HYPERGEOMETRY INSPIRED BY IRRATIONALITY QUESTIONS

CHRISTIAN KRATTENTHALER AND WADIM ZUDILIN

Abstract. We report new hypergeometric constructions of rational approximations to Catalan’s constant, log 2, and π^{2}, their connection with already known ones, and underlying ‘permutation group’ structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann’s zeta function at odd integers.

1. Introduction

Given a real (presumably irrational!) number γ, how can one prove that it is irrational? In certain cases (like for square roots of rationals) this is an easy task. A more general strategy proceeds by the construction of a sequence of rational approximations r_{n} = q_{n}γ −p_{n} ≠ 0 such that δ_{n}q_{n}, δ_{n}p_{n} are integers for some positive integers δ_{n} and δ_{n}r_{n} → 0 as n → ∞. This indeed guarantees that γ is not rational. Usually, as a bonus, such a construction also allows one to estimate the irrationality of γ in a quantitative form.

Producing such a sequence of rational Diophantine approximations, even with a weaker requirement on the growth, like r_{n} → 0 as n → ∞, is a difficult problem. For certain specific ‘interesting’ numbers γ ∈ ℝ such sequences are constructed as values of so-called hypergeometric functions; for related definitions of the latter in the ordinary and basic (q-) situations we refer the reader to the books [1, 14, 4]. One of the underlying mechanisms behind the hypergeometric settings is the existence of numerous transformations of hypergeometric functions, that is, identities that represent the same numerical (or q-) quantity in different looking ways. An arithmetic significance of such transformations is the production of identities of the form r_{n} = r̃_{n} say, where r_{n} = q_{n}γ −p_{n} and r̃_{n} = q̃_{n}γ −p̃_{n} for n = 0, 1, 2, …, while an analysis of the asymptotic behaviour of r_{n} or r̃_{n}, and of the corresponding (a priori different) denominators δ_{n} or δ̃_{n} are simpler for one of them than for the other. In several situations, the machinery can be inverted: the equality r_{n} = r̃_{n} is predicted by computing a number of first approximations, and then established by demonstrating that both sides satisfy the same linear recursion. Such instances naturally call for finding purely hypergeometric proofs, which in turn may offer more general forms of the approximations. It comes as no surprise that our computations below have been carried out using the Mathematica packages HYP and HYPQ [7].

2010 Mathematics Subject Classification. 11J72, 11M06, 11Y60, 33C20, 33D15, 33F10.

Key words and phrases. Irrationality; zeta value; π; Catalan’s constant; log 2; hypergeometric series.

The first author is partially supported by the Austrian Science Foundation FWF, grant S50-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”.

The symbiosis of arithmetic and hypergeometry is the main objective of the present note, with special emphasis on (hypergeometric) rational approximations to the following three mathematical constants (in order of their appearance below):

• Catalan’s constant G = ∑_{k=0}^{∞} \frac{(−1)^{k}}{(2k + 1)^{2}} ,

• log 2 = ∑_{k=1}^{∞} \frac{(−1)^{k−1}}{k}, and

• \frac{π^{2}}{6} = ζ(2) = ∑_{k=1}^{∞} \frac{1}{k^{2}} ,

which are discussed in Sections 2, 3, and 4, respectively. We intentionally personify these mathematical constants here, to stress their significance in the arithmetichypergeometric context.

The construction in Section 4 indicates a certain cancellation phenomenon, which we record in Lemma 1. Application of this new ingredient to a general construction of linear forms in the values of Riemann’s zeta function ζ(s) at positive odd integers leads to the following result.

Theorem 1. For any λ ∈ ℝ, each of the two collections

\{ζ(2m + 1) − λ \frac{2^{2m}(2^{2m+2} − 1)|B_{2m+2}|}{(2^{2m+1} − 1)(m + 1)(2m)!} π^{2m+1} : m = 1, 2, …, 19\}

and

\{ζ(2m + 1) − λ \frac{2^{2m}(2^{2m} − 1)|B_{2m}|}{(2^{2m+1} − 1)m(2m)!} π^{2m+1} : m = 1, 2, …, 21\}

contains at least one irrational number. Here B_{2m} denotes the 2m-th Bernoulli number.

We prove this theorem in Section 5. Notice that

\frac{2^{2m−1}|B_{2m}|}{(2m)!} = \frac{ζ(2m)}{π^{2m}} ∈ ℚ for m = 1, 2, ….

The only result in the literature we can compare our Theorem 1 with is the one given in [6, Theorems 3 and 4], which implies the irrationality of at least one number in each collection

\{ζ(2m + 1) − λ \frac{2^{2m}|B_{2m}|}{m(2m)!} π^{2m+1} : m = 1, 2, …, 169\}

and

\{ζ(2m + 1) − λ \frac{2^{2m}|B_{2m+2}|}{(m + 1)(2m)!} π^{2m+1} : m = 1, 2, …, 169\},

where λ ∈ ℝ is arbitrary.

Acknowledgement

We thank Victor Zudilin for beautifully portraying the mathematical constants involved here. We kindly acknowledge the referee’s very attentive reading of the original version.

2. Catalan’s constant

A long time ago, in joint work with T. Rivoal [11], the second author considered very-well-poised hypergeometric series that represent linear forms in Catalan’s and related constants. The approximations to Catalan’s constant itself were given by

r_{n} = ∑_{t=0}^{∞}(2t + n + 1)\frac{n! ∏_{j=1}^{n}(t + 1 − j) ∏_{j=1}^{n}(t + n + j)}{∏_{j=0}^{n}(t + j + \frac{1}{2})^{3}} (−1)^{n+t}

= \frac{\sqrt{π} Γ(3n + 2) Γ(n + \frac{1}{2})^{2}Γ(n + 1)}{4^{n} Γ(2n + \frac{3}{2})^{3}}

×_{6}F_{5}[\begin{matrix}3n + 1, \frac{3n}{2} + \frac{3}{2}, n + \frac{1}{2}, n + \frac{1}{2}, n + \frac{1}{2}, n + 1 \\ \frac{3n}{2} + \frac{1}{2}, 2n + \frac{3}{2}, 2n + \frac{3}{2}, 2n + \frac{3}{2}, 2n + 1 \end{matrix}; −1].

The approximations possess different hypergeometric forms, for example, as a_{3}F_{2}(1)- series and as a Barnes-type integral as discussed in [17] and [18].

The use of partial-fraction decomposition in [17] suggests considering a different family of approximations:

r̃_{n} = 2^{2(n+1)} ∑_{t=1}^{∞}(2t − 1)\frac{(2n + 1)! ∏_{j=0}^{2n−1}(t − n + j)}{∏_{j=0}^{2n+1}(2t − n − \frac{3}{2} + j)^{2}}

= \frac{2^{2(n+1)}Γ(2n + 2)^{2}Γ(n + \frac{1}{2})^{2}}{Γ(3n + \frac{5}{2})^{2}}

×_{6}F_{5}[\begin{matrix}2n + 1, n + \frac{3}{2}, \frac{n}{2} + \frac{1}{4}, \frac{n}{2} + \frac{1}{4}, \frac{n}{2} + \frac{3}{4}, \frac{n}{2} + \frac{3}{4} \\ n + \frac{1}{2}, \frac{3n}{2} + \frac{7}{4}, \frac{3n}{2} + \frac{7}{4}, \frac{3n}{2} + \frac{5}{4}, \frac{3n}{2} + \frac{5}{4} \end{matrix}; 1].

