Binomial sums related to rational approximations to \(\zeta(4)\).

*(English. Russian original)*Zbl 1070.11029
Math. Notes 75, No. 4, 594-597 (2004); translation from Mat. Zametki 75, No. 4, 637-640 (2004).

In “An Apéry-like difference equation for Catalan’s constant” [Electron. J. Combin. 10, No. 10, R14 (2003; Zbl 1093.11075)], the author considered the following series
\[
S_n=\sum_{k=1}^{\infty} \frac{\partial^2}{\partial k ^2} \left(\bigg(k+\frac{n}{2}\bigg) \frac{(k-1)\dots (k-n)(k+n+1)\dots(k+2n)}{k^2(k+1)^2\dots (k+n)^2} \right)
\]
and showed that there exist two explicit sequences \((u_n)_{n\geq 0}\) and \((v_n)_{n\geq 0}\) of rationals such that \(S_n=u_n \zeta(4)-v_n\) and \(d_nu_n\in {\mathbb Z}\) and \(d_n^5v_n\in {\mathbb Z}\) where \(d_n=\text{lcm}\{1,2,\dots, n\}.\) Such a series falls under the scope of a conjecture made by the reviewer [in “Séries hypergéométriques et irrationalité des valeurs de la fonction zêta de Riemann”, J. Théor. Nombres Bordx. 15, 351–365 (2003; Zbl 1041.11051)] concerning the denominators of the rational coefficients of certains linear forms in odd or even zeta values obtained by the use of very-well-poised hypergeometric series (of which \(S_n\) is an example). In this case, the conjecture asserts that in fact \(u_n\in{\mathbb Z}\) and \(d_n^4v_n\in {\mathbb Z}\), something that cannot be trivially deduced from the explicit expressions obtained for both sequences.

This conjecture was proved by C. Krattenthaler and the reviewer [in “Hypergéométrie et fonction zêta de Riemann”, preprint available at http://arxiv.org/abs/math.NT/0311114] by proving suitable alternative expressions for the sequences of rationals which appear. For example, in the case of \(\zeta(4)\), for all \(n\geq0\), one has \[ u_n=\sum_{0\leq i\leq j \leq n} \binom{n}{i}^2 \binom{n}{j}^2 \binom{n+j}{n} \binom{n+j-i}{n} \binom{2n-i}{n}. \tag{1} \] The solution relied, amongst other things, on a huge identity relating a single sum and a multiple sum, both of hypergeometric shape.

In the paper under review, another proof of (1) is given using a more compact hypergeometric identity of G. E. Andrews [in “Problems and prospects for basic hypergeometric functions”, Theory and application of special functions, 191–224 (1975; Zbl 0342.33001)], which also relates a single sum and a multiple sum: it appeared later that the huge identity and Andrews’s one are nothing but the same identity. Using certain symmetries in Andrews’s formula, the author also provides five alternative expressions for \(u_n\) similar to (1), such as \[ u_n=(-1)^n\sum_{0\leq i\leq j\leq n}(-1)^j\binom{n+i}{n}^3\binom{3n+1}{j-i}\binom{2n-j}{n}^3. \] This is an interesting observation, which could be useful to prove the general conjecture made by the author in [“Arithmetic of linear forms involving odd zeta values”, J. Théor. Nombres Bordx. 16, 251–291 (2004; Zbl 1156.11327)].

This conjecture was proved by C. Krattenthaler and the reviewer [in “Hypergéométrie et fonction zêta de Riemann”, preprint available at http://arxiv.org/abs/math.NT/0311114] by proving suitable alternative expressions for the sequences of rationals which appear. For example, in the case of \(\zeta(4)\), for all \(n\geq0\), one has \[ u_n=\sum_{0\leq i\leq j \leq n} \binom{n}{i}^2 \binom{n}{j}^2 \binom{n+j}{n} \binom{n+j-i}{n} \binom{2n-i}{n}. \tag{1} \] The solution relied, amongst other things, on a huge identity relating a single sum and a multiple sum, both of hypergeometric shape.

In the paper under review, another proof of (1) is given using a more compact hypergeometric identity of G. E. Andrews [in “Problems and prospects for basic hypergeometric functions”, Theory and application of special functions, 191–224 (1975; Zbl 0342.33001)], which also relates a single sum and a multiple sum: it appeared later that the huge identity and Andrews’s one are nothing but the same identity. Using certain symmetries in Andrews’s formula, the author also provides five alternative expressions for \(u_n\) similar to (1), such as \[ u_n=(-1)^n\sum_{0\leq i\leq j\leq n}(-1)^j\binom{n+i}{n}^3\binom{3n+1}{j-i}\binom{2n-j}{n}^3. \] This is an interesting observation, which could be useful to prove the general conjecture made by the author in [“Arithmetic of linear forms involving odd zeta values”, J. Théor. Nombres Bordx. 16, 251–291 (2004; Zbl 1156.11327)].

Reviewer: Tanguy Rivoal (Grenoble)