How to generate all possible rational Wilf-Zeilberger pairs?
Shaoshi Chen


How to generate all possible rational Wilf-Zeilberger pairs?

Dedicated to the memory of Jonathan M. Borwein and Ann Johnson
How to generate all possible rational Wilf–Zeilberger pairs?








Shaoshi Chen KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 
 Beijing 100190, (China)

School of Mathematical Sciences, University of Chinese Academy of Sciences, 
 Beijing 100049, (China)

schen@amss.ac.cn 

This work was supported by the NSFC grants 11501552, 11688101 and
by the Frontier Key Project (QYZDJ-SSW-SYS022) and the Fund of the Youth Innovation Promotion Association, CAS





*
    Shaoshi Chen
    December 30, 2023
=====================






 A Wilf–Zeilberger pair (F, G) in the discrete case satisfies the equation

    F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k).

We present a structural description of all possible rational Wilf–Zeilberger pairs and their continuous and mixed analogues.





§ INTRODUCTION
 

The Wilf–Zeilberger (abbr. WZ) theory <cit.> has become a bridge between symbolic computation and combinatorics.
Through this bridge, not only classical combinatorial identities from handbooks and long-standing conjectures in combinatorics, such as Gessel's conjecture <cit.> and q-TSPP conjecture <cit.>, are proved algorithmically, but also some new identities and conjectures related to mathematical constants, such as π and zeta values,  are discovered via computerized guessing <cit.>.

WZ-pair is one of leading concepts in the WZ theory that was originally introduced in <cit.> with a recent brief description in <cit.>.
In the discrete case,  a WZ-pair (F(n, k), G(n, k)) satisfies the WZ equation

    F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k),

where both F and G are hypergeometric terms, i.e., their shift quotients with respect to n and k are rational
functions in n and k, respectively.
Once a WZ-pair is given, one can sum on both sides of the above equation over k from 0 to ∞ to get

    ∑_k=0^∞ F(n+1, k) - ∑_k=0^∞ F(n, k)= lim_k→∞ G(n, k+1) - G(n, 0).

If G(n, 0) and lim_k→∞ G(n, k+1) are 0 then we obtain

    ∑_k=0^∞ F(n+1, k) = ∑_k=0^∞ F(n, k),

which implies that ∑_k=0^∞ F(n, k) is independent of n. Thus, we get
the identity  ∑_k=0^∞ F(n, k) = c, where the constant c can be determined by evaluating the sum for one value of n.
We may also get a companion identity by summing the WZ-equation over n.
For instance, the pair (F, G) with

    F = nk^2/2nn  and    G= ( 2k-3n-3) k^2/ 2(2n+1)( -n-1+k)^2·nk^2/2nn

leads to two identities

    ∑_k=0^∞nk^2 = 2nn  and  ∑_n=0^∞(3n-2k+1)/2(2n+1)2nnnk^2 = 1.


Besides to prove combinatorial identities, WZ-pairs have many other applications.
One of the applications can be traced back to Andrei Markov's 1890 method for convergence-acceleration of series for
computing ζ(3), which leads to the Markov-WZ method <cit.>.
WZ-pairs also play a central role in the study of finding Ramanujan-type and Zeilberger-type series for constants
involving π in <cit.>, zeta values <cit.>
and their q-analogues <cit.>. Most recent applications are related to
congruences and super congruences <cit.>.

For appreciation we select some remarkable (q)-series about π, ζ(3) together with (super)-congruences whose proofs can be obtained via WZ-pairs
as follows (this list is surely not comprehensive):



  * Ramanujan's series for 1/π:  first recorded in Ramanujan's second notebook, proved by Bauer in <cit.>, and by Ekhad and Zeilberger using WZ-pairs in <cit.>.
For a nice survey on Ramanujan's series, see <cit.>.

    2/π = ∑_k=0^∞4k+1/(-64)^k2kk^3 .



  * Guillera's series for 1/π^2: found and proved by Guillera in 2002 using WZ-pairs <cit.>. For more results on Ramanujan-type series for 1/π^2, see Zudilin's surveys <cit.>.
  
    128/π^2 = ∑_k=0^∞ (-1)^k 2kk^5 820k^2 + 180k + 13/2^20k .


  
  * Guillera's Zeilberger-type series for π^2: found and proved by Guillera using WZ-pairs in <cit.>.

  
    π^2/2 = ∑_k=1^∞(3k-1)16^k/k^32kk^3.


  
  * Markov–Apéry's series for ζ(3): first discovered by Andrei Markov in 1890, used by Apéry for his irrationality proof, and proved by Zeilberger using WZ-pairs in <cit.>.

    ζ(3)  = 5/2∑_k=1^∞(-1)^k-1/k^32kk.



  * Amdeberhan's series for ζ(3):  proved by Amdeberhan in 1996 using WZ-pairs <cit.>.

    ζ(3) = 1/4∑_k=1^∞ (-1)^k-156k^2 - 32k+5/k^3 (2k-1)^2 2kk3kk.



  
  * Bailey–Borwein–Bradley identity: experimentally discovered and proved by Bailey et al. in <cit.>, a proof using the Markov-WZ method is given in <cit.> and its q-analogue
  is presented in <cit.>.
   
