A Modified Abramov-Petkovsek Reduction and Creative Telescoping for Hypergeometric Terms

0.2 MB·English

by  Shaoshi Chen

A Modified Abramov-Petkovšek Reduction and
Creative Telescoping for Hypergeometric Terms∗
Shaoshi Chen1, Hui Huang1,2, Manuel Kauers2, Ziming Li1
1KLMM, AMSS, ChineseAcademyofSciences,Beijing100190,(China)
2RISC,JohannesKeplerUniversity,LinzA-4040,(Austria)
schen, [email protected]
5 [email protected], [email protected]
1
0
2 ABSTRACT 1. INTRODUCTION

n The Abramov-Petkovˇsek reduction computes an additive Creative telescoping is a staple of symbolic summation.
u decomposition of a hypergeometric term, which extends Its main use is to construct recurrence equations that have
J the functionality of the Gosper algorithm for indefinite a prescribed definite sum among their solutions. By using
0 hypergeometric summation. We modify the Abramov- other algorithms applicable to recurrence equations, it is
1 Petkovˇsek reduction so as to decompose a hypergeometric then possible to derive interesting facts about the original
term as the sum of a summable term and a non-summable definitesum,suchasclosedformsorasymptoticexpansions.

] one. The outputsof the Abramov-Petkovˇsekreduction and The computational problem of creative telescoping is to
C our modified version share the same required properties. construct, for a given term f(x,y), polynomials ℓ0,...,ℓr
S The modified reduction does not solve any auxiliary linear in x only,not all zero, and anotherterm g(x,y) s.t.
. difference equation explicitly. It is also more efficient
s
c than the original reduction according to computational ℓ0(x)f(x,y)+···+ℓr(x)f(x+r,y)=g(x,y+1)−g(x,y).
[ experiments. Based on this reduction, we design a new
algorithm to compute minimal telescopers for bivariate The numberr may or may not be part of theinput.
2 hypergeometric terms. The new algorithm can avoid the Wecandistinguishfourgenerationsofcreativetelescoping
v costly computation of certificates. algorithms. The first generation was based on elimination
8 techniques [15, 22, 19, 14]. The second generation started
6
with what is now known as Zeilberger’s algorithm [21,
Categories andSubject Descriptors
6
5, 23, 19]. The algorithms of this generation use the
4
I.1.2 [Computing Methodologies]: Symbolic and Alge- idea of augmenting an algorithm for indefinite summation
0
. braic Manipulation—Algebraic Algorithms (or integration) by additional parameters ℓ0,...,ℓr that
1 are carried along during the calculation and are finally
0 General Terms instantiated,ifatallpossible,suchastoensuretheexistence
5 of a term g as needed for the right-hand side. See [19] for
1 Algorithms, Theory details about thefirst two generations.
: The third generation was initiated by Apagodu and Zeil-
v
Keywords berger [17, 6]. In a sense, they applied a second-generation
i
X algorithm by hand to a generic input and worked out the
Abramov-Petkovˇsek reduction, Hypergeometric term, Tele-
r scoper, Summability resultinglinearsystemofequationsfortheparametersℓiand
a the coefficients inside the desired term g. Their algorithm
∗S.ChenwassupportedinpartbytheNSFCgrant11371143 then merely consists in solving this system. This approach
and by President Fund of Academy of Mathematics and is interesting not only because it is easier to implement
Systems Science, CAS (2014-cjrwlzx-chshsh), H. Huang and tends to run faster than earlier algorithms, but also
by the Austrian Science Fund (FWF) grant W1214-13, because it is easy to analyze. In fact the analysis of
M. Kauers by the Austrian Science Fund (FWF) grants
algorithms from this family gives rise to the best output
Y464-N18 and F50-04, H. Huang and Z. Li by two
size estimates for creative telescoping known so far [11, 12,
NSFC grants (91118001, 60821002/F02) and a 973 project
(2011CB302401). 13]. Adisadvantageisthatthesealgorithmsmaynotalways
findthe smallest possible output.
Permissiontomakedigitalorhardcopiesofallorpartofthisworkforpersonalor The fourth generation of creative telescoping algorithms
classroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributed
originates from [7]. The basic idea behind these algorithms
forprofitorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitation
is to bring each term f(x + i,y) of the left-hand side
onthefirstpage. Copyrights forcomponentsofthis workownedbyothersthan
ACMmustbehonored. Abstractingwithcreditispermitted. Tocopyotherwise,or into some kind of normal forms modulo all terms that are
republish,topostonserversortoredistributetolists,requirespriorspecificpermission differences of other terms. Then to find ℓ0,...,ℓr amounts
and/[email protected]. to finding a linear dependence among these normal forms.
ISSAC’15,July6–9,2015,Bath,UnitedKingdom. The key advantage of this approach is that it separates the
Copyrightisheldbytheowner/author(s).PublicationrightslicensedtoACM.
