Bivariate Extensions of Abramov's Algorithm for Rational Summation
aa r X i v : . [ c s . S C ] J un Bivariate Extensions of Abramov’s Algorithmfor Rational Summation
Dedicated to Professor Sergei A. Abramov on the occasion of his 70th birthday
Shaoshi Chen
Abstract
Abramov’s algorithm enables us to decide whether a univariate rationalfunction can be written as a difference of another rational function, which has beena fundamental algorithm for rational summation. In 2014, Chen and Singer gen-eralized Abramov’s algorithm to the case of rational functions in two ( q -)discretevariables. In this paper we solve the remaining three mixed cases, which completesour recent project on bivariate extensions of Abramov’s algorithm for rational sum-mation. Symbolic summation has been a powerful tool in combinatorics and mathematicalphysics, whose history is as long as that of symbolic computation. Abramov’s al-gorithm [1] for rational summation is one of the first few fundamental algorithmsin symbolic summation. The central problem in symbolic summation is whether thesum of a given sequence can be written in “closed form”. A given sequence f ( n ) belonging to some domain D is said to be summable if f = g ( n + ) − g ( n ) for somesequence g ∈ D . The problem of deciding whether a given sequence is summableor not in D is called the summability problem in D . For example, if D is the field ofrational functions, then for f = / ( n ( n + )) we can find g = / n , while for f = / n no suitable g exists in D . When f is not summable in D , there are several other ques-tions we may ask. One possibility is to ask whether there is a pair ( g , r ) in D × D such that f ( n ) = g ( n + ) − g ( n ) + r ( n ) , where r is minimal in some sense and r = f is summable. This problem is called the decomposition problem in [3]. Shaoshi ChenKLMM, AMSS, Chinese Academy of Sciences, 100190, Beijing, Chinae-mail: [email protected]
This work was supported by the NSFC grants 11501552, 11688101 and by the President Fund ofthe Academy of Mathematics and Systems Science, CAS (2014-cjrwlzx-chshsh). 1 Shaoshi Chen
For univariate sequences, extensive work has been done to solve the summabil-ity and decomposition problems. In 1971, Abramov solved the summability prob-lem for univariate rational functions in [1]. The Gosper algorithm [19] solves thesummability problem for univariate hypergeometric terms. This was then used byZeilberger [29] in 1990s to design his celebrated telescoping algorithm for hyper-geometric terms. The Gosper algorithm was extended further to the D -finite case byAbramov and van Hoeij in [6, 7], and to a more general difference-field setting byKarr [22, 23] and Schneider [28]. The decomposition problem was first consideredby Ostrogradsky [24] in 1845 and later by Hermite [20] in the continuous settingfor rational functions. The discrete case was solved by Abramov in [2], with alter-native methods later presented by Abramov himself in [3], and also by Paule [25]and Pirastu [27]. Abramov’s decomposition algorithm was later extended to the hy-pergeometric case in [4, 5], as well as to continuous extensions in [9, 13, 17].In 1993, Andrews and Paule [8] raised the general question: is it possible to pro-vide any algorithmic device for reducing multiple sums to single ones? This ques-tion is related to symbolic summation in the multivariate case. To make the problemmore tractable, we will focus on the first non-trivial case, namely the bivariate ra-tional functions. To this end, let us first introduce some notations. Throughout thepaper, let k be a field of characteristic zero and k ( x , y ) be the field of rational func-tions in x and y . For any f ∈ k ( x , y ) , we define the shift operators σ x , σ y by σ x ( f ( x , y )) = f ( x + , y ) , σ y ( f ( x , y )) = f ( x , y + ) , and the q -shift operators with q ∈ k \ { } by τ x , q ( f ( x , y )) = f ( qx , y ) , τ y , q ( f ( x , y )) = f ( x , qy ) . Let ∆ v : = σ v − ∆ v , q : = τ v , q − q -difference operatorswith respect to v ∈ { x , y } , respectively. On the field k ( x , y ) , we can also define theusual derivations D x : = ∂ / ∂ x and D y : = ∂ / ∂ y . Definition 1.