    ∑_k=0^∞ζ(2k+2) z^2k = 3 ∑_k=1^∞1/2kk(k^2-z^2)∏_m=1^k-1m^2-4z^2/m^2-z^2,   z∈ with |z|<1.




   
  * van Hamme's supercongruence I: first conjectured by van Hamme <cit.>, proved by Mortenson <cit.> using _6F_5 transformations and by Zudilin <cit.>
  using WZ-pairs.
  
    ∑_k=0^p-1/24k+1/(-64)^k2kk^3 ≡ p(-1)^p-1/2 (mod p^3),

where p is an odd prime and the multiplicative inverse of (-64)^k should be computed modulo p^3.

  
  * van Hamme's supercongruence II: first conjectured by van Hamme <cit.>, proved by Long <cit.> using hypergeometric evaluation identities, one of which is obtained by Gessel using WZ-pairs in <cit.>.
  
    ∑_k=0^p-1/26k+1/256^k2kk^3 ≡ p(-1)^p-1/2 (mod p^4),

where p>3 is a prime and the multiplicative inverse of (256)^k should be computed modulo p^4.


  
  * Guo's q-analogue of van Hamme's supercongruence I: discovered and proved recently by Guo using WZ-pairs in <cit.>.
  
    ∑_k=0^p-1/2(-1)^k q^k^2[4k+1]_q(q;q^2)_k^3/(q^2;q^2)_k^3≡ [p]_qq^(p-1)^2/4 (-1)^p-1/2[p]_q^3,

where for n∈, (a;q)_n := (1-a)(1-aq)⋯ (1-aq^n-1) with (a;q)_0 = 1, [n]_q= 1+q+⋯ + q^n-1  and p is an odd prime.
  
  * Hou–Krattenthaler–Sun's q-analogue of Guillera's Zeilberger-type series for π^2:  inspired by a recent conjecture on supercongruence  by Guo in <cit.>, and proved using WZ-pairs in <cit.>.
  This work is also connected to other emerging developments on q-analogues of series for famous constants and formulae <cit.>.
  
    2∑_k=0^∞ q^2k^2+2k(1+q^2k^2+2-2q^4k+3) (q^2;q^2)_k^3/(q;q^2)^3_k+1(-1; q)_2k+3 = ∑_k=0^∞q^2k/(1-q^2k+1)^2 .


















For applications, it is crucial to have WZ-pairs at hand. In the previous work, WZ-pairs are obtained
either by guessing from the identities to be proved using Gosper's algorithm or by certain transformations from a given WZ-pair <cit.>.
Riordan in the preface of his book <cit.> commented that the central fact developed is that identities are both
inexhaustible and unpredictable; the age-old dream of putting order in this chaos is doomed to failure . As an optimistic respond to Riordan's comment,
Gessel in his talk[The talk was given at the Waterloo Workshop in Computer Algebra (in honor of Herbert Wilf's 80th birthday), Wilfrid Laurier University, May 28, 2011.
For the talk slides, see the link: <http://people.brandeis.edu/ gessel/homepage/slides/wilf80-slides.pdf>] on the WZ method motivated with some examples that WZ forms bring order to this chaos , where WZ-forms are a multivariate generalization of WZ-pairs <cit.>. With the hope of discovering more combinatorial identities in an intrinsic and algorithmic way, it is natural and challenging to ask the following question.


How to generate all possible WZ-pairs algorithmically?


This problem seems quite open, but every promising project needs a starting point.
In <cit.>, Liu had described the structure of
a special class of analytic WZ-functions with F=G in terms of Rogers–Szegö polynomials and Stieltjes–Wigert polynomials in the q-shift case.
In <cit.>, Sun studied the relation between generating functions of F(n, k) and G(n, k) if (F, G) is a WZ-pair and applied this relation
to prove some combinatorial identities.
In this paper, we solve the problem completely for the first non-trivial case, namely, the case of rational WZ-pairs.
To this end, let us first introduce some notations. Throughout this paper, let K be a field of characteristic zero and K(x, y) be the field of rational functions
in x and y over K. Let D_x = ∂/∂_x and D_y = ∂/∂_y be the usual
derivations with respect to x and y, respectively. The shift operators _x and _y are defined respectively as

    _x(f(x, y)) = f(x+1, y)   and  _y(f(x, y)) = f(x, y+1)   for f∈ K(x, y).

For any q∈ K∖{0}, we define the q-shift operators τ_q, x and τ_q, y respectively as

    τ_q, x(f(x, y)) = f(qx, y)   and  τ_q, y(f(x, y)) = f(x, qy)   for f∈ K(x, y).

For z∈{x, y}, let Δ_z and Δ_q, z denote the difference and q-difference operators defined by Δ_z(f) = _z(f) - f and Δ_q, z(f)= τ_q, z(f)-f for f∈ K(x, y), respectively.


Let ∂_x ∈{D_x, Δ_x, Δ_q, x} and ∂_y ∈{D_y, Δ_y, Δ_q, y}. A pair (f, g) with f, g∈ K(x, y)
is called a WZ-pair with respect to (∂_x, ∂_y) in K(x, y) if ∂_x(f) = ∂_y(g).

