A method for computing the generalized hypergeometric function ${}_pF_{p-1}(a_1,\dots,a_p;b_1,\dots,b_{p-1};1)$ in terms of the riemann zeta function

A method is proposed for evaluating the generalized hypergeometric function ${}_pF_{p-1}(a_1,\dots,a_p;b_1,\dots,b_{p-1};1)=\sum_{k=0}^\infty f_k$ in terms of the Riemann zeta function $\zeta(s)$ and the Hurwitz zeta function $\zeta(1/2,s)$. By analyzing an asymptotic expansion of the coefficients $f_k$ as $k\to\infty$, an expansion of ${}_pF_{p-1}$ is constructed in the form of combinations of $\zeta(s)$ and $\zeta(1/2,s)$ with explicit coefficients expressed in terms of generalized Bernoulli polynomials. The convergence of the expansion can be considerably accelerated by choosing optimal values of two control parameters. The efficiency of the method is demonstrated through a great deal of computations and comparisons with Mathematica and Maple.