Copyright(cid:13)c 2015ACM978-1-4503-3435-8/15/07...$15.00. computation of the ℓi from the computation of g. This
is desirable in the typical situation where we are only
DOI:http://dx.doi.org/10.1145/2755996.2756648.
interested in the ℓi and their size is much smaller than According to [3, 4], every hypergeometric term T has a
the size of g. With previous algorithms there was no way multiplicativedecompositionSH,whereS isinF(y)andH
to obtain ℓi without also computing g, but with fourth isanotherhypergeometrictermwhoseshiftquotientisshift-
generation algorithms there is. So far this approach has reduced. Wecalltheshift quotientK :=σy(H)/H akernel
onlybeenworkedoutforseveralinstancesinthedifferential ofT w.r.t.yandSthecorrespondingshell. NotethatK =1
case [7, 9, 8]. The goal of the present paper is to give a if and only if T is a rational function, which is then equal
fourth-generationalgorithmforthediscretecase,namelyfor to cS for some element c∈D with σy(c)=c.
theclassical setting of hypergeometric telescoping. Let T = SH be a multiplicative decomposition, where S
Our starting point is the Abramov-Petkovˇsek reduction is a rational function and H a hypergeometric term with a
for hypergeometrictermsintroducedin[3]andsummarized kernelK. AssumethatT =∆y(G)forsomehypergeometric
in Section 3 below. Unfortunately the reduced forms term G. A straightforward calculation shows that G is
obtained by this reduction are not sufficiently“normal”for similar to T. So there exists r ∈ F(y) s.t. G = rH. One
our purpose. Therefore, in Sections 4 and 5 we present a can easily verify that
refined variant of the reduction process and show that the
corresponding normal forms are well-behaved with respect SH =∆y(rH) ⇐⇒ S =Kσy(r)−r. (1)
totakinglinearcombinations. TheninSection6wedescribe Let VK = {Kσy(r)−r | r ∈ F(y)}, which is an F-linear
thecreativetelescopingalgorithmobtainedfromthisreduc- subspaceof F(y). Then (1) translates into
tion. Thefinalsectioncontainsanexperimentalcomparison
betweenthisalgorithmandthebuilt-inalgorithmofMaple. SH ≡0modUH ⇐⇒ S ≡0modVK. (2)
These congruences enable usto shorten expressions.
2. PRELIMINARIES
Throughoutthepaper,weletFbeafieldofcharacteristic 3. ABRAMOV-PETKOVŠEK REDUCTION
zero, and F(y)bethefield of rational functionsin y overF. Reductionalgorithmshavebeendevelopedforcomputing
Let σy betheautomorphism that mapsr(y)tor(y+1) for additivedecompositionsofrationalfunctions[1],hyperexpo-
every r ∈ F(y). The pair (F(y),σy) is called a difference nentialfunctions[8],andhypergeometricterms[3,4]. These
field. A difference ring extension of (F(y),σy) is a ring D algorithms can be viewed as generalizations of the Gosper
containingF(y)togetherwithadistinguishedendomorphism algorithm [16,19] and its differential analogue [5].
σy: D → D whose restriction to F(y) agrees with the The Abramov-Petkovˇsek reduction [3, 4] is fundamental
automorphism defined before. An element c∈D is called a forthispaper. Todescribeitconcisely,weneedanotational
constant if σy(c) = c. For a nonzero polynomial p ∈ F[y], convention and a technicaldefinition.
its degree and leading coefficient are denoted by deg (p)
y
and lcy(p),respectively. Convention 3.1. Let T be a hypergeometric term whose
kernelisK andthecorrespondingshellisS. ThenT =SH,
Definition2.1. LetDbeadifferenceringextensionofF(y).
whereH isahypergeometric termwhoseshiftquotientisK.
A nonzero element T ∈ D is called a hypergeometric term
AssumethatK isunequaltoone. Moreover,writeK =u/v,
over F(y) if σ(T) = rT for some r ∈ F(y). We call r the where u,v are polynomials in F[y] with gcd(u,v)=1.
shift quotient of T w.r.t. y.
Definition3.2. AnonzeropolynomialpinF[y]issaidtobe
The product of hypergeometric terms is again hypergeo- stronglyprime with K if gcd p,σ−i(u) =gcd p,σi(v) =1
metric. Two hypergeometricterms T ,T are called similar y y
ifthereexistsarationalfunctionr∈1F(y2)s.t.T1 =rT2. By for all i≥0. (cid:0) (cid:1) (cid:0) (cid:1)
Proposition 5.6.2 in [19],thesumof similar hypergeometric The proof of Lemma 3 in [3] contains a reduction algo-
terms is either hypergeometricor zero. rithm whose inputsand outputsare given below.