The set of all rational WZ-pairs in K(x, y) with respect to (∂_x, ∂_y) forms a linear space over K, denoted by _(∂_x, ∂_y).
A WZ-pair (f, g) with respect to (∂_x, ∂_y) is said to be exact[This is motivated by the fact that a differential form ω = gdx + fdy with f, g ∈ K(x, y) is exact in K(x, y) if and only if f = D_y(h) and g = D_x(h) for some h∈ K(x, y).] if there exists h∈ K(x, y) such that f= ∂_y(h)
and g = ∂_x(h).
Let _(∂_x, ∂_y) denote the set of all exact WZ-pairs with respect to (∂_x, ∂_y), which
forms a subspace of _(∂_x, ∂_y).
The goal of this paper is to provide an explicit description of the structure of  the quotient space _(∂_x, ∂_y)/_(∂_x, ∂_y).

The remainder of this paper is organized as follows.
As our key tools, residue criteria for rational integrability and summability
are recalled in Section <ref>. In Section <ref>, we present structure theorems
for rational WZ-pairs in three different settings. This paper
ends with a conclusion along with some remarks on the future research.



§ RESIDUE CRITERIA
  

In this section, we recall the notion of residues and their (q-)discrete analogues for rational functions and
some residue criteria for rational integrability and summability from <cit.>.

Let F be a field of characteristic zero and F(z) be the field of rational functions
in z over F. Let D_z be the usual derivation on F(z) such that D_z(z)=1 and D_z(c)=0 for all c∈ F.
A rational function f∈ F(z) is said to be D_z-integrable in F(z) if f = D_z(g) for some g∈ F(z).
By the irreducible partial fraction decomposition, one can always uniquely write f∈ F(z) as

    f = q + ∑_i=1^n ∑_j=1^m_ia_i, j/ d_i^j,

where q, a_i, j, d_i ∈ F[z], _z(a_i, j)< _z(d_i) and the d_i's
are distinct irreducible and monic polynomials.
We call a_i, 1 the pseudo D_z-residue of f at d_i,
denoted by pres_D_z(f, d_i). For an irreducible polynomial p∈ F[z], we let _p denote the set

    _p := {a/b∈ F(z)|a, b ∈ F[z] with (a, b) and p∤ b},

and let _p denote the set {f∈ F(z)| pf∈_p}. If f∈_p, the pseudo-residue pres_D_z(f, p) is called the D_z-residue
of f at p, denoted by _D_z(f, p). The following example shows that pseudo-residues may not be
the obstructions for D_z-integrability in F(z).


Let F := and f = (1-x^2)/(x^2+1)^2. Then the irreducible partial fraction decomposition of f
is of the form

    f  = 2/(x^2+1)^2- 1/x^2+1.

The pseudo-residue of f at x^2+1 is -1, which is nonzero. However, f is D_z-integrable in F(z) since f = D_z(x/(x^2+1)).

The following lemma shows that D_z-residues are the only obstructions for D_z-integrability of rational functions
with squarefree denominators, so are pseudo-residues if F is algebraically closed.

<cit.>
Let f=a/b∈ F(z) be such that a, b∈ F[z], (a, b)=1.  If b is squarefree, then f is D_z-integrable
in F(z) if and only if _D_z(f, d)=0 for any irreducible factor d of b. If F is algebraically closed, then
f is D_z-integrable in F(z) if and only if
pres_D_z(f, z-α)=0 for any root α of the denominator b.


By the Ostrogradsky–Hermite reduction <cit.>, we can decompose a rational function f∈ F(z)
as  f = D_z(g) + a/b, where g∈ F(z) and a, b ∈ F[z] are such that _z(a)<_z(b), (a, b)=1,
and b is a squarefree polynomial in F[z]. By Lemma <ref>, f is D_z-integrable in F(z) if and only if a=0.







We now recall the (q-)discrete analogue of
D_z-residues introduced in <cit.>.
Let ϕ be an automorphism of F(z) that fixes F. For a polynomial p∈ F[z],
we call the set {ϕ^i(p)| i∈} the ϕ-orbit of p, denoted by [p]_ϕ.
Two polynomials p, q∈ F[z] are said to be ϕ-equivalent (denoted as p∼_ϕ q) if they are in the same ϕ-orbit, i.e., p = ϕ^i(q) for some i∈.
For any a, b ∈ F(z) and m∈, we have

    a/ϕ^m(b) = ϕ(g) - g + ϕ^-m(a)/b,

where g is equal to ∑_i=0^m-1ϕ^i-m(a)/ϕ^i(b) if m≥ 0,  and equal to -∑_i=0^-m-1ϕ^i(a)/ϕ^m+i(b) if m<0.