This is again a very-well-poised _{6}F_{5}-series, but this time evaluated at 1. In addition, it is reasonably easy to show that 2^{4n}d_{2n−1}^{2}r̃_{n} ∈ ℤ + ℤ G, where d_{N} denotes the least common multiple of 1, …, N, using an argument similar to the one in [17].

Amazingly, we have r_{n} = r̃_{n}, which accidentally came out of the recursion satisfied by r̃_{n}. Our first result is a general identity, of which the equality is a special case (namely, c = d = n + \frac{1}{2}).

4 CHRISTIAN KRATTENTHALER AND WADIM ZUDILIN

Theorem 2. We have

6F_{5}[\begin{matrix}3n + 1, \frac{3n}{2} + \frac{3}{2}, n + \frac{1}{2}, n + 1, c, d \\ \frac{3n}{2} + \frac{1}{2}, 2n + \frac{3}{2}, 2n + 1, 3n + 2 − c, 3n + 2 − d \end{matrix}; −1]

= \frac{Γ(4n + 3) Γ(3n + 2 − c) Γ(3n + 2 − d) Γ(4n + 3 − c − d)}{Γ(3n + 2) Γ(4n + 3 − c) Γ(4n + 3 − d) Γ(3n + 2 − c − d)}

×_{6}F_{5}[\begin{matrix}2n + 1, n + \frac{3}{2}, \frac{c}{2}, \frac{c}{2} + \frac{1}{2}, \frac{d}{2}, \frac{d}{2} + \frac{1}{2} \\ n + \frac{1}{2}, 2n + 2 − \frac{c}{2}, 2n + \frac{3}{2} − \frac{c}{2}, 2n + 2 − \frac{d}{2}, 2n + \frac{3}{2} − \frac{d}{2} \end{matrix}; 1]. (1)

Proof. We start with Rahman’s quadratic transformation [9, Eq. (7.8), q → 1, reversed]

8F_{7}[\begin{matrix}2a − e, 1 + a − \frac{e}{2}, \frac{1}{2} + a − e, c, d, e, \\ a − \frac{e}{2}, \frac{1}{2} + a, 1 + 2a − c − e, 1 + 2a − d − e, 1 + 2a − 2e, \end{matrix}

\begin{matrix}1 + 4a − c − d − e + n, −n \\ −2a + c + d − n, 1 + 2a − e + n \end{matrix}; 1]

= \frac{(1 + 2a − c)_{n} (1 + 2a − d)_{n} (1 + 2a − e)_{n} (1 + 2a − c − d − e)_{n}}{(1 + 2a)_{n} (1 + 2a − c − d)_{n} (1 + 2a − c − e)_{n} (1 + 2a − d − e)_{n}}

×_{11}F_{10}[\begin{matrix}a, 1 + \frac{a}{2}, e, \frac{c}{2}, \frac{1}{2} + \frac{c}{2}, \frac{d}{2}, \frac{1}{2} + \frac{d}{2}, \\ \frac{a}{2}, 1 + a − e, 1 + a − \frac{c}{2}, \frac{1}{2} + a − \frac{c}{2}, 1 + a − \frac{d}{2}, \frac{1}{2} + a − \frac{d}{2}, \end{matrix}

\begin{matrix}\frac{1}{2} + 2a − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} + \frac{n}{2}, 1 + 2a − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} + \frac{n}{2}, \frac{1}{2} − \frac{n}{2}, −\frac{n}{2} \\ \frac{1}{2} − a + \frac{c}{2} + \frac{d}{2} + \frac{e}{2} − \frac{n}{2}, −a + \frac{c}{2} + \frac{d}{2} + \frac{e}{2} − \frac{n}{2}, \frac{1}{2} + a + \frac{n}{2}, 1 + a + \frac{n}{2} \end{matrix}; 1],

in which we let n tend to ∞:

6F_{5}[\begin{matrix}2a − e, 1 + a − \frac{e}{2}, \frac{1}{2} + a − e, c, d, e \\ a − \frac{e}{2}, \frac{1}{2} + a, 1 + 2a − c − e, 1 + 2a − d − e, 1 + 2a − 2e \end{matrix}; −1]

= \frac{Γ(1 + 2a) Γ(1 + 2a − c − d) Γ(1 + 2a − c − e) Γ(1 + 2a − d − e)}{Γ(1 + 2a − c) Γ(1 + 2a − d) Γ(1 + 2a − e) Γ(1 + 2a − c − d − e)}

×_{7}F_{6}[\begin{matrix}a, 1 + \frac{a}{2}, e, \frac{c}{2}, \frac{1}{2} + \frac{c}{2}, \frac{d}{2}, \frac{1}{2} + \frac{d}{2} \\ \frac{a}{2}, 1 + a − e, 1 + a − \frac{c}{2}, \frac{1}{2} + a − \frac{c}{2}, 1 + a − \frac{d}{2}, \frac{1}{2} + a − \frac{d}{2} \end{matrix}; 1].

Now set a = 2n + 1 and e = n + 1 to deduce (1). □ The corresponding q-version, which we record here for completeness, reads

8ϕ_{7}[\begin{matrix}q^{3n+1}, q^{\frac{3n}{2} + \frac{3}{2}} , −q^{\frac{3n}{2} + \frac{3}{2}} , q^{n+ \frac{1}{2}} , −q^{n+ \frac{1}{2}} , q^{n+1}, c, d \\ q^{\frac{3n}{2} + \frac{1}{2}} , −q^{\frac{3n}{2} + \frac{1}{2}} , q^{2n+ \frac{3}{2}} , −q^{2n+ \frac{3}{2}} , q^{2n+1}, q^{3n+2}/c, q^{3n+2}/d \end{matrix}; q, −\frac{q^{3n+2}}{cd} ]

= \frac{(q^{3n+2}, q^{4n+3}/c, q^{4n+3}/d, q^{3n+2}/cd; q)_{∞}}{(q^{4n+3}, q^{3n+2}/c, q^{3n+2}/d, q^{4n+3}/cd; q)_{∞}}

×_{7}ϕ_{6}[\begin{matrix}q^{4n+2}, q^{2n+3}, −q^{2n+3}, c, cq, d, dq \\ q^{2n+1}, −q^{2n+1}, q^{4n+4}/c, q^{4n+3}/c, q^{4n+4}/d, q^{4n+3}/d \end{matrix}; q^{2}, \frac{q^{6n+4}}{c^{2}d^{2}} ].