A univariate hypergeometric term T is called hyperge-
AbramovPetkovˇsekReduction: Given K and S as de-
ometric summable if there exists another hypergeometric
finedinConvention3.1,computearationalfunctionS ∈F(y)
term G s.t. T = ∆y(G), where ∆y denotes the difference and polynomials b,w ∈ F[y] s.t. b is shift-free and st1rongly
ofσy andtheidentitymap. Weabbreviate“hypergeometric
prime with K, and thefollowing equation holds:
summable”as“summable”in this paper.
ofG{0iv}enanadhtyhpeesregteoomf seutrmicmtaebrmle hTy,pweergleeotmUeTtribcetethrmesunthioant S =Kσy(S1)−S1+ b·σy−w1(u)·v. (3)
aresimilartoT. ThenUT isanF-linearsubspaceofD. Note
thaRteUcaTll=[3,U§H1]ifthHatisaanohnyzpeerrogepoomlyentorimcitaelrimn Fsi[my]iliasrstaoidTt.o isTdehsecraiblgeodriatshmpsecuodnotacinoeddeoinntphaegepr4ooofftohfeLseammmeapa3pienr,[i3n]
which thelast tenlines aretomakethedenominatorof the
beshift-free if it is coprime with any of its nontrivialshifts.
rational function V in itsoutputminimal in some technical
A nonzero rational function is said to be shift-reduced if its
sense. Weshallnotexecutetheselines. Thenthealgorithm
numerator is coprime with any shift of its denominator.
will compute two rational functions U and U . They
A basic property of shift-reduced rational functions is correspond toS and w/ bσ−1(u)v in (13), respec2tively.
given below. 1 y
WeslightlymodifytheoutputoftheAbramov-Petkovˇsek
Lemma 2.2. Let f ∈ F(y) be shift-reduced and unequal to reduction. Note that K(cid:0)is shift-red(cid:1)uced and b is strongly
one. If there exists r∈F[y] s.t. fσy(r)−r=0, then r=0. primewithK. Thus,b,σy−1(u)andv arepairwisecoprime.
By partial fraction decomposition, (3) can berewritten as
Proof. Suppose that r 6= 0. Then f = r/σy(r). Since f is
unnoteqsuhaiflt-troedouncee,dr,daoceosnntroatdbicetlioonng.to F. It follows that f is S =Kσy(S1)−S1+(cid:18)ab + σy−p11(u) + pv2(cid:19),
where a,p ,p ∈ F[y]. Furthermore, we set r = p /σ−1(u). Lemma 4.1. Let
1 2 1 y
AUpddiraetcetSc1altcoulbaetioSn1−yierldasndr=setKpσtyo(−bre)σ−y((p−1r))++pσ2y.(Tp1h)e/nv. WK =spanF yℓ |ℓ∈N,ℓ6=degy(p) for all p∈im(φK) .
S =Kσy(S1)−S1+ ab + vp. (4) Then F[y]=imn(φK)⊕WK. o
Proof. By the definition of WK, im(φK)∩WK ={0}. The
This modification leads to shell reduction specified below.
same definition also implies that, for every non-negative
ShellReduction: Given K and S as defined in Con- integer m, there exists a polynomial fm ∈ im(φK)∪WK
vention 3.1, compute a rational function S1 ∈ F(y) and s.t. the degree of fm is equal to m. The set {f0,f1,f2,...}
polynomials a,b,p ∈ F[y] s.t. b is shift-free and strongly forms an F-basis of F[y]. Thus F[y]=im(φK)⊕WK.
prime with K, and that (4) holds.
Shellreductionprovidesuswith anecessarycondition on In view of the above lemma, we call WK the standard
summability. complementofim(φK). Apolynomialp∈Fcanbeuniquely
decomposedasp=p1+p2 withp1 ∈im(φK)andp2∈WK.
Proposition 3.3. WithConvention3.1,assumethata,b,p Lemma 4.2. With Convention 3.1, let p be a polynomial
are polynomialsinF[y]s.t. b isshift-free andstrongly prime in F[y]. Then there exists q∈WK s.t. p/v≡q/vmodVK.
with K. Assume further that (4) holds. If T is summable,
then a/b belongs to F[y]. Proof. Let q be the projection of p on WK. Then there
existsf inF[y]s.t.p=φK(f)+q,thatis,p=uσy(f)−vf+q.
Proof. Recall that T = SH by Convention 3.1 and it has Sop/v=Kσy(f)−f+q/v,thatis,p/v≡q/vmodVK.
a kernel K and the corresponding shell S. It follows from
(2) and (4) that T ≡ (a/b+p/v)H modUH. Thus, T is Remark 4.3. Replacing p in the above lemma by vp, we
summable if and only if (a/b+p/v)H is summable. see that, for every polynomial p∈F[y], there exists q∈WK
SetH′=(1/v)H,whichhasakernelK′ =u/σy(v). Note s.t. p≡q/vmodVK.
′
that b is also strongly prime with K . We can apply Theo-
′ By Lemma 4.2 and Remark 4.3,(4) implies that
rem 11 in [4] to (av/b+p)H , which equals (a/b+p/v)H.
a q
Thus, a/b is apolynomial becauseb is coprime with v. S ≡ + modVK, (5)
b v
Example 3.4. Let T = y2y!/(y+1). Then the term has where a,b,q ∈ F[y], deg (a) < deg (b), b is shift-free and
a kernel K = y +1 and the shell S = y2/(y +1). Shell strongly prime with K, yand q ∈ WKy. The congruence (5)
reduction yields S ≡−1/(y+2)+y/vmodVK where v=1. motivatesustotranslatethenotionof(continuous)residual
By Proposition 3.3, T is not summable. forms in [8] intothediscrete setting.
