  The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent
expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these
coefficients may be written as certain summations over mathematical constants
and specialized hypergeometric functions $_pF_{p+1}$. This family of results
generalizes a representation of the Euler constant in terms of a summation over
values of the trigonometric integrals Si or Ci. The series representations are
suitable for acceleration. As byproducts, we evaluate certain sine-logarithm
integrals and present the leading asymptotic form of the particular $_pF_{p+1}$
functions.
