1 Introduction
This paper is about creative telescoping for hypergeometric terms. A hypergeometric term is an expression in, say, two variables such that the two shift quotients and can be expressed as rational functions in and . The prototypical example of a hypergeometric term is the binomial coefficient . Creative telescoping is the main tool for simplifying definite sums of hypergeometric terms. The task consists in finding some nonzero recurrence operator and another hypergeometric term such that . It is required that the operator does not contain or the shift operator , i.e., it must have the form for some that only depend on .
If and are as above, we say that is a telescoper for , and is a certificate for . Once a telescoper for is known, we can extract useful information about definite sums such as from . See [13, 14] for further information. These references also contain classical algorithms for computing telescopers and certificates for given hypergeometric terms. During the past 25 years, the technique of creative telescoping has been generalized and refined in various ways [12, 6, 7, 8, 9, 10, 11]. The latest trend in this development are socalled reductionbased algorithms, first presented in [7]. One of their features is that they can find a telescoper for a given term without also computing the corresponding certificate. This is interesting because a certificate is not always needed, and it is typically much larger (and thus computationally more expensive) than the telescoper, so we may not want to compute it if we don’t have to.
Reductionbased algorithms have been first developed in the differential case, for various cases [7, 8, 9, 11]. The basic idea, formulated for the shift case, is as follows. Let be a field of characteristic zero. Suppose we know a function , called reduction, with the property that for all in the domain under consideration, say , containing , there exists a in the same domain such that , i.e., the difference is a summable term. We call a remainder of with respect to the reduction . Then in order to find a telescoper for , we can compute until we find a linear dependence over the field . If such a dependence is found, say for some in , then is a telescoper for .
In order to show that this method terminates, one possible approach is to show that the
vector space spanned by
for has a finite dimension. Then, as soon as exceeds this dimension, we can be sure that a telescoper will be found. This approach was taken in [8, 9, 11]. As a nice side result, this approach provides an independent proof of the existence of telescopers, and even a bound on their order. In the paper from last year [10], the authors used a different approach. Instead of showing that the remainders form a finitedimensional vector space, they showed that for every summable term , we have . This also ensures that the method terminates (assuming that we already know for other reasons that a telescoper exists), and in fact that it will find the smallest possible telescoper, but it does not provide a bound on its order.This discrepancy in the approaches for the differential case and the shift case is unpleasant. It is not clear why the shift case should require a different argument. The goal of the present paper is to show that it does not. We will continue the development of last year’s theory to a point where we can also show that the remainders belong to a finitedimensional vector space. As a result, we obtain new bounds for the order of telescopers for hypergeometric terms. We obtain lower as well as upper bounds. We do not find exactly the same bounds that are already in the literature [12, 2]. Comparing our bounds to the known bounds, it appears that for “generic” input, the values often agree (of course, because the known bounds are already generically sharp). However, there are some special examples in which our bounds are better than the known bounds. On the other hand, our bounds are never worse than the old bounds.
2 Preliminaries
Using the same notations as in [10], we let be a field of characteristic zero, and be the field of rational functions in over . Let be the automorphism that maps to for every . The pair is called a difference field. A difference ring extension of is a ring containing together with a distinguished endomorphism whose restriction to agrees with the automorphism defined before. An element is called a constant if . We denote by the degree of a nonzero polynomial .
Definition 2.1.
Let be a difference ring extension of . A nonzero element is called a hypergeometric term over if for some . We call the shift quotient of w.r.t. .
A univariate hypergeometric term is called hypergeometric summable if there exists another hypergeometric term s.t. , where denotes the difference of and the identity map. We abbreviate “hypergeometric summable” as “summable” in this paper.
Recall [3, §1] that a nonzero polynomial in is said to be shiftfree if no two distinct roots differ by an integer. A nonzero rational function in is said to be shiftreduced if its numerator is coprime with any shift of its denominator.
According to [3, 5], for a given hypergeometric term there always exists a rational function and another hypergeometric term whose shift quotient is shiftreduced, s.t. . This is called a multiplicative decomposition of . We call the shift quotient a kernel of and the corresponding shell.
Based on Abramov and Petkovšek’s work in [3, 5], the authors of [10] presented a modified version of AbramovPetkovšek reduction, which determines summability without solving any auxiliary difference equations. To describe it concisely, we first recall some terminology.