Notethata/b+p/vin(4)canbenonzeroforasummableT. Definition 4.4. With Convention 3.1, we further let f be
a rational function in F(y). Another rational function r
Example3.5. LetT =y·y!whosekernel isK =y+1and in F(y) is called a (discrete) residual form of f w.r.t. K
shell is S =y. Then S ≡y/vmodVK, where v=1. But T if there exist a,b,q in F[y] s.t.
is summable as it is equal to ∆y(y!). a q
f ≡rmodVK and r= + ,
Theaboveexampleillustratesthatneithershellreduction b v
northeAbramov-Petkovˇsekreductioncandecidesummabil- where deg (a) < deg (b), b is shift-free and strongly prime
y y
itydirectly. Onewaytoproceedistofindapolynomialsolu- withK,andq belongstoWK. Forbrevity,wejustsaythatr
tion ofanauxiliary first-orderlinear differenceequation[4]. is a residual form w.r.t. K if f is clear from the context.
We show howthis can beavoided in the nextsection.
Residual forms help us decide summability, as shown in
thenext proposition.
4. MODIFICATIONS
Proposition 4.5. WithConvention3.1,wefurtherassume
After the shell reduction described in (4), it remains to
that r is a nonzero residual form w.r.t. K. The hypergeo-
check the summability of (a/b+p/v)H. In the rational
metric term rH is not summable.
case, i.e. when the kernel K is one, a/b + p/v in (4)
can be further reduced to a/b with deg (a) < deg (b),
y y Proof. Suppose that rH is summable. Let r = a/b+q/v,
because all polynomials are rational summable. However,
where deg (a) < deg (b), b is shift-free and strongly prime
a hypergeometric term with a polynomial shell is not with K, anyd q belongys to WK. By Proposition 3.3, a/b is a
necessarilysummable,forexample,y!hasapolynomialshell
polynomial. Since deg (a)<deg (b), a=0. Thus, (q/v)H
but it is not summable. issummable. Itfollowsyfrom (1) tyhatthereexistsw in F(y)
Wedefinethenotionofdiscreteresidualformsforrational s.t. uσy(w)−vw = q. Thus, w ∈ F[y] by Theorem 5.2.1
functions, and present a discrete variant of the polynomial
in[19,page76]. Soqbelongstoim(φK). Butqalsobelongs
reduction for hyperexponential functions given in [8]. This to WK. By Lemma 4.1, q=0, a contradiction.
variantnotonlyleadstoadirectwaytodecidesummability,
but also reducesthenumberof terms of p in (4). With Convention 3.1, let r be a residual form of the
4.1 Discrete residual forms shell S. Then SH ≡ rH modUH by (2) and (5). By
Proposition4.5,SH issummableifandonlyifr=0. Thus,
With Convention 3.1, we define an F-linear map φK determining the summability of a hypergeometric term T
fromF[y]toitselfbysendingptouσy(p)−vpforallp∈F[y]. amountstocomputingaresidualform ofthecorresponding
We call φK themap for polynomial reduction w.r.t. K. shell w.r.t. a kernelof T, which is studied below.
4.2 Polynomialreduction Example 4.6. Let K = (y4+1)/(y+1)4, which is shift-
To compute a residual form of a rational function, we reduced. ThenτK =4. According to Case5, im(φK) hasan
project a polynomial on im(φK) and on its standard com- echelon basis
pleLmetenBtKWK=, boφtKh(dyeifi)n|eid∈inNth.epTrheevioFu-lsinseuabrsemctaiopn.φK is {φK(p)}∪{φK(ym)|m∈N,m6=4},
injective by Le(cid:8)mma 2.2. So (cid:9)BK is an F-basis of im(φK), where p = y4 +y/3+1/2, φK(p) = (5/3)y2 +2y +4/3,
whichallowsustoconstructanechelonbasis. Byanechelon and φK(ym)=(m−4)ym+3+lower terms. Therefore, WK
basis, we mean an F-basis in which distinct elements have has a basis {1,y,y7}.
distinct degrees. We can easily project a polynomial using
an echelon basis and linear elimination. Fromtheabovecasedistinctionandexample,oneobserves
To construct an echelon basis, we rewrite im(φK) as that, although the degree of a polynomial in the standard
im(φK)={u∆y(p)−(v−u)p|p∈F[y]}. cthomenpulemmbeenrtodfeiptsentdersmosndτeKp,ewndhsicmhemrealyyboneathrbeitdreagrrielyeshoigfhu,
Set α = deg (u),α = deg (v), and β = deg (v − u). and v. Werecord thisobservation in thenext proposition.
1 y 2 y y
Moreover, set τK =lcy(v−u)/lcy(u),whichis nonzerodue
toConvention3.1andletpbeanonzeropolynomialinF[y]. Proposition 4.7. With the Convention 3.1, we further let
Wemake thefollowing case distinction.
α =deg (u), α =deg (v), and β =deg (v−u).