A rational function f ∈ k ( x , y ) is said to be exact with respect tothe pair ( ∂ x , ∂ y ) ∈ { D x , ∆ x , ∆ x , q } × { D y , ∆ y , ∆ y , q } in k ( x , y ) if f = ∂ x ( g ) + ∂ y ( h ) forsome g , h ∈ k ( x , y ) .We study the following problem, which is a bivariate extension of the summabil-ity problem for univariate rational functions. Exactness Testing Problem.
Given a rational function f ∈ k ( x , y ) , decidewhether or not f is exact with respect to ( ∂ x , ∂ y ) in k ( x , y ) .According to different types of ( ∂ x , ∂ y ) , the above problem has six different casesup to the symmetry between x and y . In the pure continuous case, the problem is alsocalled integrability problem , which was first solved by Picard [26, vol 2, page 220],and see [14] for a more up-to-date presentation. Chen and Singer [16] presentedthe first necessary and sufficient condition for the exactness in the pure discrete and q -discrete cases. Based on the theoretical criterion in [16], Hou and Wang [21] then ivariate Extensions of Abramov’s Algorithm for Rational Summation 3 gave a practical algorithm for deciding the exactness in the corresponding case. Thegoal of this paper is to solve the remaining three mixed cases of the exactness testingproblem, which completes our recent project on bivariate extensions of Abramov’salgorithm for rational summation. In this section, we will prepare some basic tools for testing the exactness of bi-variate rational functions. We first introduce the classical residues and their discreteanalogue for univariate rational functions. After this we will define reduced formsfor bivariate rational functions.Let K be a field of characteristic zero and K ( z ) be the field of rational functionsin z over K . We first define residues with respect to the derivation D z on K ( z ) . Byirreducible partial fraction decomposition, we can always uniquely write a rationalfunction f ∈ K ( z ) as f = p + n ∑ i = m i ∑ j = a i , j d ji , (1)where p , a i , j , d i ∈ K [ z ] , deg z ( a i , j ) < deg z ( d i ) and all of the d i ’s are distinct irre-ducible polynomials. We call a i , the D z -residue of f at d i , denoted by res D z ( f , d i ) .We now recall the discrete analogue of D z -residues introduced in [15, 21].Let φ be an automorphism of K ( z ) that fixes K . For a polynomial p ∈ K [ z ] , wecall the set { φ i ( p ) | i ∈ Z } the φ -orbit of p , denoted by [ p ] φ . Two polynomials p , q ∈ K [ 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 ∈ Z . When φ = σ z , we can uniquely decompose arational function f ∈ K ( z ) into the form f = p ( z ) + n ∑ i = m i ∑ j = e i , j ∑ ℓ = a i , j ,ℓ σ ℓ z ( d i ) j , (2)where p , a i , j ,ℓ , d i ∈ K [ z ] , deg z ( a i , j ,ℓ ) < deg z ( d i ) and all of the d i ’s are irreduciblepolynomials such that any two of them are not σ z -equivalent. We call the sum ∑ e i , j ℓ = σ − ℓ z ( a i , j ,ℓ ) the σ z -residue of f at d i of multiplicity j , denoted by res σ z ( f , d i , j ) .The following lemma shows some commutativity properties of the residues atsome special irreducible polynomials. Lemma 1.
