv,}q)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {{array}{*{20}{c}} {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \,\,\,\,\,\,\,\,\,\,\,\,\,\,}{\,\,\,\,\,\,\,\,\,\,Σ }{( {{array}{*{20}{c}} m \\ {{l_1},{l_2},...,{l_v}} {array} } )∏j = 1^v {B_{{l_j},w}⁽ʰ⁾({y_j},q),} } {array} } \\ {{l_1},{l_2},...,{l_v} 0} \\ {{l_1} + {l_2} + ... + {l_v} = m} {array}$$ where $$Bm,w(h,v)({y_j},q)$$ is the twisted (h, q)-Bernoulli polynomials. We also define new identities involving (h, q)-Bernoulli polnomials and numbers.