3. Logarithm of 2

Another strange identity is related to the classical rational approximations to log 2:

r_{n} = (−1)^{n+1} ∑_{t=0}^{∞} \frac{∏_{j=1}^{n}(t − j)}{∏_{j=0}^{n}(t + j)} (−1)^{t}

= \frac{Γ(n + 1)^{2}}{Γ(2n + 2)} _{2}F_{1}[\begin{matrix}n + 1, n + 1 \\ 2n + 2 \end{matrix}; −1]

= ∫_{0}^{1} \frac{x^{n}(1 − x)^{n}}{(1 + x)^{n+1}} dx.

The sequence satisfies the recurrence equation (n+1)r_{n+1} −3(2n+1)r_{n} +nr_{n−1} = 0, and with the help of the latter we find out that r_{n} = r̃_{n} for

r̃_{n} = ∑_{t=0}^{∞} \frac{(2n + 1)! ∏_{j=1}^{n}(t − j)}{n! ∏_{j=0}^{2n+1}(2t − n − 1 + j)}

= \frac{Γ(n + 1) Γ(2n + 2)}{Γ(3n + 3)} _{3}F_{2}[\begin{matrix}n + 1, \frac{n}{2} + \frac{1}{2}, \frac{n}{2} + 1 \\ \frac{3n}{2} + 2, \frac{3n}{2} + \frac{3}{2} \end{matrix}; 1].

The finding is a particular case of another general identity.

Theorem 3. We have

2F_{1}[\begin{matrix}x, 2a \\ 2b − x \end{matrix}; −1] = \frac{Γ(2b − x) Γ(2b − 2a)}{Γ(2b) Γ(2b − 2a − x)} _{3}F_{2}[\begin{matrix}x, a, a + \frac{1}{2} \\ b, b + \frac{1}{2} \end{matrix}; 1].  (2)

Proof. This is a specialisation of a transformation of Whipple [1, Sec. 4.6, Eq. (3)]: set b = κ − a there and reparametrise. □

A companion q-version is

6ϕ_{7}[\begin{matrix}−b/q, \sqrt{−bq}, −\sqrt{−bq}, x, −x, a \\ \sqrt{−b/q}, \sqrt{−b/q}, −b/x, b/x, −b/a, 0, 0 \end{matrix} ; q, −\frac{b^{2}}{ax^{2}} ]

= \frac{(b^{2}, b^{2}/(ax)^{2}; q^{2})_{∞}}{(b^{2}/a^{2}, b^{2}/x^{2}; q^{2})_{∞}} _{3}ϕ_{2}[\begin{matrix}x^{2}, a, aq \\ b, bq \end{matrix}; q^{2}, \frac{b^{2}}{a^{2}x^{2}} ],

which follows from [4, Eq. (3.10.4)].

To clarify the arithmetic situation behind the right-hand side of (2), we notice that there is a permutation group for it used for producing a sharp irrationality measure of ζ(2) in [10]. As explained in [19, Section 6], a realisation of the group for a generic hypergeometric function

\frac{Γ(a_{2}) Γ(b_{2} − a_{2}) Γ(a_{3}) Γ(b_{3} − a_{3})}{Γ(b_{2}) Γ(b_{3})} _{3}F_{2}[\begin{matrix}a_{1}, a_{2}, a_{3} \\ b_{2}, b_{3} \end{matrix}; 1]  (3)

can be given by means of the ten parameters

c_{00} = (b_{2} + b_{3}) − (a_{1} + a_{2} + a_{3}) − 1,
c_{jk} = \begin{cases}a_{j} − 1, & for k = 1, \\ b_{k} − a_{j} − 1, & for k = 2, 3, \end{cases}

as follows. If the set of parameters is represented in the matrix form

c = (\begin{matrix}c_{00} \\ & c_{11} & c_{12} & c_{13} \\ & c_{21} & c_{22} & c_{23} \\ & c_{31} & c_{32} & c_{33} \end{matrix}) ,  (4)

and H(c) denotes the corresponding hypergeometric function in (3), then the quantity

\frac{H(c)}{Γ(c_{00} + 1) Γ(c_{21} + 1) Γ(c_{31} + 1) Γ(c_{22} + 1) Γ(c_{33} + 1)}  (5)

is invariant under the group 𝔊 (of order 120) generated by the four involutions

𝔞_{1} = (c_{11} c_{21}) (c_{12} c_{22}) (c_{13} c_{23}),  𝔞_{2} = (c_{21} c_{31}) (c_{22} c_{32}) (c_{23} c_{33}),
𝔟 = (c_{12} c_{13}) (c_{22} c_{23}) (c_{32} c_{33}), and 𝔥 = (c_{00} c_{22}) (c_{11} c_{33}) (c_{13} c_{31}).

Notice that the permutations 𝔞_{1}, 𝔞_{2}, and 𝔟 correspond to the rearrangements a_{1} ↔ a_{2}, a_{2} ↔ a_{3}, and b_{2} ↔ b_{3}, respectively, of the function (3), so that the invariance of (5) under their action is trivial. It is only the permutation 𝔥, underlying Thomae’s transformation [1, Sec. 3.2, Eq. (1)] and Whipple’s transformation [1, Sec. 4.4, Eq. (2)], that makes the action of the group on (5) non-trivial.

With the method in [2, Section 3.3], if

a_{1}, a_{2}, b_{2} ∈ ℤ and a_{3}, b_{3} ∈ ℤ + \frac{1}{2}  (6)

are chosen such that c_{jk} ≥ −\frac{1}{2} for all j and k, then the quantity H(c) representing (3) satisfies

H(c) ∈ ℚ log 2 + ℚ.

It is a tough task to produce a sharp integer D(c) such that D(c)H(c) ∈ ℤ log 2 + ℤ in the general case; it can be given in the particular situation where a_{3} − a_{2} = b_{3} − b_{2} = ±\frac{1}{2} with the help of (2) and the known information for the corresponding 2F_{1}(1)-series.

Observe that the group 𝔊 = ⟨𝔞_{1}, 𝔞_{2}, 𝔟, 𝔥⟩ cannot be arithmetically used in its full force when the parameters of (3) are subject to (6). However, apart from the initial representative (4), there are five more with the constraint that entries 1 3, 2 3, 3 1 and 3 2 are from ℤ + \frac{1}{2}, namely

\begin{matrix}(\begin{matrix}c_{22} \\ & c_{33} & c_{12} & c_{31} \\ & c_{21} & c_{00} & c_{23} \\ & c_{13} & c_{32} & c_{11} \end{matrix}) , (\begin{matrix}c_{12} \\ & c_{11} & c_{00} & c_{13} \\ & c_{33} & c_{22} & c_{31} \\ & c_{23} & c_{32} & c_{21} \end{matrix}) , (\begin{matrix}c_{33} \\ & c_{22} & c_{21} & c_{13} \\ & c_{12} & c_{11} & c_{23} \\ & c_{31} & c_{32} & c_{00} \end{matrix}) , \\ (\begin{matrix}c_{11} \\ & c_{00} & c_{21} & c_{31} \\ & c_{12} & c_{33} & c_{23} \\ & c_{13} & c_{32} & c_{22} \end{matrix}) , and (\begin{matrix}c_{21} \\ & c_{22} & c_{33} & c_{13} \\ & c_{00} & c_{11} & c_{31} \\ & c_{23} & c_{32} & c_{12} \end{matrix}) , \end{matrix} (7)

and another six which are obtained from (4) and (7) by further action of 𝔞_{1}.

