In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$ A variation of the above identity arises by considering the Taylor series of the Hurwitz zeta function. As $$ \frac{\partial}{\partial q} \zeta(s,q) = -s \zeta(s+1,q), \qquad \qquad (2)$$ we obtain \begin{align}\zeta(s,x+y) &= \sum_{k=0}^{\infty} \frac{y^{k}}{k!} \frac{\partial^{k}}{\partial x^{k}} \zeta(s,x) \\ &= \sum_{k=0}^{\infty} \binom{s+k-1}{s-1} (-y)^{k} \zeta(s+k,x). \qquad \qquad(3) \end{align} We could differentiate functions that are similar to the one from $(2)$ and see what happens. For instance, define $$ \zeta_{2}(s,s;a) := \sum_{n=0}^{\infty} \frac{1}{(n+1+a)^{s}} \sum_{m=0}^{n}\frac{1}{(m+a)^{s}} . \qquad \qquad (4)$$ This is a generalization of the Multiple Zeta function where we set $s_{1} = s_{2} = s$. Then:
$$\frac{\partial}{\partial a} \zeta_{2}(s,s;a) = -s(\zeta_{2}(s+1,s;a) + \zeta_{2}(s,s+1;a) ) . \qquad \qquad (5)$$ Here we see the shifts that are similar to the one found in $(2)$. We could do something like this for a generalization of the Mordell-Tornheim zeta function as well.
Question
Have equations like $$\frac{\partial}{\partial a} f(s,s;a) = -s(f(s+1,s;a) + f(s,s+1;a) ) $$ been studied before, from the perspective of differential-difference equations? Or have they perhaps been considered in the umbral calculus?
