On p. 263 of Borwein's paper entitled “Computational Strategies for the Riemann zeta function”, the following identity is stated: $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1 . \qquad \qquad (*)$$It is valid for all nonnegative integer $k$.

Can the identity be generalised and/or are there variations of it involving binomial sums which are also valid for infinitely many nonnegative lower bounds $k$?

I suppose one can rewrite the formulas of Landau: $$ \frac{1}{s-1} = \sum_{n=0}^{\infty} \binom{s+n-1}{n-1} \frac{\zeta(s+n)-1}{n} $$ and Ramaswami: $$ (1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \binom{s+n-1}{n} \zeta(s+n)$$ in a manner that is similar to $(*)$, but I'm looking for both more and more general examples of such sums. Preferably, all sums in the collection amount to the same constant.

On p. 263 of Borwein's paper entitled “Computational Strategies for the Riemann zeta function”, the following identity is stated: $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1 . \qquad \qquad (*)$$It is valid for all nonnegative integer $k$.

Can the identity be generalised and/or are there variations of it involving binomial sums which are also valid for infinitely many nonnegative lower bounds $k$?

I suppose one can rewrite the formulas of Landau: $$ \frac{1}{s-1} = \sum_{n=0}^{\infty} \binom{s+n-1}{n-1} \frac{\zeta(s+n)-1}{n} $$ and Ramaswami: $$ (1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \binom{s+n-1}{n} \zeta(s+n)$$ in a manner that is similar to $(*)$, but I'm looking for both more and more general examples of such sums. Preferably, all sums in the collection amount to the same constant and involve binomial coefficients.