1 y 2 y y
Case 1. β >α . Then β=α , and
1 2
Then there exists P ⊂{yi|i∈N} with
φK(p)=−lcy(v−u)lcy(p)yα2+degy(p)+ lower terms.
SoBK isanechelonbasisofim(φK),inwhichdegy(φK(yi)) |P|≤max{α1,α2}−Jβ≤α1−1K
iescheeqluonalbtaosiαs2 +1,yi,f.o.r.,ayllα2i−∈1 Na.ndAdcicmor(dWinKgl)y=, Wα2K. has an sto.t.anevFer-ylinpeoalryncoommibailniantioFn[yo]fcathnebeelermedeunctesdimnoPdu.loNoimte(φthKa)t
Case 2. β =α(cid:8)1. Then (cid:9) here Jβ≤α1−1K equals 1 if β≤α1−1, otherwise it is 0.
φK(p)=−lcy(v−u)lcy(p)yα1+degy(p)+ lower terms. Proof. By the above case distinction, dim(WK) is no more
SoBK isanechelonbasisofim(φK),inwhichdegy(φK(yi)) than max{α1,α2}−Jβ ≤α1−1K. The lemma follows.
is equal to α1 +i for all i ∈ N. Accordingly, WK has an
echelon basis 1,y,...,yα1−1 and dim(WK)=α1. The above case distinction enables one to findan infinite
Case 3. β<α(cid:8)1−1. If degy(p(cid:9))=0, then φK(p)=(u−v)p. sequencep0,p1,... in F[y] s.t.
Otherwise, we have
EK ={φK(pi)|i∈N} with degyφK(pi)<degyφK(pi+1),
φK(p)=degy(p)lcy(u)lcy(p)yα1+degy(p)−1+ lower terms. isanechelonbasisofim(φK). Thisbasisallowsustoproject
It follows that BK is an echelon basis of im(φK), in a polynomial on im(φK) and WK, respectively. In the first
which degy(φK(1))=β and fourcases,thepi’scanbechosenaspowersofy. Butinthe
degy(φK(yi))=α1+i−1 for all i≥1. lsahsotwcnasien,Eoxnaemopfleth4e.6p.i’s is not necessarily a monomial as
SoWK hasanechelonbasis 1,...,yβ−1,yβ+1,...,yα1−1 , PolynomialReduction:Given p∈F[y], compute f ∈F[y]
and dim(WK)=α1−1. (cid:8) (cid:9) and q∈WK s.t. p=φK(f)+q.
Case 4. β =α1−1 and τK is not a positive integer. Then
1. If p=0, thenset f =0 and q=0; return.
φK(p)= degy(p)lcy(u)−lcy(v−u) lcy(p)yα1+degy(p)−1
+(cid:0) lower terms. (cid:1) (6) 2. Set d = degy(p). Find the subset P = {pi1,...,pis}
consisting of the preimages of all polynomials in the
dSeogBy(KφKis(ayni))ec=heαlo1n+bias−is1o.fAimcc(oφrKdi)n,ginly,wWhiKch,isfosrpaanllnied∈bNy, echelon basis EK whose degrees are at most d.
anechelonbasis 1,y,...,yα1−2 ,andhasdimensionα1−1. 3. For k = s,s−1,...,1, perform linear elimination to
Case 5. β = α1−(cid:8) 1 and τK is a(cid:9)positive integer. It follows findcs,cs−1,...,c1 ∈Fs.t.p− sk=1ckφK(pik)∈WK.
fMroomreo(v6e)r,tfhoartevfoerryip6=olynτKom, idaelgpy(oφfKd(eygir)e)e τ=K,αφ1K+(pi)−is o1f. 4. Set f = sk=1ckpik and q=pP−φK(f); and return.
degreelessthanα1+τK−1. Soanyechelonbasisofim(φK) We now presePnt a modified version of the Abramov-Pet-
does not contain a polynomial of degree α1+τK−1. Set kovˇsek reduction, which determines summability without
B′K = φK(yi)|i∈N,i6=τK . solving any auxiliary differenceequations explicitly.
pbRaoesldyisuncaoinmndgiaBφl′KKp′(⊂ywτBKitKh),bndpye′g6=tyh(e0p.′p)So<olyBnαo′K1m∪−ia{l1ps.′}inSisiBnoac′Kne,eBwchKeeloiosbntaabniansFias- fhMruayntpoicoedtnriigoafienleosdmhfAye,ptrbreir∈rcagFmeteo(roymm)vePstH.rtei.ctwkrtoiitesvhrˇsmaearkekTRseirdenudoevaulelcrKftoi,rFoma(nyn:w)d,G.triw.cvto.oemnKrpa,autnaitonendiara-l
ofim(φK). Consequently,WK isspannedbyanechelonba-
sis 1,y,...,ydegy(p′)−1,ydegy(p′)+1,...,yα1−2,yα1+τK−1 . T =∆y(fH)+rH. (7)
Thendimension of WK is equalto α1−1. o 1. Find a kernelK and thecorresponding shell S of T;
2. ApplyshellreductiontoSw.r.t.K tofindb,s,t∈F[y] 5. SUM OFTWO RESIDUALFORMS
and g ∈ F(y) s.t. b is shift-free and strongly prime Tocomputetelescopersforbivariatehypergeometricterms
with K; and by the modified Abramov-Petkovˇsek reduction, we are
confronted with the difficulty that the sum of two residual
s t
T =∆y(gH)+ + H, (8) formsisnotnecessarily aresidualform. Thisisbecausethe
b v
(cid:18) (cid:19) least common multiple of two shift-free polynomials is not
where σy(H)/H =K and v is the denominatorof K. necessarily shift-free.