Let _z be the shift operator with respect to z defined by _z(f(z)) = f(z+1). Note that _z is an automorphism of F(z) that fixes F.
A rational function f∈ F(z) is said to be _z-summable in F(z) if f = _z(g)-g for some g∈ F(z).
For any f∈ F(z), we can uniquely decompose it into the form

    f = p(z) + ∑_i=1^n ∑_j=1^m_i∑_ℓ=0^e_i, ja_i, j, ℓ/_z^ℓ (d_i)^j,

where p, a_i, j, ℓ, d_i∈ F[z], _z(a_i, j, ℓ)< _z(d_i) and the d_i's
are irreducible and monic polynomials such that no two of them are _z-equivalent. We call the sum ∑_ℓ=0^e_i, j_z^-ℓ(a_i, j, ℓ)
the _z-residue of f at d_i of multiplicity j, denoted by __z(f, d_i, j).
Recently, the notion of _z-residues has been generalized to the case of rational functions over elliptic curves <cit.>.
The following lemma is a discrete analogue of Lemma <ref> which
shows that _z-residues are the only obstructions for _z-summability in the field F(z).

<cit.>
Let f=a/b∈ F(z) be such that a, b∈ F[z] and (a, b)=1. Then f is _z-summable in F(z) if and only if
__z(f, d, j)=0 for any irreducible factor d of the denominator b of any multiplicity j∈.

By Abramov's reduction <cit.>, we can decompose a rational function f∈ F(z) as

    f = Δ_z(g) + ∑_i=1^n ∑_j=1^m_ia_i, j/b_i^j,

where g∈ F(z) and a_i, j, b_i ∈ F[z] are such that _z(a_i, j)<_z(b_i) and the
b_i's are irreducible and monic polynomials in distinct _z-orbits. By Lemma <ref>, h is _z-summable in F(z) if and only if
a_i, j = 0 for all i, j with 1≤ i ≤ n and 1≤ j ≤ m_i.

Let q be a nonzero element of F such that q^m ≠ 1 for all nonzero m∈ and let τ_q, z
be the q-shift operator with respect to z defined by τ_q, z(f(z)) = f(qz). Since q is nonzero, τ_q, z
is an automorphism of F(z) that fixes F.
A rational function f∈ F(z) is said to be τ_q, z-summable in F(z) if f = τ_q, z(g)-g for some g∈ F(z).
For any f∈ F(z), we can uniquely decompose it into the form

    f = c + zp_1 + p_2/z^s + ∑_i=1^n ∑_j=1^m_i∑_ℓ=0^e_i, ja_i, j, ℓ/τ_q, z^ℓ (d_i)^j,

where c∈ F, s, n, m_i, e_i, j∈ with s≠ 0,
and p_1, p_2, a_i, j, ℓ, d_i∈ F[z] are such that _z(p_2)<s, _z(a_i, j, ℓ)< _z(d_i),
and p_2 is either zero or has nonzero constant term, i.e., p_2(0)≠ 0. Moreover, the d_i's are irreducible and monic polynomials in distinct τ_q, z-orbits and z∤ d_i for all i with 1≤ i ≤ n. We call the constant c the τ_q, z-residue of f at infinity, denoted by _τ_q, z(f, ∞) and call   the sum ∑_ℓ=0^e_i, jτ_q, z^-ℓ(a_i, j, ℓ)
the τ_q, z-residue of f at d_i of multiplicity j, denoted by _τ_q, z(f, d_i, j).
A q-analogue of Lemma <ref> is as follows.

<cit.>
Let f=a/b∈ F(z) be such that a, b∈ F[z] and (a, b)=1. Then f is τ_q, z-summable in F(z) if and only if _τ_q, z(f, ∞) =0 and
_τ_q, z(f, d, j)=0 for any irreducible factor d of the denominator b of any multiplicity j∈.


By a q-analogue of Abramov's reduction <cit.>, we can decompose a rational function f∈ F(z) as

    f = Δ_q, z(g) + c +∑_i=1^n ∑_j=1^m_ia_i, j/b_i^j,

where g∈ F(z), c∈ F, and a_i, j, b_i ∈ F[z] are such
that _z(a_i, j)<_z(b_i) and
the b_i's are irreducible and monic polynomials in distinct _z-orbits and (z, b_i)=1 for all i with 1≤ i ≤ n. By Lemma <ref>, f is τ_q,z-summable in F(z) if and only if
c=0 and a_i, j = 0 for all i, j with 1≤ i ≤ n and 1≤ j ≤ m_i.

Note that pseudo-residues are essentially different from residues in the
differential case, but not needed in the shift and q-shift cases.




§ STRUCTURE THEOREMS







In this section, we present structure theorems for rational WZ-pairs in terms of some special pairs.
Throughout this section, we will assume that K is an algebraically closed field of characteristic zero and let ∂_x ∈{D_x, Δ_x, Δ_q, x} and ∂_y ∈{D_y, Δ_y, Δ_q, y}.

We first consider the special case that q∈ K is a root of unity.
Assume that m is the minimal positive integer such that q^m=1. For any f∈ K(x, y),
it is easy to show that τ_q, y(f) = f if and only if f∈ K(x)(y^m).
Note that K(x, y) is a finite algebraic extension of K(x)(y^m) of degree m.
In the following theorem, we show that WZ-pairs in this special case
are of a very simple form.