Let be a hypergeometric term whose kernel is and the corresponding shell is . Then , where is a hypergeometric term whose shift quotient is . Write , where are polynomials in with .
Definition 2.2.
A nonzero polynomial in is said to be strongly prime with if for all .
Now define the linear map from to itself by sending to for all . We call the map for polynomial reduction w.r.t. . Let
Then , and thus we call the standard complement of .
Definition 2.3.
Let be a rational function in . Another rational function in is called a (discrete) residual form of w.r.t. if there exists and in s.t.
where , , is shiftfree and strongly prime with , and belongs to . For brevity, we just say that is a residual form w.r.t. if is clear from the context. We call the significant denominator of .
The modified AbramovPetkovšek reduction [10, Theorem 4.8] can be stated as follows.
Theorem 2.4.
With the notations given above, the modified version of the AbramovPetkovšek reduction computes a rational function in and a residual form w.r.t. , such that
(1) 
Moreover, is summable if and only if .
3 Properties of residual forms
In this section, we will explore important properties of residual forms, which enables us to derive nontrivial relationship among remainders in Section 5.
Unlike the differential case, a rational function may have more than one residual form in shift case. However, these residual forms are related to each other in some way. Before describing it, let us recall some technology.
Recall [5, §2] that polynomials are said to be shiftequivalent (w.r.t. ) if for some , denoted by . It is an equivalence relation.
Let be a rational function in . We call the rational function pair a rational normal form (RNF) of if and is shiftreduced. By [5, Theorem 1], every rational function has at least one RNF. Let be a hypergeometric term over . It is not hard to see that is an RNF of if and only if and are a kernel and the corresponding shell of .
Definition 3.1.
Two shiftfree polynomials are called shiftrelated (w.r.t. ), denoted by , if for any nontrivial monic irreducible factor of , there exists a unique monic irreducible factor of with the same multiplicity as in s.t. , and vice versa.
One can show that is an equivalence relation.
Proposition 3.2.
Let be a shiftreduced rational function in . Assume that are both residual forms of the same rational function in w.r.t. . Then the significant denominators of and are shiftrelated to each other.
Proof.
Assume that are of the forms
where , , , is shiftfree and strongly prime with , for , and is the denominator of . Since are both residual forms of the same rational function, there exists s.t.
It follows that
(2) 
Let be a nontrivial monic irreducible factor of with multiplicity . If divides , then we are done. Otherwise, let be the denominator of . Then divides or as . If divides , let
Then and . Since is strongly prime with , . Apparently, neither nor is divisible by as is shiftfree and is maximal. Hence (2) implies is the required factor of . Similarly, we can show that with
is the required factor of , if divides .
In summary, there always exists a monic irreducible factor of with multiplicity at least s.t. it is shiftequivalent to . Due to the shiftfreeness of , this factor is unique. Conversely, the proof proceeds in a similar way as above. According to the definition, . ∎
Given a hypergeometric term, it is readily seen that the above proposition reveals the relationship between two residual forms w.r.t. the same kernel. To study the case with different kernels, we need to develop two lemmas.
Lemma 3.3.
Let be an RNF of a rational function in and be a residual form of w.r.t. . Write
Assume that is a nontrivial monic irreducible factor of with multiplicity . Then
is an RNF of , in which . Moreover, there exists a residual form of w.r.t. whose significant denominator is equal to that of .
Proof.
Since is shiftreduced, so is . Then the first assertion follows by noticing
Let be of the form where , , , is shiftfree and strongly prime with , and . Then there exists s.t.
which implies that
Since is strongly prime with and , we have . According to Lemma 4.2 and Remark 4.3 in [10], there exist with and , and s.t.
Note that is strongly prime with , so is also strongly prime with . By the shiftfreeness of ,
is a residual form of w.r.t. . The lemma follows. ∎
Lemma 3.4.
Let be an RNF of a rational function in and be a residual form of w.r.t. . Write
Assume that is a nontrivial monic irreducible factor of with multiplicity . Then
is an RNF of , in which . Moreover, there exists a residual form of w.r.t. whose significant denominator is equal to that of .
Proof.
Similar to Lemma 3.3. ∎
Proposition 3.5.
Let be an RNF of a rational function in and be a residual form of w.r.t. . Then there exists another RNF of such that

has shiftfree numerator and shiftfree denominator;

there exists a residual form of w.r.t. whose significant denominator is equal to that of .
Proof.
Let with and , and be the significant denominator of .
