Decomposition theorems produced from differential operators

Abstract

In an attempt to improve Ramanujan's unsuccessful evaluation of double sums on Hurwitz zeta functions, we introduce more general multiple zeta values on Hurwitz zeta functions defined as∑k1=0∞∑k2=0∞⋯∑kr=0∞(k1+x1)−α1×[(k1+x1)+(k2+x2)]−α2×⋯×[(k1+x1)+(k2+x2)+⋯+(kr+xr)]−αr, with α1,α2,…,αr positive integers, αr≥2 and positive numbers x1,x2,…,xr. Especially, we extend Euler decomposition theorem which expressed a product of two Riemann zeta values in terms of Euler double sums, to a more general decomposition theorem which expressed products of n Hurwitz zeta values in terms of multiple zeta values on Hurwitz zeta functions as mentioned before. Furthermore, we apply various differential operators to the resulted decomposition theorem to produce more decomposition theorems concerning products of multiples of values of Hurwitz zeta function.