The goal of this section is to show that the sum of two
3. Set p and a to be the quotient and remainder of s residual forms is congruent to aresidual form modulo VK.
and b, respectively.
Example5.1. LetK=1/y,r=1/(2y+1)ands=1/(2y+3).
4. Apply polynomial reduction to vp+t to findh∈F[y]
Thenbothrandsareresidualformsw.r.t.K,buttheirsum
and q∈WK s.t. vp+t=φK(h)+q. isnot, because thedenominator (2y+1)(2y+3) isnotshift-
5. Setf :=g+handr:=a/b+q/vandreturnH,f andr. free. However, we can still find an equivalent residual form.
Forexample,wehaver+s≡−1/(2(2y+1))+1/2y modVK.
Theorem 4.8. With Convention 3.1, the modified version Note that the residual form is not unique. Another possible
of the Abramov-Petkovˇsek reduction computes a rational choice is r+s≡1/(3(2y+3))+1/3y modVK.
function f in F(y) and a residual form r w.r.t. K s.t. (7)
Let f and g be two nonzero polynomials in F[y]. We say
holds. Moreover, T is summable if and only if r=0.
that f and g are shift-coprime if gcd f,σℓ(g) = 1 for all
y
nonzero integer ℓ. Assume that both f and g are shift-free.
Proof. Recall thatT =SH,whereH hasakernelK andS (cid:0) (cid:1)
By polynomial factorization and dispersion computation,
isarationalfunction. ApplyingshellreductiontoSw.r.t.K
one can uniquelydecompose
yields (8), which can berewritten as
g=g˜σℓ1(pm1)···σℓk(pmk), (9)
y 1 y k
a vp+t
T =∆y(gH)+ b + v H, where g˜ is shift-coprime with f, p1,...,pk are distinct,
(cid:18) (cid:19) monicandirreduciblefactorsoff,ℓ ,...,ℓ arenonzeroin-
1 k
whereaandparegiveninstep3ofthemodifiedAbramov- tegers,m1,...,mk aremultiplicitiesofσyℓ1(p1),...,σyℓk(pk)
Petkovˇsek reduction. The polynomial reduction in step 4 in g, respectively. We refer to (9) as the shift-coprime
yields that vp+t = uσy(h)− vh+q. Substituting this decomposition of g w.r.t. f.
into (8), we see that Remark 5.2. The factors g˜,σyℓ1(pm11), ..., σyℓk(pmkk)
a q in (9) are pairwise coprime, since f and g are shift-free.
T =∆y(gH)+(Kσy(h)−h)H+ + H
b v Toconstructaresidualform congruenttothesumoftwo
(cid:16) (cid:17)
=∆y((g+h)H)+rH, given residual ones, we need three technical lemmas. The
first one corresponds to thekernelreduction in [8].
wherer=a/b+q/v. Thus,(7)holds. ByProposition4.5,T
is summable if and only r is equal tozero. Lemma 5.3. With Convention 3.1, assume that p1,p2 are
in F[y] and m in N. Then there exist q1,q2 in WK s.t.
EinxEamxapmlpele4.39..4.LeTthTenbeKth=e syam+e1hyapnedrgeSom=etryi2c/(tyer+m1a)s. mi=0p1σyi(v)≡qv1 mod VK and mj=1pσ2y−j(u)≡qv2 mod VK.
Set H =y!. By the shell reduction in Example 3.4,
−1 −1 y PQroof. To provethefirst congruQence, let wm = mi=0σyi(v).
T =∆y y+1H + y+2 + v H, conWceluspioronceheodldbsybyinLduemctmioan 4o.n2.mA.ssuIfmme th=at0Qt,htehelenmtmhae
(cid:18) (cid:19) (cid:18) (cid:19)
holds for m−1. Consider theequality
where v = 1. Applying the polynomial reduction to (y/v)H
dyieeclodmsp(oys/ev)THa=s ∆Ty=(1∆·Hy)(.y/C(yom+b1in)Hin)g−th(e1/a(byov+e2s)te)pHs,. wSoe wpm1 =Kσy(cid:18)wms−1(cid:19)− wms−1 + wmt−1,
the input term T is not summable.
where s,t∈F[y] are to be determined. This equality holds
Example 4.10. Let T be the same hypergeometric term as ifandonlyifσy(s)u+(t−s)σym(v)=p1. Sinceuandσym(v)
inExample3.5. ThenK =y+1andS =y. SetH =y!. By arecoprime,suchsandtcanbecomputedbytheextended
theshellreductioninExample3.5,T =yH. Thepolynomial Euclidean algorithm. Thus, p1/wm ≡ t/wm−1 modVK.
reduction yields yH =∆y(y!), hence T =∆y(y!). Consequently, p1/wm has a required residual form by the
induction hypothesis.