Assume that is not shiftfree. Then there exist two nontrivial monic irreducible factors and of with multiplicity and , respectively. W.L.O.G., suppose further that is not a factor of for all and . By Lemma 3.3, has an RNF , in which has a denominator , where , and the numerator remains to be . Moreover, there exists a residual form of w.r.t. whose significant denominator is . If , is an irreducible factor of with multiplicity . Otherwise, it is an irreducible factor of with multiplicity . More importantly, is not a factor of for all . Iteratively using the argument, we arrive at an RNF of such that divides the denominator of the new kernel with certain multiplicity but does not whenever . Moreover, there exists a residual form of the new shell with respect to the new kernel whose significant denominator is equal to . Applying the same argument to each irreducible factor, we can obtain an RNF of whose kernel has a shiftfree denominator and whose shell has a residual form with significant denominator .
With Lemma 3.4, one can obtain an RNF of whose kernel has a shift free numerator whose shell has a residual form with significant denominator . ∎
A nonzero rational function is said to be shiftfree if it is shiftreduced and its denominator and numerator are both shiftfree. The main result is given below.
Proposition 3.6.
Let and be two RNF’s of a rational function in , and be residual forms of (w.r.t. ) and (w.r.t. ), respectively. Then the significant denominators of and are shiftrelated.
Proof.
Let and be the significant denominators of and , respectively. By the above proposition, there exist two RNF’s and of such that their kernels are shiftfree and their shells have residual forms whose significant denominators are and , respectively.
According to [5, Theorem 2], the respective denominators and of and are shiftrelated. Thus, for a nontrivial monic irreducible factor of with multiplicity , there exists a unique factor of with the same multiplicity. W.L.O.G., we may assume . Otherwise, we can switch the roles of and . If , a repeated use of Lemma 3.3 leads to a new RNF from such that is shiftfree, is a factor of the denominator of with the same multiplicity.
Applying the above argument to each irreducible factor and using Lemma 3.4 for numerators in the same fashion, we can obtain two new RNF’s whose kernels are equal and whose shells have respective residual forms with significant denominators and . It follows that and are shiftrelated by Proposition 3.2. ∎
4 Telescoping via reductions
We now translate terminology concerning univariate hypergeometric terms to bivariate ones. Let be a field of characteristic zero, and be the field of rational functions in and over . Let be the shift operators w.r.t. and , respectively, defined by,
for any in . Then the pair forms a partial difference field.
Definition 4.1.
Let be a partial difference ring extension of . A nonzero element is called a hypergeometric term over if there exist s.t. and . We call and the shift quotient and shift quotient of , respectively.
An irreducible polynomial in is called integerlinear over if there exists a univariate polynomial and two integers s.t. . A polynomial in is called integerlinear over if all its irreducible factors are integerlinear. A rational function in is called integerlinear over if its denominator and numerator are both integerlinear.
Let be the ring of linear recurrence operators in , in which the commutation rule is that for all . The application of an operator to a hypergeometric term is defined as .
Given a hypergeometric term over , the computational problem of creative telescoping is to construct a nonzero operator s.t.
for some hypergeometric term . We call a telescoper for w.r.t. and a certificate for . To avoid unnecessary duplication, we make a convention.
Convention 4.2.
Let be a hypergeometric term over with a multiplicative decomposition , where is in and is a hypergeometric term whose shift quotient is shiftreduced w.r.t. . By [4, Theorem 8], we know is integerlinear over . Write where and .
For hypergeometric terms, telescopers do not always exist. Abramov presented a criterion for determining the existence of telescopers in [1, Theorem 10]. With Convention 4.2, applying the modified AbramovPetkovšek reduction to w.r.t. yields (1). By Abramov’s criterion, has a telescoper if and only if the significant denominator of in (1) is integerlinear over . Based on this criterion and the modified reduction, the authors of [10] developed a reductionbased telescoping algorithm, named ReductionCT, which either finds a minimal telescoper for , or proves that no telescoper exists. The key advantage of this algorithm is that it separates the computation of telescopers from that of certificates. This is desirable in the typical situation where we are only interested in the telescopers and their size is much smaller than that of certificates.
When the existence of telescopers for is guaranteed, we summarize below the idea of the algorithm ReductionCT.