Remarkably enough, the choices x = 12n + 1, a = 14n + 1, b = 28n + 2 and x = 14n + 1, a = 12n + 1, b = 28n + 2 in (2), which originate from the trivial transformation of the _{2}F_{1}(−1)-side and which correspond to an early (‘pre-Raffaele’ [8]) irrationality measure record [5, 13, 15, 20], produce 𝔊-disjoint collections

( \begin{matrix}14n+1 \\ & 12n+1 & 8n+1 & 8n+ \frac{1}{2} \\ & 7n+1 & 13n+1 & 13n+ \frac{1}{2} \\ & 7n+ \frac{1}{2} & 13n+ \frac{1}{2} & 13n+1 \end{matrix} ) and ( \begin{matrix}16n+1 \\ & 14n+1 & 7n+1 & 7n+ \frac{1}{2} \\ & 6n+1 & 15n+1 & 15n+ \frac{1}{2} \\ & 6n+ \frac{1}{2} & 15n+ \frac{1}{2} & 15n+1 \end{matrix} )

on the _{3}F_{2}(1)-side.

4. π squared

Our next hypergeometric entry a priori produces linear forms not only in 1 and ζ(2) = π^{2}/6 but also in ζ(4) = π^{4}/90, with rational coefficients. It originates from the well-poised hypergeometric series

r_{n} = ∑_{t=1}^{∞} \frac{2^{8n}n!^{4}(2n)!^{2} ∏_{j=0}^{4n−1}(t − n + j)}{(4n)! ∏_{j=0}^{2n}(t − \frac{1}{2} + j)^{4}}

= \frac{π^{2}Γ(2n + 1)^{6}}{Γ(3n + \frac{3}{2})^{4}} _{5}F_{4}[\begin{matrix}4n + 1, n + \frac{1}{2}, n + \frac{1}{2}, n + \frac{1}{2}, n + \frac{1}{2} \\ 3n + \frac{3}{2}, 3n + \frac{3}{2}, 3n + \frac{3}{2}, 3n + \frac{3}{2} \end{matrix}; 1].

It is standard to sum the rational function

R_{n}(t) = \frac{2^{8n}n!^{4}(2n)!^{2} ∏_{j=0}^{4n−1}(t − n + j)}{(4n)! ∏_{j=0}^{2n}(t − \frac{1}{2} + j)^{4}}

by expanding it into the sum of partial fractions; the well-poised symmetry R_{n}(t) = R_{n}(2n − 1 − t) (and the residue sum theorem) imply then that

r_{n} ∈ ℚπ^{4} + ℚπ^{2} + ℚ

for n = 0, 1, 2, …. At the same time,

r_{0} = \frac{1}{6} π^{4}, r_{1} = \frac{19}{6} π^{4} − \frac{125}{4} π^{2},

and the sequence r_{n} satisfies a second order recurrence equation, so that r_{n} = a_{n}π^{4} − b_{n}π^{2} ∈ ℚπ^{4} + ℚπ^{2} for all n. This happens because the function R_{n}(t) vanishes at t = 1, 0, −1, …, −n + 2 so that

r_{n} = ∑_{t=−n+1}^{∞} R_{n}(t),

and in view of the following result.

Lemma 1. Assume that a rational function

R(t) = ∑_{i=1}^{s} ∑_{k=0}^{n} \frac{a_{i,k}}{(t + k)^{i}}

satisfies R(t) = R(−n − t). Put m = ⌊(n − 1)/2⌋. Then a_{i,n−k} = (−1)^{i}a_{i,k} and

∑_{t=−m}^{∞} R(t − \frac{1}{2}) = ∑_{\begin{matrix}i=2 \\ i even \end{matrix}}^{s} a_{i} ∑_{ℓ=1}^{∞} \frac{1}{(ℓ − \frac{1}{2})^{i}} + a_{0}

= ∑_{\begin{matrix}i=2 \\ i even \end{matrix}}^{s} a_{i}(2^{i} − 1)ζ(i) + a_{0} ∈ ℚ + ℚ π^{2} + ℚ π^{4} + ⋯ + ℚ π^{2⌊s/2⌋},

where

a_{i} = ∑_{k=0}^{n} a_{i,k}, for i = 2, …, s, and a_{0} = \begin{cases}0, & for n even, \\ \frac{1}{2} R(−m − \frac{1}{2}), & for n odd. \end{cases}

Proof. The property a_{i,n−k} = (−1)^{i}a_{i,k} is straightforward to see from R(t) = R(−n−t). Furthermore, we have

∑_{t=−m}^{∞} R(t − \frac{1}{2}) = ∑_{i=1}^{s} ∑_{k=0}^{n} a_{i,k} ∑_{t=−m}^{∞} \frac{1}{(t + k − \frac{1}{2})^{i}} = ∑_{i=1}^{s} ∑_{k=0}^{n} a_{i,k} ∑_{ℓ=k−m}^{∞} \frac{1}{(ℓ − \frac{1}{2})^{i}}

= ∑_{i=1}^{s} ∑_{k=0}^{n} a_{i,k} · ∑_{ℓ=1}^{∞} \frac{1}{(ℓ − \frac{1}{2})^{i}}

+ ∑_{i=1}^{s}(∑_{k=0}^{m} a_{i,k} ∑_{ℓ=k−m}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k=m+1}^{n} a_{i,k} ∑_{ℓ=1}^{k−m−1} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

= ∑_{\begin{matrix}i=2 \\ i even \end{matrix}}^{s} a_{i} ∑_{ℓ=1}^{∞} \frac{1}{(ℓ − \frac{1}{2})^{i}} + a_{0},

HYPERGEOMETRY INSPIRED BY IRRATIONALITY QUESTIONS 9

with the constant term equal to

a_{0} = ∑_{i=1}^{s}(∑_{k=0}^{m} a_{i,k} ∑_{ℓ=k−m}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k=m+1}^{n} a_{i,k} ∑_{ℓ=1}^{k−m−1} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

= ∑_{i=1}^{s}(−1)^{i}(∑_{k=0}^{m} a_{i,n−k} ∑_{ℓ=k−m}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k=m+1}^{n} a_{i,n−k} ∑_{ℓ=1}^{k−m−1} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

(take k′ = n − k)

= ∑_{i=1}^{s}(−1)^{i}( ∑_{k′=n−m}^{n} a_{i,k′} ∑_{ℓ=n−m−k′}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k′=0}^{n−m−1} a_{i,k′} ∑_{ℓ=1}^{n−m−1−k′} \frac{1}{(ℓ − \frac{1}{2})^{i}} ).

