  In this paper, we prove a new identity for values of the Hurwitz zeta
function which contains as particular cases Koecher's identity for odd zeta
values, the Bailey-Borwein-Bradley identity for even zeta values and many other
interesting formulas related to values of the Hurwitz zeta function. We also
get an extension of the bivariate identity of Cohen to values of the Hurwitz
zeta function. The main tool we use here is a construction of new Markov-WZ
pairs. As application of our results, we prove several conjectures on
supercongruences proposed by J. Guillera, W. Zudilin, and Z.-W. Sun.