Remark 4.11. With the notation given in the step 5 To provethesecond congruence, we use theidentity
of the modified version, we can rewrite rH as (s /s )G,
where s1 = av+bq, s2 = b, and G= H/v. It follo1ws2from σ−p12(u) =Kσy −σ−p12(u) − −σ−p12(u) + σyv(p2),
the case distinction at the beginning of this section that the y (cid:18) y (cid:19) (cid:18) y (cid:19)
degree of s1 is bounded by λ given in [3, Theorem 8]. The which implies that p2/σy−1(u) ≡ σy(p2)/v mod VK. By
polynomial s2 is equal to b in (3) whose degree is minimal Lemma 4.2, there exists q2 ∈ WK s.t. q2/v is a residual
by [3, Theorem 3]. Moreover, σy(G)/G is shift-reduced, form of p2/σy−1(u) w.r.t. K. Assume that the congruence
because σy(H)/H is. These are exactly the same required holds for m−1. The induction can be completed as in the
properties of the output of the original version [3]. proof for p1/wm.
The next lemma provides us with flexibility to rewrite a and partial fraction decomposition, we have
rational function modulo VK. k
Lemma 5.4. Let K ∈ F(y) be nonzero and shift-reduced. gb = bg˜0 + Pbmii, (10)
Then, for every f ∈F(y) and every ℓ∈Z+, Xi=1 i
ℓ−1 ℓ where b0,b1,...,bk ∈ F[y]. Note that pi = σy−ℓi(Pi), which
f ≡σyℓ(f)i=0σyi(K)≡σy−ℓ(f)i=1σy−i(cid:18)K1 (cid:19)modVK. icsanaafapcptloyrLoefmfm.aT5h.5ustoiteiaschstfrroancgtliyonpbriim/Peimwiitinh(K10.)Stoogweet
Y Y
k ′ ′
b b b q
PFororoℓf.=Le1t,utshsehoidwentthietyfifrst=coKngσryu(e−ncfe)b−y(i−ndfu)c+tioσny(ofn)Kℓ. g ≡ g˜0 +i=1 pmiii + v modVK, (11)
timhaptlieitshthoaldtsffoisrcℓo−ng1ru.enStettowσy=(f)Kℓ−m1oσdiu(Klo)V.KT.hAenssufmies where b′1,...,b′k ∈F[y]aXnd q′ ∈WK.
tpchoonethginreusdieusn.cttMitooonrσbeyℓoa−vs1ee(,rf,inσ)wyℓw−ℓh−1i(1cfhm)wfoℓdi−ℓsu1rloeispVQclaoKcin=egbd0ryuwetyihnthettiσonyℓd−σuy1ℓc((tffi))owwnℓℓh−by1y-. psaolrliSmiiestteshhf.waciL=ttheotrg˜KstQabrakiee=s1tsbhhpoeimiftthis-.ucfompTarohinfmednqe/ghwvaiaitrshne.dsfh,tSihfsiteno-fcrraeearetefioatnihnsaedlsphfsuiit’frnsto,-cnftarigenoledny,
Hence, f is congruent toσyℓ(f)wℓ modulo VK. in the right-hand side of (11). Then there exist b∗ in F[y]
theThideesnetcitoyndfc=onKgrσuye(nrc)e−carn+breswhiotwhnrs=imσily−a1r(lyf.)σFy−or1(ℓ1=/K1), wSiinthcedfeagnyd(bh∗)a<redeshgyif(th-c)opanridmeq,∗thineirWleKasts.cto.mt=mbo∗n/hm+uqlt∗i/pvle.
implies that f is congruent to r modulo VK. We can then is shift-free. Therefore, λr+µt is a residual form w.r.t. K,
proceed as in the proof of thefirst congruence. and λr+µt is congruent to λr+µsmodVK.
Lemma 5.5. With Convention 3.1, let a,b ∈ F[y] with 6. TELESCOPING VIA REDUCTIONS
b6=0. Assume that b isshift-free and strongly prime with K.
Assumefurtherthatσℓ(b)isstronglyprimewithK forsome Let C be a field of characteristic zero, and C(x,y) be the
y
integerℓ,thena/bhasaresidualformc/σℓ(b)+q/vw.r.t.K, field of rational functions in x and y over C. Let σx,σy be
where c∈F[y] with degy(c)<degy(b) andy q∈WK. theshift operators w.r.t. x and y,respectively,definedby,
σx(f(x,y))=f(x+1,y) and σy(f(x,y))=f(x,y+1),
Proof. First,considerthecaseinwhichℓ≥0. Ifℓ=0,then
thereexistc,p∈Fwithdeg (c)<deg (b)s.t.a/b=c/b+p. for any f ∈C(x,y). Then the pair (C(x,y),{σx,σy}) forms
y y a partial difference field.
The lemma follows from Remark 4.3.
