1 Main binomial and binomial-harmonic identities

Very well-poised hypergeometric series provide a clue in the study of diophantine properties of the values of the Riemann zeta function \(\zeta (s)\) at positive integers, see e.g. [4, 7, 8], to quote only a few papers dealing with this important topic. Further references can be found in the bibliography of [4].

In the recent paper [5], we observed that the linear forms

and

$$\begin{aligned} n! \sum _{k=1}^\infty \frac{(k-n)_n}{(k)_{n+1}^2} = A_n \zeta (2) + B_n, \qquad A_n, B_n \in \mathbb {Q}, \end{aligned}$$

have two common coefficients, namely

$$\begin{aligned} A_n=\alpha _n \qquad \text {and} \qquad B_n=\beta _n. \end{aligned}$$

Here and in the sequel, if \(m>0\) is an integer, and \((x)_0=1\). We denote by \(\zeta (s)\) and \(\widetilde{\zeta }(s)\) the following:

The equality \(A_n=\alpha _n\) was noted earlier by Paule and Schneider [6], and is a special case of [4, Proposition 1].

In the present paper we adapt the methods of [5] to the linear forms

and

It seems reasonable to expect that one can solve [9, Problem 1] by similar methods. We assume that the reader is familiar with the background contained in [4, Section 2]. In particular, we have

$$\begin{aligned} \sigma _n&= -\sum _{j=0}^n \frac{\mathrm{d}}{\mathrm{d} j} \biggl (\frac{n}{2} - j\biggr ) \left( {\begin{array}{c}n\\ j\end{array}}\right) {\phantom {\biggr )}}^{6} \nonumber \\&=\sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) {\phantom {\biggr )}}^{6} \biggl ( 1 - 6\, \biggl ( \frac{n}{2} - j \biggr ) ( H_{n-j} - H_{j} ) \biggr ), \end{aligned}$$
(1)
$$\begin{aligned} \tau _n&=-\,\frac{1}{3!}\sum _{j=0}^n \frac{\mathrm{d^3}}{\mathrm{d} j^3} \biggl (\frac{n}{2} - j \biggr )\left( {\begin{array}{c}n\\ j\end{array}}\right) {\phantom {\biggr )}}^{6} \nonumber \\&= - \sum _{j=0}^n \left( {\begin{array}{c}n\\ j\end{array}}\right) {\phantom {\biggr )}}^{6} \biggl ( 36 \,\biggl ( \frac{n}{2} - j \biggr ) ( H_{n-j} - H_{j})^3\nonumber \\&\quad \qquad \qquad \qquad \qquad + 18\, \biggl ( \frac{n}{2} - j \biggr ) ( H_{n-j} - H_{j} ) \bigl (H_{n-j}^{(2)} + H_{j}^{(2)}\bigr )\nonumber \\&\quad \qquad \qquad \qquad \qquad + 2\,\biggl ( \frac{n}{2} - j \biggr ) \bigl ( H_{n-j}^{(3)} - H_{j}^{(3)}\bigr ) \nonumber \\ {}&\quad \qquad \qquad \qquad \qquad - 18 ( H_{n-j} - H_{j} )^2 - 3\bigl ( H_{n-j}^{(2)} + H_{j}^{(2)}\bigr )\biggr ), \end{aligned}$$
(2)
$$\begin{aligned} S_n&= (-1)^n \sum _{j=0}^n \left( {\begin{array}{c}n+j\\ n\end{array}}\right) \left( {\begin{array}{c}n\\ j\end{array}}\right) {\phantom {\biggr )}}^{2} \left( {\begin{array}{c}2n-j\\ n\end{array}}\right) , \end{aligned}$$
(3)
$$\begin{aligned} T_n&= (-1)^n \sum _{j=0}^n \left( {\begin{array}{c}n+j\\ n\end{array}}\right) \left( {\begin{array}{c}n\\ j\end{array}}\right) {\phantom {\biggr )}}^{2} \left( {\begin{array}{c}2n-j\\ n\end{array}}\right) \nonumber \\&\qquad \qquad \qquad \,\cdot \bigl ( H_{j}^{(2)} + H_{j} \bigl (3(H_{j} - H_{n-j}) + H_{2n-j} - H_{n+j}\bigr )\bigr ). \end{aligned}$$
(4)

Throughout this paper,

and the notation for the derivative \(\mathrm{d}/\mathrm{d} j\) is taken from [4, (7.2)].

