$\gamma_0$ in terms of $\gamma_{1-Inf}$

Question

I’ve numerically observed that $\sum_{n=1}^{\infty}\gamma_n/n!$ = $1/2 - \gamma_0$ , where $\gamma$ are the Stieltjes constants.

Is there a recurrence explanation for this or a known proof?

Answer 1

Well, we have $$\zeta(z) = \frac1{z-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!}\gamma_n (z-1)^n$$ (see MathWorld) and so setting $z=0$ we get $$-\frac12 = -1 + \sum_{n=0}^\infty \frac{\gamma_n}{n!}.$$ The first term of the sum is $\gamma_0$, so moving it to the other side along with the constants gives us $$\frac12 - \gamma_0 = \sum_{n=1}^\infty \frac{\gamma_n}{n!}.$$