Let f = a / b ∈ k ( x , y ) and d ∈ k [ y ] be an irreducible factor of b. Then thefollowing commutativity formulae hold: ( i ) res D y ( σ x ( f ) , d ) = σ x ( res D y ( f , d )) ; ( ii ) res D y ( τ x , q ( f ) , d ) = τ x , q ( res D y ( f , d )) ; ( iii ) res σ y ( τ x , q ( f ) , d , j ) = τ x , q ( res σ y ( f , d , j )) for all j ∈ N .Proof. To show the first formula, we decompose f ∈ k ( x , y ) into the form Shaoshi Chen f = p + n ∑ i = m i ∑ j = a i , j d ji , where p , a i , j ∈ k ( x )[ y ] , d i ∈ k [ x , y ] with deg y ( a i , j ) < deg y ( d i ) and the d i ’s are distinctirreducible polynomials with d = d ∈ k [ y ] . Since σ x is an automorphism of k ( x , y ) ,we have that σ x ( f ) = σ x ( p ) + m ∑ j = σ x ( a , j ) d j + n ∑ i = m i ∑ j = σ x ( a i , j ) σ x ( d i ) j is the irreducible partial fraction decomposition of σ x ( f ) with respect to y over k ( x ) . Then res D y ( σ x ( f ) , d ) = σ x ( a , ) = σ x ( res D y ( f , d )) . The second formula canbe proved similarly. To show the third formula, we decompose f into the form f = p + n ∑ i = m i ∑ j = e i , j ∑ ℓ = a i , j ,ℓ σ ℓ y ( d i ) j , where p , a i , j ,ℓ ∈ k ( x )[ y ] , d i ∈ k [ x , y ] with deg y ( a i , j ,ℓ ) < deg y ( d i ) and the d i ’s are ir-reducible polynomials in distinct σ y -orbits with d = d ∈ k [ y ] . Since σ y is an auto-morphism of k ( x , y ) , the polynomial d ∈ k [ y ] is not σ y -equivalent to any irreduciblepolynomial d ′ ∈ k [ x , y ] with deg x ( d ′ ) =
0. Then we can decompose τ x , q ( f ) into theform τ x , q ( f ) = τ x , q ( p ) + m ∑ j = e , j ∑ ℓ = τ x , q ( a , j ,ℓ ) σ ℓ y ( d ) j + st , where s ∈ k ( x )[ y ] and t ∈ k [ x , y ] satisfying that deg y ( s ) < deg y ( t ) and any irreduciblefactor of t is not σ y -equivalent to d . Then for all j ∈ N we haveres σ y ( τ x , q ( f ) , d , j ) = e , j ∑ ℓ = σ − ℓ y τ x , q ( a , j ,ℓ ) = τ x , q e , j ∑ ℓ = σ − ℓ y ( a , j ,ℓ ) ! = τ x , q ( res σ y ( f , d , j )) . This completes the proof. ⊓⊔ Let φ be any automorphism of k ( x , y ) that fixes k ( y ) which will be taken as τ x , q or σ x in the next section. Then φ commutes with D y . To study the exactness test-ing problem with respect to the pair ( φ , D y ) , we define reduced forms for rationalfunctions in k ( x , y ) as follows. Definition 2.
A rational function r = ∑ mi = a i d i with a i ∈ k ( x )[ y ] and d i ∈ k [ x , y ] is said to be ( φ , D y ) -reduced if deg y ( a i ) < deg y ( d i ) and the d i ’s are irreduciblepolynomials in distinct φ -orbits. Let f ∈ k ( x , y ) . We call the decomposition f = φ ( g ) − g + D y ( h ) + r with g , h , r ∈ k ( x , y ) and r being ( φ , D y ) -reduced a ( φ , D y ) -reduced form of f .We next show that ( φ , D y ) -reduced forms always exist for rational functions in k ( x , y ) . For any rational function f ∈ k ( x , y ) , Ostrogradsky–Hermite reduction [24,20] decomposes f into the form ivariate Extensions of Abramov’s Algorithm for Rational Summation 5 f = D y ( h ) + m ∑ i = a i d i , (3)where h ∈ k ( x , y ) , a i ∈ k ( x )[ y ] , d i ∈ k [ x , y ] satisfying that deg y ( a i ) < deg y ( d i ) and the d i ’s are irreducible over k ( x ) . Let φ , φ be two automorphisms of k ( x , y ) such that φ ( φ ( f )) = φ ( φ ( f )) for all f ∈ k ( x , y ) . Then for any a , d ∈ k ( x )[ y ] , m , n ∈ N , wehave the following reduction formula a φ m φ n ( d ) = φ ( u ) − u + φ ( v ) − v + φ − m φ − n ( a ) d (4)where u = m − ∑ j = φ j − m ( a ) φ j φ n ( d ) and v = n − ∑ k = φ k − n φ − m ( a ) φ k ( d ) . By applying the above reduction formula to (3) with φ = φ and φ = id , we canfurther decompose f as f = φ ( g ) − g + D y ( h ) + ˜ m ∑ i = ˜ a i ˜ d i , where g ∈ k ( x , y ) and the ˜ d i ’s are in distinct φ -orbits, which is a ( φ , D y ) -reducedform of f . The above process for obtaining such a ( φ , D y ) -reduced form of f iscalled a ( φ , D y ) -reduction .Next we will define reduced forms for rational functions in k ( x , y ) with respect tothe pair ( τ x , q , σ y ) . Two polynomials p , p ′ ∈ k [ x , y ] are said to be ( τ x , q , σ y ) -equivalentif p = τ mx , q σ ny ( p ′ ) for some m , n ∈ Z . The set { τ ix , q σ jy ( p ) ∈ k [ x , y ] | i , j ∈ Z } is calledthe ( τ x , q , σ y ) -orbit of p , denoted by [ p ] ( τ x , q , σ y ) . Definition 3.