The main result of the present paper is the following:

Theorem 1.1

We have

$$\begin{aligned} \sigma _n&= S_n, \end{aligned}$$
(5)
$$\begin{aligned} \tau _n&= 2 T_n. \end{aligned}$$
(6)

Despite the analogy between the equalities \(\tau _n=2T_n\) in (6) and \(\beta _n=B_n\) in the paper [5] quoted above, we currently miss a unified proof. However, both (5) and (6), and similar observations in [5], are implicitely connected to the period structure of some multiple integrals (see [3, Section 9.5]). In particular, the linear forms \(\mathrm{\Lambda }_n\) (respectively, \(\mathrm{\Theta }_n\)) in Sect. 2, and even more general linear forms, are equal to suitable 3-fold (respectively, 5-fold) multiple integrals over (respectively, over ), and similar remarks hold for the linear forms in 1 and \(\zeta (2)\) and in \(1,\zeta (2)\) and \(\zeta (4)\) in [5]. All the integrals alluded to above are period integrals on moduli spaces (see [3, Section 1.3]).

In Sect. 2 we provide more details on the above linear forms and coefficients, and in Sects. 3–4 we prove Theorem 1.1.

By combining (5) with another special case of [4, Proposition 1], we have

Theorem 1.2

(7)

We give an independent proof of (7) in the last section of the present paper.

2 Linear forms in \(1,\zeta (2)\) and in \(1,\zeta (3)\) and \(\zeta (5)\)

The following series is a linear form in 1 and \(\zeta (2)\) with rational coefficients:

It is worth noticing that \(\mathrm{\Lambda }_n\) is a Saalschützian \({}_4 F_3\) well-poised hypergeometric series

Throughout the present paper we use identities between values of the function \({}_{q+1} F_q\) with the argument \(z=1\), which is customary omitted.

We have

whence \(S_n\) is the right-hand side of (3) and, similarly, \(T_n\) is given as the right-hand side of (4).

The next series is a linear form in \(1,\zeta (3)\) and \(\zeta (5)\) with rational coefficients:

where

By exchanging j with \(n-j\), i.e. by inverting the order of summation, we have

Therefore \(\sigma _n\) equals the left-hand side of (1), and similarly \(\tau _n\) is given by one of the two equivalent sums in (2).

3 Application of Whipple’s transformation

We apply the following transformation formula, due to Whipple (see [2, 4.3 (4)]):

(8)

The coefficient \(\sigma _n\) can be written as

By applying (8), we obtain

Since

(9)

we have

Therefore (5) is proved.

Let \(a,b,c,d,\alpha ,\beta ,\gamma ,\delta \) be eight complex parameters to be chosen later. We consider the following functions of \(\varepsilon \):

We have

and similar expressions for the second and third order derivatives of \(f_{n,j}(\varepsilon )\) at \(\varepsilon =0\). With the choice \(a=\beta =1+i\), \(b=\alpha =1-i\), \(c=d=\gamma =\delta =1\), where \(i=\sqrt{-1}\), we have

Therefore,

Application of (8) yields

Using (9) again,

Taking

and computing its first and second derivatives at \(z=0\), after a few simplifications we obtain

4 End of the proof of Theorem 1.1

In this section we denote by \(a,b,\alpha ,\beta \) four real parameters to be chosen later. Let \(h_n(\varepsilon ,\omega )\) be defined by

By applying (see [2, 7.2 (1)])

valid for \(u+v+w=x+y+z-n+1\), with

and

we have

Here we used \((\xi )_n = (-1)^n (1-n-\xi )_n\) with \(\xi =v-z\) and \(\xi =w-z\). By choosing \(a=-1\), \(b=-2\), \(\alpha =1\), \(\beta =-1\) and comparing the two expressions of

we find out that

By using

with \(k=0\) and \(k=n\), the above sum simplifies to

hence

Since

we have

and (6) is proved.

5 Application of Sheppard’s transformation

In this section we give a direct proof of (7). A similar argument was applied in [4, Section 7] to the double binomial sum in the middle of [4, (7.1)].

We start with the double sum in (7), and rewrite it in the form

(10)

Let us apply Sheppard’s transformation (see [1, Corollary 3.3.4] and [2, Section 3.9]):

We obtain

Hence the sum (10) is equal to

Exchanging the order of summation and using

the last double sum becomes

The inner sum \({}_2 F_1\) can be evaluated by the Chu–Vandermonde convolution formula (see e.g. [2, Section 1.3]):

Therefore (7) is established.