Let ∂_x ∈{D_x, Δ_x, Δ_q, x} and f, g∈ K(x, y) be such that ∂_x(f) = Δ_q, y(g).
Then there exist rational functions h∈ K(x, y) and a, b∈ K(x, y^m) such that ∂_x(a)=0 and

    f  = Δ_q, y(h) + a    and   g  = ∂_x(h) + b.

Moreover, we have a ∈ K(y^m) if ∂_x∈{D_x, Δ_x} and a∈ K(x^m, y^m) if ∂_x = Δ_q, x.


By Lemma 2.4 in <cit.>, any rational function f∈ K(x, y) can be decomposed as

    f = Δ_q, y(h)+ a,   where h∈ K(x, y) and a∈ K(x)(y^m).

Moreover, f is τ_q, y-summable in K(x, y) if and only if a=0. Then

    ∂_x(f) = Δ_q, y(∂_x(h))+ ∂_x(a).

Note that ∂_x(a) ∈ K(x)(y^m), which implies that ∂_x(a)=0 because ∂_x(f) is τ_q, y-summable in K(x, y).
Then Δ_q, y(g) = Δ_q, y(∂_x(h)). So g = ∂_x(h) + b for some b∈ K(x, y^m). This completes the proof.


From now on, we assume that q is not a root of unity. We will investigate WZ-pairs in three different cases according to the choice of the pair (∂_x, ∂_y).





















 §.§ The differential case
 

In the continuous setting, we consider WZ-pairs with respect to (D_x, D_y), i.e., the
pairs of the form (f, g) with f, g∈ K(x, y) satisfying D_x(f)= D_y(g).

A WZ-pair (f, g) with respect to (D_x, D_y) is called a log-derivative pair if there exists nonzero h∈ K(x, y) such that f = D_y(h)/h
and g = D_x(h)/h.

The following theorem shows that any WZ-pair in the continuous case is a linear combination of exact and log-derivative pairs,
which was first proved by Christopher in <cit.> and then extended to the multivariate case in <cit.>.


Let f, g∈ K(x, y) be such that D_x(f) = D_y(g). Then there exist rational functions a, b_1, …, b_n ∈ K(x, y) and nonzero
constants c_1, …, c_n∈ K such that

    f  = D_y(a) + ∑_i =1^n c_i D_y(b_i)/b_i  and   g  = D_x(a) + ∑_i =1^n c_i D_x(b_i)/b_i.



The proof in the case when K is the field of complex numbers can be found in <cit.> and in the case when K is any algebraically closed
field of characteristic zero can be found in <cit.>.



The quotient space _(D_x, D_y)/_(D_x, D_y) is spanned over K by the set

    {(f, g) + _(D_x, D_y)|f, g ∈ K(x, y) such that (f, g) is a log-derivative pair}.




A differentiable function h(x, y) is said to be hyperexponential over (x, y) if D_x(h)= f h and D_y(h)= gh  for some f, g∈(x, y).
The above theorem enables us to obtain the multiplicative structure of hyperexponential functions, i.e., any hyperexponential function h(x, y)
can be written as
h= exp(a) ·∏_i=1^n b_i^c_i for some a, b_i∈(x, y) and c_i∈.




 §.§ The (q)-shift case
 
In the discrete setting,  we consider WZ-pairs with respect to (∂_x, ∂_y) with ∂_x ∈{Δ_x, Δ_q, x}
and ∂_y ∈{Δ_y, Δ_q, y},  i.e., the
pairs of the form (f, g) with f, g∈ K(x, y) satisfying ∂_x(f)= ∂_y(g).

Let θ_x∈{_x, τ_q, x} and θ_y∈{_y, τ_q, y}. For any nonzero m∈, θ_x^m is also an automorphism on K(x, y) that fixes K(y), i.e., for any f∈ K(x, y), θ_x^m(f) = f if and only if f∈ K(y).
The ring of polynomials in θ_x and θ_y over K is denoted by
K[θ_x, θ_y]. For any p = ∑_i, j c_i, jθ_x^iθ_y^j ∈ K[θ_x, θ_y] and f∈ K(x, y), we define the action p ∙ f = ∑_i, j c_i, jθ_x^i(θ_y^j(f)). Then K(x, y) can be viewed as a K[θ_x,  θ_y]-module. Let G = ⟨θ_x, θ_y⟩ be the free abelian group generated by θ_x and θ_y.
Let f∈ K(x, y) and H be a subgroup of G. We call the set {cθ(f)| c∈ K∖{0}, θ∈ H} the H-orbit at f, denoted by [f]_H.
Two elements f, g∈ K(x, y) are said to be H-equivalent if [f]_H = [g]_H, denoted by f ∼_H g. The relation
∼_H is an equivalence relation. A rational function f∈ K(x, y) is said to be (θ_x, θ_y)-invariant if there exist m, n∈, not all zero, such that θ_x^mθ_y^n(f) = f.
All possible (θ_x, θ_y)-invariant rational functions have been completely characterized in <cit.>. We summarize the characterization as follows.