A rational function r = ∑ ni = ∑ m i j = a i , j d ji ∈ k ( x , y ) with a i , j ∈ k ( x )[ y ] and d i ∈ k [ x , y ] is said to be ( τ x , q , σ y ) -reduced if deg y ( a i , j ) < deg y ( d i ) and the d i ’s are irreducible polynomials in distinct ( τ x , q , σ y ) -orbits. The decomposition f = ∆ x , q ( g ) + ∆ y ( h ) + r with g , h , r ∈ k ( x , y ) and r being ( τ x , q , σ y ) -reduced is calleda ( τ x , q , σ y ) -reduced form of f .The existence of ( τ x , q , σ y ) -reduced forms for rational functions relies on Abr-ramov’s reduction [3] that decomposes a rational function f ∈ k ( x , y ) into the form f = ∆ y ( h ) + n ∑ i = m i ∑ j = a i , j d ji , where h ∈ k ( x , y ) , a i , j ∈ k ( x )[ y ] , d i ∈ k [ x , y ] satisfying that deg y ( a i , j ) < deg y ( d i ) andthe d i ’s are irreducible polynomials in distinct σ y -orbits. Using the formula (4) with φ = τ x , q and φ = σ y , we can further decompose f as Shaoshi Chen f = ∆ x , q ( g ) + ∆ y ( h ) + ˜ n ∑ i = m i ∑ j = a i , j d ji , where g ∈ k ( x , y ) and the d i ’s are in distinct ( τ x , q , σ y ) -orbits, which is a ( τ x , q , σ y ) -reduced form of f . The above process for obtaining such a ( τ x , q , σ y ) -reduced formof f is called a ( τ x , q , σ y ) -reduction . We first solve the exactness testing problem for the case in which q ∈ k is a root ofunity. Assume that m is the minimal positive integer such that q m = k containsall m th roots of unity. For any f ∈ k ( x , y ) , it is easy to show that τ x , q ( f ) = f if andonly if f ∈ k ( y )( x m ) . Note that k ( x , y ) is a finite algebraic extension of k ( y )( x m ) ofdegree m . We recall a lemma in [16] on reduced forms for rational functions withrespect to τ x , q . Lemma 2.
Let q be such that q m = with m minimal and let f ∈ k ( x , y ) .(a) f = τ x , q ( g ) − g for some g ∈ k ( x , y ) if and only if the trace Tr k ( x , y ) / k ( y )( x m ) ( f ) = .(b) Any rational function f ∈ k ( x , y ) can be decomposed intof = τ x , q ( g ) − g + c , where g ∈ k ( x , y ) and c ∈ k ( y )( x m ) . (5) Moreover, f is τ x , q -summable in k ( x , y ) if and only if c = . We call this decom-position a τ x , q -reduced form for f . Theorem 1.
