New Bounds for Hypergeometric Creative Telescoping

04/27/2016 ∙ by Hui Huang, et al. ∙ 0

Based on a modified version of Abramov-Petkovšek reduction, a new algorithm to compute minimal telescopers for bivariate hypergeometric terms was developed last year. We investigate further in this paper and present a new argument for the termination of this algorithm, which provides an independent proof of the existence of telescopers and even enables us to derive lower as well as upper bounds for the order of telescopers for hypergeometric terms. Compared to the known bounds in the literature, our bounds are sometimes better, and never worse than the known ones.

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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 so-called reduction-based 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.

Reduction-based 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 finite-dimensional 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 finite-dimensional 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 shift-free if no two distinct roots differ by an integer. A nonzero rational function in is said to be shift-reduced if its numerator is co-prime 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 shift-reduced, 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 Abramov-Petkovšek 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 shift-free 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 Abramov-Petkovšek reduction [10, Theorem 4.8] can be stated as follows.

Theorem 2.4.

With the notations given above, the modified version of the Abramov-Petkovšek 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 shift-equivalent (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 shift-reduced. 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 shift-free polynomials are called shift-related (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 shift-reduced 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 shift-related to each other.

Proof.

Assume that  are of the forms

where , is shift-free 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 shift-free 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 shift-equivalent to . Due to the shift-freeness 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 shift-reduced, so is . Then the first assertion follows by noticing

Let  be of the form  where , is shift-free 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 shift-freeness 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

  1. has shift-free numerator and shift-free denominator;

  2. 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 shift-free. 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 shift-free 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 shift-free if it is shift-reduced and its denominator and numerator are both shift-free. 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 shift-related.

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 shift-free and their shells have residual forms whose significant denominators are  and , respectively.

According to [5, Theorem 2], the respective denominators  and  of  and  are shift-related. 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 shift-free, 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 shift-related 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 integer-linear over if there exists a univariate polynomial and two integers s.t. . A polynomial in  is called integer-linear over if all its irreducible factors are integer-linear. A rational function in is called integer-linear over if its denominator and numerator are both integer-linear.

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 shift-reduced w.r.t. . By [4, Theorem 8], we know  is integer-linear 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 Abramov-Petkovšek 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 integer-linear over . Based on this criterion and the modified reduction, the authors of [10] developed a reduction-based 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 shift-free w.r.t.  and strongly prime with , and belongs to . Moreover, the least common multiple  of  is shift-free w.r.t. . Let

Then , is shift-free 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 finite-dimensional vector space over . This is exactly what we are going to do in the next section.

5 Finite-dimensional remainders

In this section, we will show that some sequence of  satisfying (3) has a common multiple , provided that  has a telescoper. Moreover, is shift-free and strongly prime with . The existence of this common multiple implies that the corresponding from (3) span a finite-dimensional 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 Shift-homogeneous decomposition

Recall [3] that irreducible polynomials in  are said to be shift-equivalent 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 shift-homogeneous if all non-constant monic irreducible factors of its denominator and numerator belong to the same shift-equivalence class.

By grouping together the factors in the same shift-equivalence class, every rational function can be decomposed into the form

(5)

where , , each is a shift-homogeneous rational function, and any two non-constant monic irreducible factors of and are pairwise shift-inequivalent whenever . We call (5) a shift-homogeneous decomposition of . The shift-homogeneous decomposition is unique up to the order of the factors and multiplication by nonzero constants.

Let be an irreducible integer-linear 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 integer-linear polynomials as univariate ones. For a Laurent polynomial  in  with  and , define

It is readily seen that for any two irreducible integer-linear polynomials  of the forms and  with , , and , we have  if and only if , and for some integer , in which .

Adapt from (5), every integer-linear rational function  in  admits the following decomposition

(6)

where , , each  is irreducible, monic and integer-linear 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 shift-reduced w.r.t. , which can be proven by observing that, for any two polynomials , if and only if . Let , where  belong to , , , is shift-free and strongly prime with , and . Clearly, we have and . The shift-freeness 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.

With Convention 4.2, for every nonnegative integer , assume that can be decomposed into (3), where , with , , shift-free w.r.t.  and strongly prime with , and  belongs to . Then .

Proof.

It suffices to show . The rest follows by a direct induction on .

Applying to both sides of (3) with gives

It follows from Lemma 5.1 that is an RNF of the -shift quotient of , and is a residual form of  w.r.t. . Let . Then is also an RNF of the -shift quotient of . By (3) with , is a residual form of  w.r.t. . By Proposition 3.6, we have . ∎

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 shift-reduced w.r.t. . Let

One can see that  is strongly prime with .

Case 3. There exists an integer  s.t.