the Stieltjes constants [@coffeyjmaa; @coffeyprsa; @ivic; @liang; @stieltjes; @wilton2] and $\gamma_0(a)=-\psi
(a)$, where $\psi=\Gamma'/\Gamma$ is the digamma function. The constants $\gamma_k(a)$ can be written in the form $$\gamma_k(a)=\lim_{N \to \infty}\left (\sum_{m=0}^N {1 \over {m+a}}\ln^k (m+a)-
{{\ln^{k+1} {(N+a)}} \over {k+1}}\right ). \eqno(2)$$ In certain special cases including $a \geq 1$ a positive integer and $a=1/2$ the Hurwitz zeta function may be expressed in terms of the Riemann zeta function $\zeta(s)$. In the case of $a=1$, one puts $\gamma_k(1)=\gamma_k$ by convention.

In this paper we present an array of series representations of the Stieltjes and related constants. Moreover we obtain series and integral representations of the sum $$S_\gamma(n) \equiv \sum_{k=1}^n {{(-1)^k} \over {(k-1)!}} {n \choose k}
\gamma_{k-1}.  \eqno(3)$$ Putting $P_1(t)=B_1(t-[t])=t-[t]-1/2$ the first periodized Bernoulli polynomial (e.g., [@ivic; @titch]) we may also write $$S_\gamma(n)=\int_1^\infty {1 \over t} L_{n-1}^1(\ln t) dP_1(t), \eqno(4)$$ where $L_n^\alpha$ is the Laguerre polynomial of degree $n$. We have shown elsewhere [@inprep] that $S_\gamma(n)$ is a component sum of the sum $S_2(n)$ arising in application of the Li criterion for the Riemann hypothesis [@coffeympag; @coffey08]. We have demonstrated [@inprep] that $S_\gamma(n)+n=O(n^{1/4})$. Whereas the Stieltjes constants are very closely related to derivatives of the zeta function at $s=1$, we develop a 