Let q be such that q m = with m minimal and let f ∈ k ( x , y ) . Assumethat f = τ x , q ( g ) − g + c with g ∈ k ( x , y ) and c ∈ k ( y )( x m ) is a τ x , q -reduced form of f .Then f is exact with respect to ( τ x , q , ∂ y ) with ∂ y ∈ { ∆ y , D y } if and only if c = ∂ y ( d ) for some d ∈ k ( y )( x m ) .Proof. The sufficiency is clear. To show the necessity, we assume that f is exactwith respect to ( τ x , q , ∂ y ) with ∂ y ∈ { ∆ y , D y } , so is c , i.e., c = ∆ x , q ( u ) + ∂ y ( v ) forsome u , v ∈ k ( x , y ) . Write u = ∑ m − i = u i x i and v = ∑ m − i = v i x i with u i , v i ∈ k ( y , x m ) .Then we have c = u ( q − ) x + · · · + u m − ( q m − − ) x m − + m − ∑ i = ∂ y ( v i ) x i . Since 1 , x , . . . , x m − are linearly independent in k ( x , y ) over k ( y , x m ) , we get that c = ∂ y ( v ) . ⊓⊔ From now on, we assume that q ∈ k \ { } is not a root of unity. For any f ∈ k ( x , y ) ,we have τ x , q ( f ) = f if and only if f ∈ k ( y ) . We next solve the exactness testingproblem in the case when ∂ x ∈ { ∆ x , ∆ x , q } and ∂ y = D y . ivariate Extensions of Abramov’s Algorithm for Rational Summation 7 Theorem 2.
Let φ ∈ { σ x , τ x , q } and f ∈ k ( x , y ) . Assume that f = φ ( g ) − g + D y ( h ) + ∑ mi = a i / d i with a i ∈ k ( x )[ y ] and d i ∈ k [ x , y ] be a ( φ , D y ) -reduced form of f . Then f isexact with respect to ( ∂ x , D y ) with ∂ x = φ − if and only if for each i ∈ { , . . . , m } ,d i ∈ k [ y ] and a i = ∂ x ( b i ) for some b i ∈ k ( x )[ y ] .Proof. The sufficiency is clear. To show the necessity, we assume that f is exactwith respect to ( ∂ x , D y ) . This implies that r = ∑ mi = a i / d i is also exact with respectto ( ∂ x , D y ) , i.e., r = φ ( u ) − u + D y ( v ) for some u , v ∈ k ( x , y ) . By the Ostrogradsky–Hermite reduction, we first decompose u into the form u = D y ( ˜ u ) + s ∑ i = v i w i , where ˜ u ∈ k ( x , y ) , v i ∈ k ( x )[ y ] , and the w i ’s are irreducible polynomials in k [ x , y ] .Then we have r = m ∑ i = a i d i = T + D y ( ˜ v ) with T = s ∑ i = (cid:18) φ ( v i ) φ ( w i ) − v i w i (cid:19) and ˜ v = φ ( ˜ u ) − ˜ u + v . Since φ is an automorphism of k [ x , y ] , the polynomials φ ( w i ) are also irreducibleand all of the simple fractions in the irreducible partial fraction decomposition of T have simple poles.We first show that all of the d i ’s are in k [ y ] . Set D : = { d , . . . , d m } and W : = { w , . . . , w s } . Note that all of the simple fractions in D y ( ˜ v ) have at least doublepoles. This implies that r = T and each simple fraction a i / d i can only be cancelledwith some simple fractions of T . Then for each i ∈ { , ..., m } , d i is equal to w j or φ ( w j ) for some j ∈ { , . . . , s } . Assume that d i = w j . If φ ( w j ) = w j , then w j ∈ k [ y ] by [15, Lemma 3.4]. Otherwise, φ ( w j ) = w j for some j ∈ { , . . . , s } \ { j } .Indeed, If φ ( w j ) = d j with i = j , then d i is φ -equivalent to d j , which contradictswith the assumption that the d i ’s are in distinct φ -orbits. If w j = φ ( w j ) , we alsoget that w j is in k [ y ] and so is d i . Otherwise φ ( w j ) = w j for some j ∈ { , . . . , s } \{ j , j } . Continuing this process, we either conclude that d i ∈ k [ y ] or get a series ofequalities d i = w j , φ ( w j ) = w j , φ ( w j ) = w j , . . . . Since the set W is finite, there exists t with 1 ≤ t ≤ s such that φ ( w j t ) = w j ˜ t with1 ≤ ˜ t ≤ t . Then w j ˜ t = φ t − ˜ t + ( w j ˜ t ) , which implies that w j ˜ t is in k [ y ] and so is d i .Similarly, we have d i ∈ k [ y ] when d i = φ ( w j ) .Since d i ∈ k [ y ] , applying the commutativity formulae in Lemma 1 yields a i = res D y ( r , d i ) = res D y ( φ ( u ) − u + D y ( v ) , d i ) = res D y ( φ ( u ) − u , d i ) = φ ( b i ) − b i , where b i = res D y ( u , d i ) ∈ k ( x )[ y ] . ⊓⊔ Example 1.