If n is odd, then n − m = m + 1 and

a_{0} = ∑_{i=1}^{s}(−1)^{i}( ∑_{k′=m+1}^{n} a_{i,k′} ∑_{ℓ=m+1−k′}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k′=0}^{m} a_{i,k′} ∑_{ℓ=1}^{m−k′} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

= ∑_{i=1}^{s}( ∑_{k′=m+1}^{n} a_{i,k′} ∑_{ℓ=1}^{k′−m} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k′=0}^{m} a_{i,k′} ∑_{ℓ=k′−m+1}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

= ∑_{i=1}^{s}( ∑_{k′=m+1}^{n} a_{i,k′} ∑_{ℓ=1}^{k′−m−1} \frac{1}{(ℓ − \frac{1}{2})^{i}} + ∑_{k′=m+1}^{n} a_{i,k′} \frac{1}{(k′ − m − \frac{1}{2})^{i}}

− ∑_{k′=0}^{m} a_{i,k′} ∑_{ℓ=k′−m}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} + ∑_{k′=0}^{m} a_{i,k′} \frac{1}{(k′ − m − \frac{1}{2})^{i}} )

= −a_{0} + ∑_{i=1}^{s} ∑_{k′=0}^{n} \frac{a_{i,k′}}{(k′ − m − \frac{1}{2})^{i}} = −a_{0} + R(−m − \frac{1}{2}).

Similarly, if n is even, then n − m = m + 2 and

a_{0} = ∑_{i=1}^{s}(−1)^{i}( ∑_{k′=m+2}^{n} a_{i,k′} ∑_{ℓ=m+2−k′}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k′=0}^{m+1} a_{i,k′} ∑_{ℓ=1}^{m+1−k′} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

= ∑_{i=1}^{s}( ∑_{k′=m+2}^{n} a_{i,k′} ∑_{ℓ=1}^{k′−m−1} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k′=0}^{m+1} a_{i,k′} ∑_{ℓ=k′−m}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

= ∑_{i=1}^{s}( ∑_{k′=m+1}^{n} a_{i,k′} ∑_{ℓ=1}^{k′−m−1} \frac{1}{(ℓ − \frac{1}{2})^{i}} − ∑_{k′=0}^{m} a_{i,k′} ∑_{ℓ=k′−m}^{0} \frac{1}{(ℓ − \frac{1}{2})^{i}} )

= −a_{0}. This implies the required formula for a_{0}. □

The characteristic polynomial of the recursion for r_{n} as above is λ^{2}−123λ+1, and its zeroes are quite recognisable: ((1±\sqrt{5})/2)^{10}. After performing some experiments, it turns out that

r_{n} = \frac{π^{2}(2n)!^{4}}{(4n + 1)!^{2}} _{3}F_{2}[\begin{matrix}2n + 1, 2n + 1, 2n + 1 \\ 4n + 2, 4n + 2 \end{matrix}; 1],

where the latter is a ‘rarified’ sequence of the Apéry approximations to ζ(2). This follows as a consequence of the hypergeometric identity

\frac{Γ(4n + 2)^{2}Γ(2n + 1)^{2}}{Γ(3n + \frac{3}{2})^{4}} _{5}F_{4}[\begin{matrix}4n + 1, n + \frac{1}{2}, n + \frac{1}{2}, n + \frac{1}{2}, n + \frac{1}{2} \\ 3n + \frac{3}{2}, 3n + \frac{3}{2}, 3n + \frac{3}{2}, 3n + \frac{3}{2} \end{matrix}; 1]

= _{3}F_{2}[\begin{matrix}2n + 1, 2n + 1, 2n + 1 \\ 4n + 2, 4n + 2 \end{matrix}; 1], (8)

which is in turn the particular case where a = b = c = 2n+1 of the following general transformation.

Theorem 4. We have

3F_{2}[ \begin{matrix}a, b, c \\ −a + 2b + c, −a + b + 2c \end{matrix}; 1] = \frac{Γ(−\frac{a}{2} + b + c + \frac{1}{2}) Γ(−\frac{3a}{2} + 2b + c + \frac{1}{2})}{Γ(−a + b + c + \frac{1}{2}) Γ(−a + 2b + c + \frac{1}{2})}

× \frac{Γ(−a + b + 2c) Γ(−3a + 2b + 2c) Γ(−2a + 2b + 2c) Γ(−2a + 4b + 2c)}{Γ(−\frac{3a}{2} + b + 2c) Γ(−\frac{5a}{2} + 2b + 2c) Γ(−a + 2b + 2c) Γ(−3a + 4b + 2c)}

×_{5}F_{4}[\begin{matrix}−2a + 2b + 2c − 1, c − \frac{a}{2}, −\frac{3a}{2} + b + c, \frac{a}{2}, b − \frac{a}{2} \\ −\frac{3a}{2} + 2b + c, −\frac{a}{2} + b + c, −\frac{5a}{2} + 2b + 2c, −\frac{3a}{2} + b + 2c \end{matrix}; 1].  (9)

Proof. We start with the transformation formula (cf. [4, Eq. (3.5.10), q → 1, reversed])

3F_{2}[\begin{matrix}a, b, c \\ d, d − b + c \end{matrix}; 1] = \frac{Γ(2d) Γ(2d − 2b − a) Γ(d − b + c) Γ(d − a + c)}{Γ(2d − 2b) Γ(2d − a) Γ(d + c) Γ(d − b − a + c)}

×_{7}F_{6}[\begin{matrix}d − \frac{1}{2}, \frac{d}{2} + \frac{3}{4}, \frac{d}{2} − \frac{c}{2}, b, \frac{a}{2}, \frac{a}{2} + \frac{1}{2}, −\frac{c}{2} + \frac{d}{2} + \frac{1}{2} \\ \frac{d}{2} − \frac{1}{4}, \frac{c}{2} + \frac{d}{2} + \frac{1}{2}, −b + d + \frac{1}{2}, −\frac{a}{2} + d + \frac{1}{2}, d − \frac{a}{2}, \frac{c}{2} + \frac{d}{2} \end{matrix}; 1]. (10)

To the very-well-poised_{7}F_{6}-series on the right-hand side we apply the transformation formula (cf. [1, Sec. 7.5, Eq. (2)])

7F_{6}[\begin{matrix}a, \frac{a}{2} + 1, b, c, d, e, f \\ \frac{a}{2}, a − b + 1, a − c + 1, a − d + 1, a − e + 1, a − f + 1 \end{matrix}; 1]

= \frac{Γ(a − c + 1) Γ(a − d + 1) Γ(a − e + 1) Γ(a − f + 1)}{Γ(a + 1) Γ(b) Γ(2a − b − c − d − e + 2) Γ(2a − b − c − d − f + 2)}

× \frac{Γ(3a − 2b − c − d − e − f + 3) Γ(2a − b − c − d − e − f + 2)}{Γ(2a − b − c − e − f + 2) Γ(2a − b − d − e − f + 2)}