Let f∈ K(x, y) be (θ_x, θ_y)-invariant, i.e., there exist m, n∈, not all zero, such that
θ_x^mθ_y^n(f) = f.  Set n̅ = n/(m, n) and m̅ = m/ (m ,n). Then


  1. if θ_x = _x and θ_y = _y, then f = g(n̅x-m̅y) for some g∈ K(z);

  2. if θ_x = τ_q, x, θ_y = τ_q, y, then f = g(x^n̅y^-m̅) for some g∈ K(z);

  3. if θ_x = _x, θ_y = τ_q, y, then f ∈ K(x) if m=0, f∈ K(y) if n=0, and f∈ K if mn≠ 0.



















We introduce a discrete analogue of the log-derivative pairs.

A WZ-pair (f, g) with respect to (∂_x, ∂_y) is called a cyclic pair if there exists a (θ_x, θ_y)-invariant h∈ K(x, y)
such that

    f = θ_x^s - 1/θ_x-1∙ h   and   g = θ_y^t - 1/θ_y-1∙ h,

where s, t∈ are not all zero satisfying that
θ_x^s(h)= θ_y^t(h).

In the above definition, we may always assume that s≥ 0.
Note that for any n∈ we have

    θ_y^n - 1/θ_y-1 = {[  ∑_j=0^n-1θ_y^j,            n≥ 0;; -∑_j=1^-nθ_y^-j,             n<0. ].



 Let a∈ K(y) and b∈ K(x). Then both (a, 0) and (0, b) are cyclic  by taking h =a, s=1, t=0  and h=b, s=0, t=1, respectively.
Let p = 2x + 3y. Then the pair (f, g) with

    f = 1/p + 1/_x(p) + 1/_x^2(p)  and   g = 1/p + 1/_y(p)

is a cyclic WZ-pair with respect to (Δ_x, Δ_y).




Let V_0 = K(x)[y] and V_m be
the set of all rational functions of the form ∑_i=1^I a_i/b_i^m,
where m∈_+, a_i, b_i, ∈ K(x)[y], _y(a_i) < _y(b_i) and the b_i's are distinct
irreducible polynomials in the ring K(x)[y]. By definition, the set V_m forms a subspace of
K(x, y) as a vector spaces over K(x). By the irreducible partial fraction decomposition,
any f∈ K(x, y) can be uniquely decomposed
into f = f_0 + f_1 + ⋯ + f_n with f_i ∈ V_i and so K(x, y) = ⊕_i=0^∞ V_i. The following lemma shows that
the space V_m is invariant under certain shift operators.

 
Let f∈ V_m and P∈ K(x)[θ_x, θ_y]. Then P(f)∈ V_m.


Let f = ∑_i=1^I a_i/b_i^m and P= ∑_i, j p_i, jθ_x^i θ_y^j.
For any θ=θ_x^iθ_y^j with i, j, k ∈, θ(b_i) is still
irreducible and _y(θ(a_i))< _y(θ(b_i)). Then all of the simple fractions
p_i, jθ_x^iθ_y^j(a_i)/θ_x^iθ_y^j(b_i)^n appearing in P(f) are
proper in y and have irreducible denominators. If some of denominators are the same,
we can simplify them by adding the numerators to get a simple fraction. After this simplification,
we see that P(f) can be written in the same form as f, so it is in V_m. 



Let p be a monic polynomial in K(x)[y]. If θ_x^m(p) = cθ_y^n(p) for some c∈ K(x) and m, n∈ with m, n being not both zero, then c∈ K.


Write p = ∑_i=0^d p_i y^i with p_i∈ K(x) and p_d =1. Then

    θ_x^m(p) =  ∑_i=0^d θ_x^m(p_i) y^i  = c ∑_i=0^d p_i θ_y^n(y^i)= cθ_y^n(p).

Comparing the leading coefficients in y yields c=1 if θ_y = _y and c = q^-nd if θ_y = τ_q, y. Thus, c∈ K because q∈ K.



Let f∈ K(x, y) be a rational function of the form

    f = a_0/b^m + a_1/θ_x(b^m) + ⋯ + a_n/θ_x^n(b^m),

where m∈_+, n∈, a_0, a_1, …, a_n ∈ K(x)[y] with a_n≠ 0 and b∈ K(x)[y] are such that _y(a_i)<_y(b) and b is an
irreducible and monic polynomial in K(x)[y] such that θ_x^i(b)
and θ_x^j(b) are not θ_y-equivalent for all i, j ∈{0, 1, …, n} with i≠ j.
If θ_x(f)- f = θ_y(g) - g for some g∈ K(x, y), then (f, g) is cyclic.


By a direct calculation, we have

    θ_x(f)- f  = θ_x(a_n)/θ_x^n+1(b^m) - a_0/b^m + θ_x(a_0)-a_1/θ_x(b^m) + ⋯ + θ_x(a_n-1)-a_n/θ_x^n(b^m).

If θ_x(f) - f = θ_y(g) - g for some g∈ K(x, y), then all of the θ_y-residues at distinct θ_y-orbits of θ_x(f) - f
are zero by residue criteria in Section <ref>. Since b^m, θ_x(b^m), …, θ_x^n(b^m) are in distinct θ_y-orbits, θ_x^n+1(b^m)
must be θ_y-equivalent to one of them. Otherwise, we get

    a_0 = 0,   θ_x(a_0)-a_1 = 0,   …,   θ_x(a_n-1)-a_n = 0,  and  θ_x(a_n) = 0.

