A reducible case of double hypergeometric series involving the riemann $zeta$-function

Usng the Pochhammer symbol $(lambda)_n$ given by $$ (1.1) (lambda)_n = {1, if n = 0 {lambda(lambda + 1) cdots (lambda + n - 1), if n in N = {1, 2, 3, ldots}, $$ we define a general double hypergeometric series by [3, pp.27] $$ (1.2) F_{q:s;upsilon}^{p:r;u} [alpha_1, ldots, alpha_p : gamma_1, ldots, gamma_r; lambda_1, ldots, lambda_u;_{x,y}][eta_1, ldots, eta_q : delta_1, ldots, delta_s; mu_1, ldots, mu_v; ] = sum_{l,m = 0}^{infty} frac {prod_{j=1}^{q} (eta_j)_{l+m} prod_{j=1}^{s} (delta_j)_l prod_{j=1}^{v} (mu_j)_m)}{prod_{j=1}^{p} (alpha_j)_{l+m} prod_{j=1}^{r} (gamma_j)_l prod_{j=1}^{u} (lambda_j)_m} frac{l!}{x^l} frac{m!}{y^m} $$ provided that the double series converges.