HYPERGEOMETRY INSPIRED BY IRRATIONALITY QUESTIONS 11

×_{7}F_{6}[\begin{matrix}3a − 2b − c − d − e − f + 2, \frac{3a}{2} − b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} − \frac{f}{2} + 2, a − b − c + 1, \\ \frac{3a}{2} − b − \frac{c}{2} − \frac{d}{2} − \frac{e}{2} − \frac{f}{2} + 1, 2a − b − d − e − f + 2, \end{matrix}

\begin{matrix}a − b − d + 1, a − b − e + 1, \\ 2a − b − c − e − f + 2, 2a − b − c − d − f + 2, \end{matrix}

\begin{matrix}a − b − f + 1, 2a − b − c − d − e − f + 2 \\ 2a − b − c − d − e + 2, a − b + 1 \end{matrix}; 1]. (11)

Thus, we obtain

3F_{2}[\begin{matrix}a, b, c \\ d, d − b + c \end{matrix}; 1] = \frac{Γ(2d) Γ(d − \frac{a}{2}) Γ(−\frac{a}{2} + d + \frac{1}{2}) Γ(−b + d + \frac{1}{2}) Γ(\frac{c}{2} + \frac{d}{2})}{Γ(d + \frac{1}{2}) Γ(2d − a) Γ(2d − 2b) Γ(\frac{d}{2} − \frac{c}{2}) Γ(c + d)}

× \frac{Γ(−a − 2b + 2d) Γ(−b + c + d) Γ(−a − b + \frac{3c}{2} + \frac{3d}{2} + \frac{1}{2})}{Γ(−\frac{a}{2} − b + c + d) Γ(−\frac{a}{2} − b + c + d + \frac{1}{2}) Γ(−a − b + \frac{c}{2} + \frac{3d}{2} + \frac{1}{2})}

×_{7}F_{6}[\begin{matrix}−a − b + \frac{3c}{2} + \frac{3d}{2} − \frac{1}{2}, −\frac{a}{2} − \frac{b}{2} + \frac{3c}{4} + \frac{3d}{4} + \frac{3}{4}, −\frac{a}{2} + \frac{c}{2} + \frac{d}{2}, c, \\ −\frac{a}{2} − \frac{b}{2} + \frac{3c}{4} + \frac{3d}{4} − \frac{1}{4}, −\frac{a}{2} − b + c + d + \frac{1}{2}, −a − b + \frac{c}{2} + \frac{3d}{2} + \frac{1}{2}, \end{matrix}

\begin{matrix}−a − b + c + d, −b + \frac{c}{2} + \frac{d}{2} + \frac{1}{2}, −\frac{a}{2} + \frac{c}{2} + \frac{d}{2} + \frac{1}{2} \\ \frac{c}{2} + \frac{d}{2} + \frac{1}{2}, −a + c + d, −\frac{a}{2} − b + c + d \end{matrix}; 1].

Next we apply the transformation formula (cf. [1, Sec. 7.5, Eq. (1)])

7F_{6}[\begin{matrix}a, \frac{a}{2} + 1, b, c, d, e, f \\ \frac{a}{2}, a − b + 1, a − c + 1, a − d + 1, a − e + 1, a − f + 1 \end{matrix}; 1]

= \frac{Γ(a − e + 1) Γ(a − f + 1) Γ(2a − b − c − d + 2) Γ(2a − b − c − d − e − f + 2)}{Γ(a + 1) Γ(a − e − f + 1) Γ(2a − b − c − d − e + 2) Γ(2a − b − c − d − f + 2)}

×_{7}F_{6}[\begin{matrix}2a − b − c − d + 1, a − \frac{b}{2} − \frac{c}{2} − \frac{d}{2} + \frac{3}{2}, a − c − d + 1, a − b − d + 1, \\ a − \frac{b}{2} − \frac{c}{2} − \frac{d}{2} + \frac{1}{2}, a − b + 1, a − c + 1, \end{matrix}

\begin{matrix}a − b − c + 1, e, f \\ a − d + 1, 2a − b − c − d − e + 2, 2a − b − c − d − f + 2 \end{matrix} ; 1].(12)

We arrive at

3F_{2}[\begin{matrix}a, b, c \\ d, d − b + c \end{matrix}; 1] = \frac{Γ(2d) Γ(d − \frac{a}{2}) Γ(\frac{c}{2} + \frac{d}{2}) Γ(−a − 2b + 2d)}{Γ(d + \frac{1}{2}) Γ(2d − a) Γ(2d − 2b) Γ(c + d)}

× \frac{Γ(−a + c + d) Γ(−b + c + d) Γ(−\frac{a}{2} − b + \frac{c}{2} + \frac{3d}{2} + 1)}{Γ(−\frac{a}{2} + \frac{c}{2} + \frac{d}{2} − \frac{1}{2}) Γ(−\frac{a}{2} − b + c + d + \frac{1}{2}) Γ(−a − b + \frac{c}{2} + \frac{3d}{2} + \frac{1}{2})}

×_{7}F_{6}[\begin{matrix}−\frac{a}{2} − b + \frac{c}{2} + \frac{3d}{2}, −\frac{a}{4} − \frac{b}{2} + \frac{c}{4} + \frac{3d}{4} + 1, −\frac{a}{2} + \frac{c}{2} + \frac{d}{2} + \frac{1}{2}, −\frac{c}{2} + \frac{d}{2} + \frac{1}{2}, \\ −\frac{a}{4} − \frac{b}{2} + \frac{c}{4} + \frac{3d}{4}, −b + d + \frac{1}{2}, −\frac{a}{2} − b + c + d + \frac{1}{2}, \end{matrix}

\begin{matrix}\frac{a}{2} + \frac{1}{2}, −\frac{a}{2} − b + d + \frac{1}{2}, −b + \frac{c}{2} + \frac{d}{2} + \frac{1}{2} \\ −a − b + \frac{c}{2} + \frac{3d}{2} + \frac{1}{2}, \frac{c}{2} + \frac{d}{2} + \frac{1}{2}, −\frac{a}{2} + d + \frac{1}{2} \end{matrix}; 1].

Now we apply again (11). As a result, we obtain

3F_{2}[\begin{matrix}a, b, c \\ d, d − b + c \end{matrix}; 1] = \frac{Γ(2d) Γ(−\frac{a}{2} + d + \frac{1}{2}) Γ(\frac{c}{2} + \frac{d}{2} + \frac{1}{2}) Γ(−a − 2b + 2d)}{Γ(d + \frac{1}{2}) Γ(2d − a) Γ(2d − 2b) Γ(c + d)}

× \frac{Γ(−a + c + d) Γ(−b + c + d) Γ(−\frac{a}{2} − b + \frac{c}{2} + \frac{3d}{2})}{Γ(−\frac{a}{2} + \frac{c}{2} + \frac{d}{2} + \frac{1}{2}) Γ(−\frac{a}{2} − b + c + d) Γ(−a − b + \frac{c}{2} + \frac{3d}{2})}

