On some weighted sum formulas involving general multiple zeta-type series

Abstract

The purpose of this paper is a study of the general finite sums Phi N,d(K) := Sigma(N >= n1 >=...>= nk >= 1) Pi(K)(j=1) A (inverted right perpendicular nj/d inverted left perpendicular), d is an element of N, that generalize the truncated multiple harmonic sums zeta(star)(N)({s}(K)) corresponding to d = 1 and A(n) = 1/n(s) with s is an element of N. Surprisingly, when specializing our general transformation result concerning Phi(dN,d)(K), such a type of finite sums can be used for generating and closed-form evaluation of new linear combinations of multiple Hurwitz zeta-star values of the form Sigma(s proves K max(s)<= d) zeta(star) (cs; a). Pi(l(s))(r=1) ((-1)(sr-1).(d s(r))), assuming (a, c, d, K) is an element of R x N-3 with a > -1, c > 1, where the sum is extended over all compositions of K with maximal part not exceeding d.