The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-called Gregory coefficients or the so-called Schröder's integral formula. Having these identities that I evoke, as a relationships for the integer arguments of the Riemann zeta function $\zeta(n)$ the best mathematical content.
Gregory coefficients are an interesting sequence. See if you need it the Wikipedia article for Gregory coefficients. In my aproach I didn't exploit the asymptotic, or other elements as recurrence formulas.
Question. My attempts are showed in next examples, as I've said I would to know if it is possible to state formulas similar than mine, with the best mathematical content or interesting in themselves. I don't know if my approach and elaboration is satisfactory (my strategy was exploit well known integral representations for $\zeta(3)$ and try make more calculations hoping to find sometthing more interesting). What is your appoach to get relationships for $\zeta(3)$, or $\zeta(n)$, and Gregory coefficients $G_n$ (integral representation ofr Gregory coefficients or series involving Gregory coefficients)? Many thanks.
Example 1. Take the derivative of the generating function of Gregory coefficients, after we multiply by $(\log z)^r$ and integrating over the unit interval with the purpose to exploit the formula cited by Janous in [1] in the last paragraph of the proof of his Theorem 2.1 (he cited it as [2]), we get $$-(1)^r r!=(-1)^r r!\zeta(r+1)-\sum_{n=1}^\infty|G_n|\int_0^1 z^n\left(\frac{n\log(1-z)}{z}-\frac{1}{1-z}\right)dz.$$
Example 2. A more elaborated example is write the generating function for the Gregory coefficients as $$-1=\frac{\log(1-z)}{z}-\sum_{n=1}^\infty|G_n|\log(1-z)z^{n-1},$$ after we multiply by $\log z$ and integrating over the unit interval exploiting Beukers integrals ([3]) we get
$$\zeta(3)=1-\sum_{n=1}^\infty|G_n|\sum_{k=1}^\infty\frac{1}{k}\int_0^1(\log z)z^{n+k-1}dz.$$ Finally after integrating we use the definition of the Hurwitz Lerch transcendent and Schröder's integral formula ([4]) I wrote previous identity as $$\zeta(3)=1+\int_0^\infty\left(\sum_{k=1}^\infty\frac{1}{k}\Phi\left(\frac{1}{1+x},2,k+1\right)\right)\frac{dx}{(1+x)(\pi^2+\log^2 x)}.$$
Example 3 (Similar than first example, we conclude that it is the same than previous example). Similarly if there aren't mistakes, from the derivative of the generating function of the Gregory coefficients
$$-1=\frac{-1}{1-\omega}-\sum_{n=1}^\infty|G_n|\omega^n\left(\frac{n\log(1-\omega)}{\omega}-\frac{1}{1-\omega}\right),$$ for $|\omega|<1$, we make the specialization $\omega=xyz$, then integrating for the unit cube, exploiting the known integral representation for the Apéry's constant (see the first formula from this section of the Wikipedia Apéry's constant) we obtain
$$-1=-\zeta(3)+\sum_{n=1}^\infty\left(\sum_{k=1}^\infty\frac{n}{k}\frac{1}{(n+k)^3}+\sum_{k=1}^\infty\frac{1}{(n+k)^3}\right).$$ We observe that this identity is the same that the series showed in Example 2.
References:
[1] Whalther Janous, AROUND APÉRY’S CONSTANT, Journal of Inequalities in Pure and Applied Mathematics, Volume 7, Issue 1, Article 35 (2006).
[2] S.R. Finch, Mathematical Constants, page 43, Cambridge University Press, Cambridge (2003).
[3] F. Beukers, A Note on the Irrationality of $\zeta(2)$ and $\zeta(3)$, Bulletinf of the London Mathematical Society, 11 (3), 268–272 (1979).
[4] I. V. Blagouchine, A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind, Journal of Integer Sequences, Vol. 20, No. 3 (2017).
or
E. Schröder, Bestimmung des infinitären Werthes des Integrals $\int_0^1 (u)_n du$, Z. Math. Phys., 25 (1880), 106–117.