We begin by fixing the order of a telescoper for , say , and then look for a telescoper of that order. If none exists, we look for one of the next higher order. We make an ansatz
with undetermined coefficients . For from to , iteratively applying the modified reduction to and manipulating the resulting residual forms according to Theorem 5.6 in [10] lead to
(3) 
where , with , , is shiftfree w.r.t. and strongly prime with , and belongs to . Moreover, the least common multiple of is shiftfree w.r.t. . Let
Then , is shiftfree w.r.t. and strongly prime with . Moreover, is a linear space over , so is in . A direct calculation shows that
According to Theorem 2.4, is summable w.r.t. if and only if . Equivalently, is a telescoper for if and only if the linear system
(4) 
has a nontrivial solution in . A linear dependence among these residual forms , for minimal , gives rise to a minimal telescoper for .
The termination of the algorithm ReductionCT is guaranteed by Abramov’s criterion, see Theorem 6.3 in [10] for more details. However, instead of using Abramov’s criterion, one could prove the algorithm ReductionCT terminates by showing that the residual forms from (3) form a finitedimensional vector space over . This is exactly what we are going to do in the next section.
5 Finitedimensional remainders
In this section, we will show that some sequence of satisfying (3) has a common multiple , provided that has a telescoper. Moreover, is shiftfree and strongly prime with . The existence of this common multiple implies that the corresponding from (3) span a finitedimensional vector space over , and lead to upper and lower bounds on the order of minimal telescopers. To this end, we need some preparations.
5.1 Shifthomogeneous decomposition
Recall [3] that irreducible polynomials in are said to be shiftequivalent w.r.t. , denoted by , if there exist two integers such that . Clearly is an equivalence relation. Choosing the pure lexicographic order , we say a polynomial is monic if its highest term has coefficient . A rational function is said to be shifthomogeneous if all nonconstant monic irreducible factors of its denominator and numerator belong to the same shiftequivalence class.
By grouping together the factors in the same shiftequivalence class, every rational function can be decomposed into the form
(5) 
where , , each is a shifthomogeneous rational function, and any two nonconstant monic irreducible factors of and are pairwise shiftinequivalent whenever . We call (5) a shifthomogeneous decomposition of . The shifthomogeneous decomposition is unique up to the order of the factors and multiplication by nonzero constants.
Let be an irreducible integerlinear polynomial. Then for and . W.L.O.G., we further assume that and . By Bézout’s relation, there exist unique integers with and such that . Define to be . For brevity, we just write if is clear from the context. Note that with , which allows us to treat integerlinear polynomials as univariate ones. For a Laurent polynomial in with and , define
It is readily seen that for any two irreducible integerlinear polynomials of the forms and with , , and , we have if and only if , and for some integer , in which .
Adapt from (5), every integerlinear rational function in admits the following decomposition
(6) 
where , , each is irreducible, monic and integerlinear over , and then for , with , , and . Moreover, whenever . W.L.O.G., we further assume that belongs to .
5.2 Relationship among remainders
With Proposition 3.6, we can describe an inherent relationship among any residual forms satisfying (3).
Lemma 5.1.
With Convention 4.2, let be a residual form of w.r.t. . Then and are a kernel and the corresponding shell of w.r.t. . Moreover, is a residual form of w.r.t. .
Proof.
According to Convention 4.2, and is the shift quotient of . To prove the first assertion, one needs to show that is shiftreduced w.r.t. , which can be proven by observing that, for any two polynomials , if and only if . Let , where belong to , , , is shiftfree and strongly prime with , and . Clearly, we have and . The shiftfreeness and strong primeness w.r.t. of easily follows by the above observation.
Note that and , where is the leading coefficient of a polynomial in . So the standard complements and for polynomial reduction have the same echelon basis according to the case study in [10, §4.2]. It follows from that . Accordingly, is a residual form of w.r.t. . ∎
Proposition 5.2.
Proof.
It suffices to show . The rest follows by a direct induction on .
The following lemma says that, with Convention 4.2, for any polynomial in , there always exists s.t. and is strongly prime with .
Lemma 5.3.
With Convention 4.2, assume that is an irreducible polynomial in . Then there exists an integer s.t. is strongly prime with .
Proof.
It suffices to consider the following three cases according to the definition of strong primeness.
Case 1. is strongly prime with . Then the lemma follows by letting .
Case 2. There exists an integer s.t. . Then for every integer , we have , since is shiftreduced w.r.t. . Let
One can see that is strongly prime with .
Case 3. There exists an integer s.t.