×_{7}F_{6}[\begin{matrix}−\frac{a}{2} − b + \frac{c}{2} + \frac{3d}{2} − 1, −\frac{a}{4} − \frac{b}{2} + \frac{c}{4} + \frac{3d}{4} + \frac{1}{2}, −b + \frac{c}{2} + \frac{d}{2}, −\frac{a}{2} − b + d, \\ −\frac{a}{4} − \frac{b}{2} + \frac{c}{4} + \frac{3d}{4} − \frac{1}{2}, d − \frac{a}{2}, \frac{c}{2} + \frac{d}{2}, \end{matrix}

\begin{matrix}\frac{a}{2}, \frac{d}{2} − \frac{c}{2}, −\frac{a}{2} + \frac{c}{2} + \frac{d}{2} − \frac{1}{2} \\ −a − b + \frac{c}{2} + \frac{3d}{2}, −\frac{a}{2} − b + c + d, −b + d + \frac{1}{2} \end{matrix}; 1]. (13)

Here, we equate the second upper parameter and the last lower parameter in the 7F_{6}-series, that is,

−\frac{a}{4} − \frac{b}{2} + \frac{c}{4} + \frac{3d}{4} + \frac{1}{2} = −b + d + \frac{1}{2},

or, equivalently, d = c + 2b − a. If we make this substitution in (13), then the _{7}F_{6}- series reduces to a _{5}F_{4}-series. The corresponding transformation formula is (9). □

q-Analogues of (8), (9), and (13) can be obtained by going through the analogous computations when using the _{8}ϕ_{7}-transformation formula [4, Eq. (3.5.10)] instead of (10), the _{8}ϕ_{7}-transformation formula [4, Appendix (III.24)] instead of (11), and the 8ϕ_{7}-transformation formula [4, Eq. (2.10.1)] instead of (12). The q-analogue of (13) obtained in this way is

8ϕ_{7}[\begin{matrix}−cd/q, i\sqrt{cdq}, −i\sqrt{cdq}, b, −b, c, −c, a \\ i\sqrt{cd/q}, −i\sqrt{cd/q}, −cd/b, cd/b, −d, d, −cd/a \end{matrix}; q, \frac{d^{2}}{ab^{2}} ]

= \frac{(d^{2}q, cdq/a, cd^{3}/a^{2}b^{2}, c^{2}d^{2}/ab^{2}; q^{2})_{∞} (d^{2}/b^{2}, d^{2}/a, cd, −cd, cd/ab, −cd/ab; q)_{∞}}{(c^{2}d^{2}/a^{2}b^{2}, cdq, d^{2}q/a, cd^{3}/ab^{2}; q^{2})_{∞} (d^{2}, d^{2}/ab^{2}, cd/a, −cd/a, cd/b, −cd/b; q)_{∞}}

×_{8}ϕ_{7}[\begin{matrix}cd^{3}/ab^{2}q^{2}, \sqrt{cd^{3}q^{2}/ab^{2}}, −\sqrt{cd^{3}q^{2}/ab^{2}}, cd/b^{2}, d^{2}/ab^{2}, a, d/c, cd/aq \\ \sqrt{cd^{3}/ab^{2}q^{2}}, −\sqrt{cd^{3}/ab^{2}q^{2}}, d^{2}/a, cd, cd^{3}/a^{2}b^{2}, c^{2}d^{2}/ab^{2}, d^{2}q/b^{2} \end{matrix}; q^{2}, \frac{cdq}{a} ]. (14)

Similarly to before, we equate the second upper parameter and the last lower parameter in the _{8}ϕ_{7}-series on the right-hand side, that is,

\sqrt{\frac{cd^{3}q^{2}}{ab^{2}}} = \frac{d^{2}q}{b^{2}} ,

or, equivalently, d = cb^{2}/a. If we substitute this in (14), then we obtain

8ϕ_{7}[\begin{matrix}−b^{2}c^{2}/aq, ibc\sqrt{q}/\sqrt{a}, −ibc\sqrt{q}/\sqrt{a}, b, −b, c, −c, a \\ ibc/\sqrt{a}\sqrt{q}, −ibc/\sqrt{a}\sqrt{q}, −bc^{2}/a, bc^{2}/a, −b^{2}c/a, b^{2}c/a, −b^{2}c^{2}/a^{2} \end{matrix}; q, \frac{b^{2}c^{2}}{a^{3}} ]

= \frac{(b^{4}c^{2}q/a^{2}, b^{2}c^{2}q/a^{2}, b^{4}c^{4}/a^{5}, b^{2}c^{4}/a^{3}; q^{2})_{∞}}{(b^{2}c^{4}/a^{4}, b^{2}c^{2}q/a, b^{4}c^{2}q/a^{3}, b^{4}c^{4}/a^{4}; q^{2})_{∞}}

× \frac{(b^{4}c^{2}/a^{3}, b^{2}c^{2}/a, −b^{2}c^{2}/a, bc^{2}/a^{2}, −bc^{2}/a^{2}; q)_{∞}}{(b^{4}c^{2}/a^{2}, b^{2}c^{2}/a^{3}, −b^{2}c^{2}/a^{2}, bc^{2}/a, −bc^{2}/a; q)_{∞}}

×_{6}ϕ_{5}[\begin{matrix}b^{4}c^{4}/a^{4}q^{2}, −b^{2}c^{2}q/a^{2}, c^{2}/a, b^{2}c^{2}/a^{3}, a, b^{2}/a \\ −b^{2}c^{2}/a^{2}q, b^{4}c^{2}/a^{3}, b^{2}c^{2}/a, b^{4}c^{4}/a^{5}, b^{2}c^{4}/a^{3} \end{matrix}; q^{2}, \frac{b^{2}c^{2}q}{a^{2}} ],

a q-analogue of (9). Setting all of a, b, c equal to q^{2n+1}, we arrive at a q-analogue of (8), namely

8ϕ_{7}[\begin{matrix}−q^{6n+2}, iq^{3n+2}, −iq^{3n+2}, q^{2n+1}, −q^{2n+1}, q^{2n+1}, −q^{2n+1}, q^{2n+1} \\ iq^{3n+1}, −iq^{3n+1}, −q^{4n+2}, q^{4n+2}, −q^{4n+2}, q^{4n+2}, −q^{4n+2} \end{matrix}; q, q^{2n+1}]

= \frac{(q^{6n+3}, q^{6n+3}, −q^{6n+3}, −q^{2n+1}; q)_{∞} (q^{8n+5}, q^{4n+3}, q^{6n+3}, q^{6n+3}; q^{2})_{∞}}{(q^{8n+4}, −q^{4n+2}, q^{4n+2}, −q^{4n+2}; q)_{∞} (q^{4n+2}, q^{6n+4}, q^{6n+4}, q^{8n+4}; q^{2})_{∞}}

×_{6}ϕ_{5}[\begin{matrix}q^{8n+2}, −q^{4n+3}, q^{2n+1}, q^{2n+1}, q^{2n+1}, q^{2n+1} \\ −q^{4n+1}, q^{6n+3}, q^{6n+3}, q^{6n+3}, q^{6n+3} \end{matrix}; q^{2}, q^{4n+3}].

5. Zeta values

In this section we prove Theorem 1.

