Simultaneous generation for zeta values by the Markov-WZ method

By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Ap\'ery-like formulae for odd zeta values. As a consequence, we get a new identity producing Ap\'ery-like series for all $\zeta(2n+4m+3),$ $n,m\ge 0,$ convergent at the geometric rate with ratio $2^{-10}.$


Introduction
The Riemann zeta function is defined by the series Apéry's irrationality proof of ζ(3) [14] operates with the faster convergent series first obtained by A. A. Markov in 1890 [10].The general formula giving analogous series for all ζ(2s+3), s ≥ 0, was proved by Koecher [7] (and independently in an expanded form by Leshchiner [9]).It reads A similar identity generating fast convergent series for all ζ(4s + 3), s ≥ 0, which for s > 1 is different from Koecher's result (3), was experimentally discovered in [3] and proved by G. Almkvist  There exists a bivariate unifying formula for identities (3) and (4), which was first conjectured by H. Cohen and then proved by D. Bradley [5] and, independently, by T. Rivoal [15].This identity implies (3) if y = 0, and gives (4) if x = 0.The proof of (5) given in [5,15] relies on Borwein and Bradley's method [3] and consists of reduction of ( 5) to a finite non-trivial combinatorial identity which can be proved on the basis of Almkvist and Granville's work [1].Recently, in [6] it was shown that Koecher's formula (3), and similarly Leschiner's and the identities of Bailey, Borwein and Bradley [9,4] generating accelerated series for even zeta values ζ(2n + 2), can be proved by means of the WZ method.
Formulas (3)-( 5) generate accelerated series for odd zeta values and, in particular, series (2) for ζ(3) which converge at a geometric rate with ratio 1/4.Many other more rapidly convergent expressions for ζ(3) can be proved on the basis of the WZ method.The following series, for example, convergent at the geometric rate with ratio 2 −10 , was obtained by T. Amdeberhan and D. Zeilberger [2] by application of WZ-pairs.There are even faster convergent representations for ζ(3) with ratios 10 −5 , 10 −8 (see [11]).In [6] it was shown how to get such fast convergent series explicitly for other values ζ(n), n > 3.This can be accomplished by applying the WZ method not to the series (1) itself but to a generating function of a sequence of zeta values.
In this note, we prove a more general form of the bivariate identity (5) by application of the Markov-WZ method.We show that identity (5) and the series (6) of Amdeberhan and Zeilberger can be proved with the help of the same Markov-WZ pair, but using different summation formulas.Moreover, we get a new identity generating accelerated series for all ζ(2n + 4m + 3), n, m ≥ 0, convergent at a geometric rate with ratio 2 −10 .

Statement of the main results
We start by giving several definitions, and by reviewing known facts related to the Markov-Wilf-Zeilberger theory (see [8,10,11,12]).
A function H(n, k), in the integer variables n and k, is called hypergeometric or closed form (CF) if the quotients are both rational functions of n and k.A hypergeometric function that can be written as a ratio of products of factorials is called pure-hypergeometric.A pair of CF functions A P-recursive function is a function that satisfies a linear recurrence relation with polynomial coefficients.If for a given hypergeometric function H(n, k), there exists a polynomial P (n, k) in k of the form for some non-negative integer L, and P-recursive functions a 0 (n), . . ., a L (n) such that F (n, k) := H(n, k)P (n, k) satisfies ( 7) with some function G, then a pair (F, G) is called a Markov-WZ pair associated with the kernel H(n, k) (MWZ-pair for short).We call G(n, k) an MWZ mate of F (n, k).
In 2005, M. Mohammed [11] showed that for any pure-hypergeometric kernel H(n, k), there exists a non-negative integer L and a polynomial P (n, k) as above such that From relation (7) we get the following summation formulas.
whenever both sides converge.
whenever both sides converge.
Formulas ( 8), ( 9) with an appropriate choice of MWZ-pairs can be used to convert a given hypergeometric series into a different rapidly converging one.
Let (λ) ν be the Pochhammer symbol (or the shifted factorial) defined by Let a, b be complex numbers such that |a| < 1, |b| < 1.In Section 3, we construct a Markov-WZ pair associated with the kernel and then apply Propositions 1, 2 to get the following two theorems.
Theorem 1 Let a, b be complex numbers, with |a| < 1, |b| < 1.Then for arbitrary complex numbers A 0 , B 0 , C 0 we have , with where L n is a solution of the second order difference equation whose growth is described by the inequality If in Theorem 1 we take B 0 = 1, A 0 = C 0 = 0, then L n = 0 for all n ≥ 0 and we get If we now put we get the bivariate identity (5) conjectured by H. Cohen.
Let a, b be complex numbers such that |a| < 1, |b| < 1.Let We are interested in finding a Markov-WZ pair associated with H(n, k).For this purpose, we define the function F (n, k) = H(n, k)P (n, k), where P (n, k) is a polynomial in k of degree L 1 with unknown coefficients as functions of n.Then where P 1 (n, k) is a polynomial in k of degree L 1 + 4. From ( 15) it follows that we can determine a MWZ mate of with unknown coefficients as functions of n.Indeed, for such a choice we have where This implies that L 2 = L 1 + 1.On the other hand, equating coefficients of powers of k on both sides of (16), we get a system of L 1 + 5 linear homogeneous equations with L 1 + L 2 + 2 = 2L 1 + 3 unknowns.In order to guarantee a solution, we should at least have that 2L 1 + 3 ≥ L 1 + 5 and hence L 1 ≥ 2, L 2 ≥ 3.
We now show that there is a non-zero solution of ( 16) with the optimal choice L 1 = 2, L 2 = 3.To see this, define two functions Substituting F, G into (17) and cancelling common factors we get that (17) is equivalent to the following equation of degree 6 in the variable k : To satisfy condition (17), all the coefficients of the powers of (k + 1) in the equation ( 18) must be identically zero.This leads to a system of first order linear recurrence equations with polynomial coefficients for A Finally, substitution of (25), (26) into (24) gives the second-order difference equation 20), ( 22), and polynomials p(n), q(n) defined in (10), (11).
we get the following series for ζ(4) mentioned by Markov in[10,  p.18]: