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Dating back to Euler, in classical analysis and number theory, the Hurwitzzeta function $$ \zeta(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ theRiemann zeta function $\zeta(z)$, the generalized Stieltjes constants$\gamma_k(q)$, the Euler constant $\gamma$, Euler's gamma function $\Gamma(q)$and the digamma function $\psi(q)$ have many close connections on theirdefinitions and properties. There are also many integrals, series or infiniteproduct representations of them along the history.  In this note, we try to provide a parallel story for the alternating Hurwitzzeta function (also known as the Hurwitz-type Euler zeta function)$$\zeta_{E}(z,q)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(n+q)^{z}},$$ the alternatingzeta function $\zeta_{E}(z)$ (also known as the Dirichlet's eta function$\eta(z)$), the modified Stieltjes constants $\tilde\gamma_k(q)$, the modifiedEuler constant $\tilde\gamma_{0}$, the modified gamma function$\tilde\Gamma(q)$ and the modified digamma function $\tilde\psi(q)$ (also knownas the Nielsen's $\beta$ function). Many new integrals, series or infiniteproduct representations of these constants and special functions have beenfound. By the way, we also get two new series expansions of $\pi:$\begin{equation*} \frac{\pi^2}{12}=\frac34-\sum_{k=1}^\infty(\zeta_E(2k+2)-1)\end{equation*} and \begin{equation*} \frac{\pi}{2}=\log2+2\sum_{k=1}^\infty\frac{(-1)^k}{k!}\tilde\gamma_k(1)\sum_{j=0}^kS(k,j)j!.\end{equation*}

On the Stieltjes constants and gamma functions with respect to alternating Hurwitz zeta functions