Proof of Theorem 1. Fix an even integer s ≥ 8 and define the rational functions

R(t) = R_{n}(t) = \frac{n!^{s−6} · 2^{12n+1}(t + \frac{n}{2}) ∏_{j=1}^{3n}(t − n − \frac{1}{2} + j)^{2}}{∏_{j=0}^{n}(t + j)^{s}} ,

R̂(t) = R̂_{n}(t) = \frac{n!^{s−6} · 2^{12n} ∏_{j=1}^{3n}(t − n − \frac{1}{2} + j)^{2}}{∏_{j=0}^{n}(t + j)^{s}} ,

both vanishing together with their derivatives at t = ν−n+ \frac{1}{2} for ν = 0, 1, …, 3n−1. Then Lemma 1 and the results from [21, Section 2] apply, and we obtain the linear forms

r_{n} = ∑_{ν=1}^{∞} R_{n}(ν − \frac{1}{2}) = ∑_{\begin{matrix}i=2 \\ i odd \end{matrix}}^{s} a_{i}(2^{i} − 1)ζ(i) + a_{0},

r′_{n} = − ∑_{ν=1}^{∞} \frac{dR_{n}}{dt} (ν − \frac{1}{2}) = ∑_{\begin{matrix}i=2 \\ i odd \end{matrix}}^{s} a_{i}i(2^{i+1} − 1)ζ(i + 1),

r̂_{n} = ∑_{ν=1}^{∞} R̂_{n}(ν − \frac{1}{2}) = ∑_{\begin{matrix}i=2 \\ i even \end{matrix}}^{s} â_{i}(2^{i} − 1)ζ(i),

r̂′_{n} = − ∑_{ν=1}^{∞} \frac{dR̂_{n}}{dt} (ν − \frac{1}{2}) = ∑_{\begin{matrix}i=2 \\ i even \end{matrix}}^{s} â_{i}i(2^{i+1} − 1)ζ(i + 1) + â_{0},

with the following inclusions available:

d_{n}^{s−i}a_{i}, d_{n}^{s−i}â_{i} ∈ ℤ for i = 2, 3, …, s, and d_{n}^{s}a_{0}, d_{n}^{s+1}â_{0} ∈ ℤ.

Here d_{n} denotes the least common multiple of 1, …, n. Its asymptotic behaviour d_{n}^{1/n} → e as n → ∞ follows from the prime number theorem.

The standard asymptotic machinery [16, Section 2] implies that

lim_{n→∞} |r_{n}|^{1/n} = lim_{n→∞} |r̂_{n}|^{1/n} = g(x_{0})

and

lim_{n→∞} |r′_{n}|^{1/n} = lim_{n→∞} |r̂′_{n}|^{1/n} = g(x′_{0}),

where

g(x) = \frac{2^{12}(x + 3)^{6}(x + 1)^{s}}{(x + 2)^{2s}} ,

and x_{0}, x′_{0} are the real zeroes of the polynomial

x^{2}(x + 2)^{s} − (x + 3)^{2}(x + 1)^{s}

on the intervals x > 0 and −1 < x < 0, respectively. It can also be observed numerically for each choice of even s that 0 < g(x′_{0}) < g(x_{0}), so that

lim_{n→∞} |r_{n} − μr′_{n}|^{1/n} = lim_{n→∞} |r̂_{n} − μ̂r̂′_{n}|^{1/n} = g(x_{0})

for any real μ and μ̂. Theorem 1 follows from taking μ = λ/π for the first collection, μ̂ = 4λπ for the second one, and noticing that, when s = 40, we obtain

g(x_{0}) = exp(−40.54232882 …) and g(x′_{0}) = exp(−40.54234026 …),

while for s = 42 we get

g(x_{0}) = exp(−43.31492040 …) and g(x′_{0}) = exp(−43.31492612 …). □

Finally, we remark that further variations on Theorem 1 are possible by combining the two hypergeometric constructions from this section and [21] (see also related applications in [3] and [12]). As the corresponding results remain similar in spirit to the theorem, we do not pursue this line here.

References

[1] W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Math. 32 (Cambridge Univ. Press, Cambridge, 1935).

[2] S. Fischler and W. Zudilin, A refinement of Nesterenko’s linear independence criterion with applications to zeta values, Math. Ann. 347:4 (2010), 739–763.

[3] S. Fischler, J. Sprang and W. Zudilin, Many values of the Riemann zeta function at odd integers are irrational, Comptes Rendus Math. Acad. Sci. Paris 356:7 (2018), 707–711.

[4] G. Gasper and M. Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia Math. Appl. 96 (Cambridge Univ. Press, Cambridge, 2004).

[5] M. Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99–125.

[6] Kh. Hessami Pilehrood and T. Hessami Pilehrood, On the irrationality of the sums of zeta values, Math. Notes 79:3-4 (2006), 561–571.

[7] C. Krattenthaler, HYP and HYPQ — Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively q-binomial sums and basic hypergeometric series, J. Symbol. Comput. 20 (1995), 737–744; the packages are freely available at

http://www.mat.univie.ac.at/~kratt/.

[8] R. Marcovecchio, The Rhin–Viola method for log 2, Acta Arith. 139:2 (2009), 147–184.

[9] M. Rahman and A. Verma, Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc. 335:1 (1993), 277–302.

[10] G. Rhin and C. Viola, On a permutation group related to ζ(2), Acta Arith. 77:3 (1996), 23–56.

[11] T. Rivoal and W. Zudilin, Diophantine properties of numbers related to Catalan’s constant, Math. Ann. 326:4 (2003), 705–721.

[12] T. Rivoal and W. Zudilin, A note on odd zeta values, Preprint

arχiv:1803.03160 [math.NT] (2018).

[13] E. A. Rukhadze, A lower bound for the approximation of ln 2 by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1987), no. 6, 25–29. (Russian)

[14] L. J. Slater, Generalized hypergeometric functions (Cambridge Univ. Press, Cambridge, 1966).

[15] C. Viola, Hypergeometric functions and irrationality measures, in: “Analytic number theory” (Kyoto, 1996), London Math. Soc. Lecture Note Ser. 247 (Cambridge Univ. Press, Cambridge, 1997), 353–360.

[16] W. Zudilin, Irrationality of values of the Riemann zeta function, Izv. Math. 66:3 (2002), 489–542.

[17] W. Zudilin, A few remarks on linear forms involving Catalan’s constant, Chebyshevskiǐ Sb. 3:2 (4) (2002), 60–70; English transl., arχiv:math.NT/0210423 (2002).

[18] W. Zudilin, An Apéry-like difference equation for Catalan’s constant, Electron. J. Combin. 10:1 (2003), #R14, 10 pages.

[19] W. Zudilin, Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux 16:1 (2004), 251–291.

[20] W. Zudilin, An essay on irrationality measures of π and other logarithms, Chebyshevskiǐ Sb. 5:2 (2004), 49–65; English transl., arχiv:math.NT/0404523 (2004).

[21] W. Zudilin, One of the odd zeta values from ζ(5) to ζ(25) is irrational. By elementary means, SIGMA 14 (2018), no. 028, 8 pages.

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

URL: http://www.mat.univie.ac.at/~kratt

Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, Netherlands

URL: http://www.math.ru.nl/~wzudilin