Since θ_x is an automorphism on K(x, y), we have a_0 = a_1 = ⋯ = a_n =0, which contradicts the assumption that a_n≠ 0.
If θ_x^n+1(b^m) is θ_y-equivalent to θ_x^i(b^m) for some 0<i≤ n, so is
θ_x^n+1-i(b^m), which contradicts the assumption. Thus,  θ_x^n+1(b^m) = cθ_y^t(b^m) for some c∈ K(x)∖{0} and t∈.
By Lemma <ref>, we have c∈ K∖{0}. A direct calculation leads to

    θ_x(f)- f    =θ_x(a_n)/θ_x^n+1(b^m)-a_0/b^m + ∑_i=1^nθ_x(a_i-1)-a_i/θ_x^i(b^m)=θ_x(a_n)/cθ_y^t(b^m)-a_0/b^m + ∑_i=1^nθ_x(a_i-1)-a_i/θ_x^i(b^m)
       =θ_y^-tθ_x(a_n/c)-a_0/b^m  + ∑_i=1^nθ_x(a_i-1)-a_i/θ_x^i(b^m) + θ_y(u) - u

for some u∈ K(x, y) using the formula (<ref>). By the residue criteria,  we then get a_0 = θ_y^-tθ_x(a_n/c), a_1 = θ_x(a_0),  …, and a_n = θ_x(a_n-1).
This implies that θ_x^n+1(a_0) = c θ_y^t(a_0) and a_i = θ_x^i(a_0) for i∈{1, …, n}. So f = θ_x^n+1-1/θ_x-1∙ h
with h= a_0/b^m, which leads to

    θ_x(f)- f = θ_x^n+1(h)- h = θ_y^t(h) - h = θ_y(g) - g   with    g = θ_y^t-1/θ_y-1∙ h.

Thus, (f, g) is a cyclic WZ-pair. 
































The following theorem is a discrete analogue of Theorem <ref>.


Let f, g∈ K(x, y) be such that ∂_x(f)  = ∂_y(g). Then there exist rational functions a, b_1, …, b_n ∈ K(x, y) such that

    f  = ∂_y(a) + ∑_i =1^n θ_x^s_i-1/θ_x - 1∙ b_i   and   g  = ∂_x(a) + ∑_i =1^n θ_y^t_i-1/θ_y - 1∙ b_i ,

where for each i∈{1, …, n} we have θ_x^s_i(b_i) = θ_y^t_i(b_i) for some s_i∈ and t_i∈ with s_i, t_i not all zero.



By Abramov's reduction and its q-analogue, we can decompose f as

    f = ∂_y(a) + c + ∑_j=1^J f_j    with f_j = ∑_i=1^I ∑_ℓ=0^L_i, ja_i, j, ℓ/θ_x^ℓ(b_i^j),

where a∈ K(x, y), c∈ K(x),  and a_i, j, ℓ b_i ∈ K(x)[y] such that
c=0 if θ_y = _y, _y(a_i, j, ℓ)<_y(b_i), and the b_i's are irreducible and monic polynomials belonging to distinct G-orbits where G=⟨θ_x, θ_y ⟩.
Moreover, θ_x^ℓ_1(b_i^j) and θ_x^ℓ_2(b_i^j) are in distinct θ_y-orbits if ℓ_1 ≠ℓ_2.





By applying Lemma <ref> to the equation θ_x(f)-f = θ_y(g)-g, we get that θ_x(c) - c is θ_y-summable and so is θ_x(f_j)-f_j for each multiplicity j∈{1, …, J}.
By residue criteria for θ_y-sumability and the assumption that the b_i's are in distinct ⟨θ_x, θ_y ⟩-orbits,
we have θ_x(c)-c = 0 and for each i∈{1, …, I}, the rational function f_i, j := ∑_ℓ=0^L_i, ja_i, j, ℓ/θ_x^ℓ(b_i^j)
is either equal to zero or there exists g_i, j∈ K(x, y) such that θ_x(f_i, j)-f_i, j = θ_y(g_i, j)-g_i, j. Then (f_i, j, g_i, j)
is cyclic by Lemma <ref> for every i, j with 1≤ i ≤ I and 1≤ j ≤ J. So the pair (f, g) can be written as

    (f, g) = (∂_y(a), ∂_x(a)) + (c, 0) + ∑_i=1^I∑_j=1^J (f_i, j, g_i, j).

This completes the proof. 


The quotient space _(∂_x, ∂_y)/_(∂_x, ∂_y) is spanned over K by the set

    {(f, g)+_(∂_x, ∂_y)|f, g∈ K(x, y) such that (f, g) is a cyclic pair}.











 §.§ The mixed case
 

In the mixed continuous-discrete setting, we consider the rational WZ-pairs with respect to (θ_x-1, D_y) with θ_x∈{_x, τ_q, x}.