By Theorem 2, the rational function 1 / ( x + y ) is not exact with respectto ∆ x and D y since x + y is not in k [ y ] . So is the rational function 1 / ( xy ) since1 / x = ∆ x ( g ) for any g ∈ k ( x , y ) . Shaoshi Chen
We now consider the exactness testing problem in the case when ∂ x = ∆ x , q and ∂ y = ∆ y . To this end, we first recall a lemma which is a special case of Lemma 5.4in [10]. Lemma 3.
Let p be an irreducible polynomial in k [ x , y ] . Assume that τ ix , q σ jy ( p ) = pfor some i , j ∈ Z with i = . Then p ∈ k [ y ] . Let f ∈ k ( x , y ) . We assume that f = ∆ x , q ( g ) + ∆ y ( h ) + r is a ( τ x , q , σ y ) -reducedform of f . Write r = ∑ ni = ∑ m i j = a i , j d ji , where a i , j ∈ k ( x )[ y ] and d i ∈ k [ x , y ] satisfyingthat deg y ( a i , j ) < deg y ( d i ) and the d i ’s are in distinct ( τ x , q , σ y ) -orbits. Then f is exactwith respect to ( ∆ x , q , ∆ y ) if and only if r is exact with respect to ( ∆ x , q , ∆ y ) . Notethat the operators τ x , q and σ y preserve the multiplicities of irreducible factors in thedenominators of rational functions. Therefore the rational function r is exact withrespect to ( ∆ x , q , ∆ y ) if and only if for each j , the rational function r j = m ∑ i = a i , j d ji (6)is exact with respect to ( ∆ x , q , ∆ y ) . By the same argument in the proof of Lemma 3.2in [21], r j is exact with respect to ( ∆ x , q , ∆ y ) if and only if each simple fraction a i , j / d ji is exact with respect to ( ∆ x , q , ∆ y ) . We now give an exactness criterion for rationalfunctions of the form a / d m . Lemma 4.
Let f = a / d m , where m ∈ N , d ∈ k [ x , y ] is an irreducible polynomialand a ∈ k ( x )[ y ] is nonzero and deg y ( a ) < deg y ( d ) . Then f is exact with respect to ( ∆ x , q , ∆ y ) if and only if d ∈ k [ y ] and a = ∆ x , q ( b ) for some b ∈ k ( x )[ y ] .Proof. The sufficiency is clear. For the necessity, we will outline the same argumentused in the proof of Theorem 3.7 in [16] or that of Proposition 3.4 in [21]. Weassume that f is exact with respect to ( ∆ x , q , ∆ y ) , i.e., there exist g , h ∈ k ( x , y ) suchthat f = ∆ x , q ( g ) + ∆ y ( h ) . (7)We decompose the rational function g into the form g = σ y ( g ) − g + g + λ τ µ x , q d m + · · · + λ s τ µ s x , q d m , (8)where λ k ∈ k ( x )[ y ] , µ k ∈ Z , g , g ∈ k ( x , y ) such that g is a rational function havingno terms of the form λ / ( τ µ x , q d m ) in its partial fraction decomposition with respectto y , and the ( τ µ i x , q d m ) ’s are irreducible polynomials in distinct σ y -orbits.The following claim can be shown by the same argument as in [16, 21]. Claim 1.
