Sum Formula of Multiple Hurwitz-Zeta Values

Let s_1,…,s_d be d positive integers and consider the multiple Hurwitz-zeta value zeta(s_1,…,s_d;-1/2,…,-1/2)/2^w where w=s_1+…+s_d is called the weight. For d<n+1, let T(2n,d) be the sum of all these values with even arguments whose weight is 2n and whose depth is d. Recently Shen and Cai gave formulas for T(2n,d) for d<6 in terms of t(2n), t(2)t(2n-2) and t(4)t(2n-4). In this short note we generalize Shen-Cai’s results to arbitrary depth by using the theory of symmetric functions established by Hoffman.