Let p be an irreducible and monic polynomial in K(x)[y].
Then for any nonzero m∈,  we have either (p, θ_x^m(p))=1 or p∈ K[y].


Since θ_x is an automorphism on K(x, y), θ_x^i(p) is irreducible in K(x)[y] for any i∈.
If (p, θ_x^m(p)) ≠ 1, then θ_x^m(p)= c p for some c∈ K(x). Write p = ∑_i=0^d p_i y^i with p_i∈ K(x) and p_d= 1.
Then θ_x^m(p)= c p implies that θ_x^m(p_i) = c p_i for all i with 0≤ i ≤ d. Then c=1 and p_i∈ K for all i with 0≤ i ≤ d-1.
So p∈ K[y]. 

The structure of WZ-pairs in the mixed setting is as follows.


Let f, g∈ K(x, y) be such that θ_x(f) -f   = D_y(g). Then there exist h∈ K(x, y), u∈ K(y) and v∈ K(x) such that

    f  = D_y(h) + u   and   g  = θ_x(h) - h  + v.




By the Ostrogradsky–Hermite reduction, we decompose f into the form

    f = D_y(h) + ∑_i=1^I ∑_j=0^J_ia_i, j/θ_x^j(b_i),

where h∈ K(x, y) and a_i, j, b_i ∈ K(x)[y] with a_i, J_i≠ 0, _y(a_i, j)<_y(b_i) and
b_i being irreducible and monic polynomials in y over K(x) such that the b_i's are in distinct θ_x-orbits.
By a direct calculation, we get

    θ_x(f)  - f = D_y(θ_x(h)- h)+ ∑_i=1^I (θ_x(a_i, J_i)/θ_x^J_i+1(b_i) - a_i, 0/b_i + ∑_j=1^J_iθ_x(a_i, j-1)-a_i, j/θ_x^j(b_i)) .

For all i, j with 1≤ i ≤ I and 0≤ j ≤ J_i + 1, the θ_x^j(b_i)'s are irreducible and monic polynomials in y over K(x).
We first show that for each i∈{1, …, I}, we have b_i∈ K[y]. Suppose that there exists i_0∈{1, …, I}, b_i_0∉ K[y]. Then (θ_x^m(b_i_0), b_i_0)=1 for any nonzero m∈ by Lemma <ref>. Since θ_x(f)-f is D_y-integrable in K(x, y), we have
θ_x(a_i_0, J_i_0) = 0 by Lemma <ref>. Then a_i_0, J_i_0=0, which contradicts the assumption that a_i, J_i≠ 0 for all i
with 1≤ i ≤ I. Since b_i∈ K[y], f can be written as

    f = D_y(h) + ∑_i=1^I a_i/b_i,   where a_i := ∑_j=0^J_i a_i, j.

Since θ_x(f)-f is D_y-integrable in K(x, y) and since

    θ_x(f)- f = D_y(θ_x(h)-h) + ∑_i=1^Iθ_x(a_i)-a_i/b_i,

we have  θ_x(a_i)-a_i = 0 for each i∈{1, …, I} by Lemma <ref>. This implies that a_i∈ K(y) and f = D_y(h) + u with u = ∑_i=1^I a_i/b_i∈ K(y). Since θ_x(f)-f = D_y(g), we get D_y(g - (θ_x(h)-h))=0. Then g = θ_x(h)-h + v for some v∈ K(x). 



The quotient space _(θ_x-1, D_y)/_(θ_x-1, D_y) is spanned over K by the set

    {(f, g)+_(θ_x-1, D_y)|f∈ K(y) and g∈ K(x)}.

















§ CONCLUSION
 
We have explicitly described the structure of rational WZ-pairs in terms of special pairs.
With structure theorems, we can easily generate rational WZ-pairs, which solves Problem <ref>
in the rational case completely.
For the future research, the next direction is to solve
the problem in the cases of more general functions.
Using the terminology of Gessel in <cit.>, a hypergeometric term F(x, y) is said to be a WZ-function if
there exists another hypergeometric term G(x, y) such that (F, G) is a WZ-pair. In the scheme of creative telescoping,
(F, G) being a WZ-pair with respect to (∂_x, ∂_y) is equivalent to that
∂_x being a telescoper for F with certificate G. Complete criteria for the
existence of telescopers for hypergeometric terms and their variants are known <cit.>.
With the help of existence criteria for telescopers, one can show that F(x, y) can be decomposed as the sum
F = ∂_y(H_1) + H_2 with H_1, H_2 being hypergeometric terms and H_2 is of proper form (see definition in <cit.>)
if F is a WZ-function.
So it is promising to apply the ideas in the study of the existence problem of telescopers to explore the structure of WZ-pairs.




Acknowledgment. 
I would like to thank Prof. Victor J.W. Guo and Prof. Zhi-Wei Sun for many discussions on series for special
constants, (super)-congruences and their q-analogues that
can be proved using the WZ method. I am also very grateful to Ruyong Feng and Rong-Hua Wang for many constructive
comments on the earlier version of this paper. I also thank the anonymous reviewers for their constructive
and detailed comments.

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Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger.
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