New summation relations for the Stieltjes constants

The Stieltjes constants gamma(k)(a) have been of interest for over a century, yet their detailed behaviour remains under investigation. These constants appear in the Laurent expansion of the Hurwitz zeta function zeta(s, a) about s = 1. We obtain novel single and double summatory relations for gamma(k)(a), including single summation relations for gamma(k)(a - b) and gamma(k)(a+p/q), where a and b are real and p and q are positive integers. In addition, we obtain new integration formulae for the Hurwitz zeta function and a new expression for the Stieltjes constants gamma(k)(1) gamma(k). Portions of the presentation show an intertwining of the theory of the hypergeometric function with that of the Hurwitz zeta function.