Referring to this question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry.

For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ and $\zeta(s+1)$, or $\zeta(s)$, $\zeta(s+1)$, and $\zeta(s+2)$ or in general $\zeta(s)$, $\zeta(s+1)$, $\zeta(s+2)$, ..., $\zeta(s+n)$ for $\Re(s) > 0$ and for $n \in \mathbb{N}$?

*My main motivation behind asking this question is I have found such an equation, but I do not know whether such an equation exists in literature. Also, I do not want to appear as if I am promoting my formula here, but rather I am more interested in the works that have been done in such directions.*

*My formula goes as follows: For any $n \in \mathbb{N}$ and $\Re(s) > 0$ we have*
$$\frac{1}{s-1} + \sum_{r=1}^n \binom{n}{r} (-1)^r \left(\frac{\zeta(s+r-1)}{s+n-1} + \int_1^\infty\frac{\rho(x)^r}{x^{s+r}}\mathrm{d}x\right) = 0$$
*where $\binom{n}{r}$ is the binomial coefficient and $\rho(x)$ is the fractional part of $x$*

*For example, putting $n=1$ we get the well known identity,*
$$\frac{1}{s-1} - \frac{\zeta(s)}{s} - \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x = 0$$
*putting $n=2$,*
$$\frac{1}{s-1} - 2\left(\frac{\zeta(s)}{s+1} + \int_1^\infty\frac{\rho(x)}{x^{s+1}}\mathrm{d}x\right) + \left(\frac{\zeta(s+1)}{s+1} + \int_1^\infty\frac{\rho(x)^2}{x^{s+2}}\mathrm{d}x\right) = 0$$
*and so on...*

EDIT: Classic answer by Juan. This question is now solved.