Let Λ : = { τ µ x , q d , . . . , τ µ s x , q d , τ µ + x , q d , . . . , τ µ s + x , q d } . Then: (1) at least one element of Λ is in the same σ y -orbit as d ; (2) for each η ∈ Λ ,there is one element of ( Λ \{ η } ) ∪ { d } that is σ y -equivalent to η . ivariate Extensions of Abramov’s Algorithm for Rational Summation 9 Claim 1 implies that either d ∼ σ y τ µ ′ x , q d or d ∼ σ y τ µ ′ + x , q d for some µ ′ ∈ { µ , . . . , µ s } .Assume that d ∼ σ y τ µ ′ x , q d . By the same argument as in [16, 21], we can show thatthere exists a positive integer t ≤ s and j ∈ Z such that τ tx , q σ jy ( d ) = d , which implies d ∈ k [ y ] by Lemma 3. Similarly, if d ∼ σ y τ µ ′ + x , q d , then we also have d ∈ k [ y ] .Since d ∈ k [ y ] , applying the commutativity formulae in Lemma 1 yields a = res σ y ( f , d , m ) = res σ y ( ∆ x , q ( g ) + ∆ y ( h ) , d , m ) = res σ y ( ∆ x , q ( g ) , d , m ) = ∆ x , q ( b ) , where b = res σ y ( g , d , m ) ∈ k ( x )[ y ] . ⊓⊔ We conclude the above discussions by the following theorem.
Theorem 3.
Let f ∈ k ( x , y ) and assume thatf = ∆ x , q ( g ) + ∆ y ( h ) + n ∑ i = m i ∑ j = a i , j d ji with a i , j ∈ k ( x )[ y ] and d i ∈ k [ x , y ] is a ( τ x , q , σ y ) -reduced form of f . Then f is exactwith respect to the pair ( ∆ x , q , ∆ y ) if and only if for each i ∈ { , . . . , n } , d i ∈ k [ y ] andfor each j ∈ { , . . . , m i } , a i , j = ∆ x , q ( b i , j ) for some b i , j ∈ k ( x )[ y ] .Example 2. By Theorem 3, the rational function 1 / ( x + y ) is not exact with respectto ∆ x , q and ∆ y since x + y is not in k [ y ] . But the rational function 1 / ( xy ) is exact withrespect to ∆ x , q and ∆ y . In fact, xy = ∆ x , q (cid:16) q ( − q ) xy (cid:17) . Remark 1.
The exactness criteria given above reduce the exactness testing problemin the bivariate case to two subproblems: one is testing whether an irreducible poly-nomial p ∈ k [ x , y ] is free of x , the other is testing whether a rational function is ( q ) -summable or not with respect to x . The first subproblem is easy and the secondone can be solved by Abramov’s algorithm and its q -analogue for univariate rationalsummation. We conclude this paper by recalling the following open problem proposed in [12]:
Problem 1.
Develop an algorithm which takes as input a multivariate hypergeo-metric term h in m discrete variables k , . . . , k m , and decides whether there existhypergeometric terms g , . . . , g m such that h = ∆ ( g ) + · · · + ∆ m ( g m ) . Here, ∆ i is the forward difference operator with respect to the variable k i , i.e., ∆ i f ( k , . . . , k m ) = f ( k , . . . , k i + , . . ., k m ) − f ( k , . . . , k i , . . . , k m ) . A solution of this problem would be an important step towards the developmentof a Zeilberger-like algorithm for multisums. Together with the results in [16, 21],the exactness criteria in previous section enable us to completely solve the aboveproblem in the case of bivariate rational functions. The summability criteria in [16,21] were used in [11] to derive some conditions on the existence of telescopersfor trivariate rational functions. Hopefully, the results in this paper can be used tosolve the corresponding existence problems for the three mixed cases. An answerto the above open problem may analogously allow for the formulation of existencecriteria for telescopers in the multivariate setting. In the long run, we would hopethat a multivariate Gosper algorithm serves as a starting point for the developmentof a reduction-based creative telescoping algorithm for the multivariate setting. Anecessary condition for bivariate hypergeometric summability has been given in [18]with many applications but the summability criterion in this case is still missing andfurther new ideas and tools are needed to be developed.
Acknowledgment.
I would like to thank Hui Huang and Rong-Hua Wang for theirconstructive comments on the early version of this paper.
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