*[This question is copied from math.stackexchange, it didn't get answers so far]*

For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as $$ f(x) = \zeta(1-x) + 1/x \qquad x \in \mathbb N $$ and then the scaled derivatives $$ g(x,d) = x^d \cdot {f^{(d)}(x)\over d!} $$.

In Pari/GP there is a very nice procedure for the computation of $\zeta$ implemented, but I don't see anything for the derivatives except to implement numerical differentiation along the line ${f(x+\epsilon/2) - f(x-\epsilon/2)\over \epsilon}$ and so on, but which becomes inaccurate for higher derivatives.

For small *x* I can use the representation of the function $f(x)$ as a power series involving the Stieltjes-constants themselves, but a suitable computation is then possible only for a small range of *x*-values if I use only, say 100 or 200 terms of the power series.

Q: What would be a more efficient representation of $f(x)$ and/or $g(x)$ for the actual computation, where -say- *x* and *d* both can go up to 100 or 200 ? Note, that for high *c* (the index for the partial sums, see below) there is the significant effect , that either (x is small and d is high) or converse, such that *d+x* in the terms of one partial sum is constant.

More context: what I'm finally after is to be able to explore the partial sums and its single terms in

$$ S \underset{\mathfrak E}{=} \lim_{c\to \infty} S(c) = \sum_{k=0}^{c} (-1)^k g( c+1-k ,k) $$ which should be a representation for the (divergent) sum of the Stieltjes constants, taken by some

*"Eulerian"*summation procedure $\mathfrak E$ (see an earlier post here on MO).