Assumethatℓ>0. BythefirstcongruenceofLemma5.4, Definition6.1. LetDbeapartialdifferenceringextension
a a ℓ−1 σℓ(a) ℓ−1σi(u) of C(x,y). A nonzero element T ∈ D is called a hyper-
b ≡σyℓ(cid:16)b(cid:17) Yi=0σyi(K)!= σyyℓ(b)Qiℓi==−001σyyi(v) modVK. gs.eto.mσext(rTic) t=ermfToavenrd Cσ(yx(T,y))=ifgtThe.reWeexisctallf,fg,g∈theC(xsh,iyft)
quotients of T w.r.t. x and y, respectively.
Note that σℓ(b) is strongly prime wQith v by assumption.
y
pTahretnialitfriascctoiopnrimdeecowmitphotshiteiopnr,owdeucgtetvσy(v)···σyℓ−1(v). By linAeanrirorveedrucCibifletphoerlyeneoxmisitalfp∈∈CC[[zx],,ym]i,snsa∈idZtowbiethinnteg≥er0-
and gcd(m,n) = 1, s.t. p = f(mx+ny). A polynomial
a a˜ q˜
b ≡ σyℓ(b) + ℓi=−01σyi(v) modVK. iinrreCd[uxc,iyb]leisfascatiodrstoarebeinitnetgeegre-rli-nlienaera.r AoverratCioniaflaflulnocftiiotns
inC(x,y)issaidtobeinteger-linear overCifitsdenominator
BythefirstcongruenceofQLemma5.3,thesecondsummand
and numeratorare integer-linear.
in the right-hand side of the above congruence can be Let F be the field C(x), and FhSxi be the ring of linear
replacedbyaresidualformwhosedenominatorisequaltov.
recurrence operators in x, in which the commutation rule
The first assertion holds. is that Sxr = σx(r)Sx for all r ∈ F. The application of
inTwhheicchatsheeinsewcohnicdhcℓon<gr0uceanncebseofhLanemdlmedasin5.t4heansadm5e.3wwaiyll, an operator L = ρi=0ℓiSxi to a hypergeometric term T is
be used instead of thefirst ones in these lemmas. definedas L(T)=P ρi=0ℓiσxi(T).
Definition 6.2. LPet T be ahypergeometric term over F(y).
Weare ready topresent themain result of this section. A nonzero operator L ∈ FhSxi is called a telescoper for T
if there exists a hypergeometric term G s.t. L(T)=∆y(G).
Theorem 5.6. With Convection 3.1, let r and s be two
We call G the certificate of L.
residual forms w.r.t. K. Then there exists a residual form t
congruent to s modulo VK s.t., for all λ,µ∈F, λr+µt is a Forhypergeometricterms,telescopersdonotalwaysexist.
residual form w.r.t. K congruent to λr+µs modulo VK. Abramovpresentedacriterionfordeterminingtheexistence
of telescopers in [2, Theorem 10]. Let K=u/v be a kernel
Proof. Letr=a/f+p/vands=b/g+q/v,wherea,f,b,g∈ of σy(T)/T and S the corresponding shell. Applying the
F[y], degy(a) < deg(f), degy(b) < degy(g), p,q ∈ WK, modified Abramov-Petkovˇsek reduction w.r.t. y to T yields
and f,g are shift-free and strongly primewith K. T = ∆y(uH) + rH, where u ∈ F(y), H = T/S, and
Assume that (9) is the shift-coprime decomposition of g r = a/b + q/v is the residual form of S w.r.t. K. By
w.r.t. f. Set Pi =σyℓi(pi) for i=1, ..., k. By Remark 5.2 Abramov’s criterion, T has a telescoper if and only if b is
integer-linear over C. When telescopers exist, Zeilberger’s a residual form for all ℓ ,ℓ ∈ F. Indeed, the proofs of the
0 1
algorithm [21] constructs a telescoper for T by iteratively lemmas and Theorem 5.6 enable us to obtain not only r
1
using the Gosper algorithm to detect the summability but also a rational function g1 s.t. r˜1 =Kσy(g1)−g1+r1.
of L(T) for an ansatzL= ρi=0ℓiSxi ∈FhSxi. Settingu1 =u˜1+g1, wesee that(14)holdsfor i=1. Bya
Following the creative telescoping algorithms based on direct inductionon i, (14) holdsin theloop of step 4.
Hermite reductions [7, 10,P9, 8] in the continuous case, we Assume that L = ρi=0ciSxi is a telescoper of minimal
use the modified Abramov-Petkovˇsek reduction to develop order for T with ci ∈ F and cρ 6= 0. Then L(T) is
a telescoping algorithm, which is outlined below. summable. By TheoPrem 4.8, ρi=0ciri is equal to zero.
Thus, the linear homogeneous system (over F) obtained by
qRueodtiuecnttisofnC=Tσ:xG(Tiv)/eTn aanhdypge=rgeσoym(Tet)r/icTtienrmF(Ty),wciothmpshuitfet yeqieuldatsinagteleρis=co0pℓierriotfomzienriomhaalsorPadnero.ntrivial solution, which
a telescoper of minimal order for T and its certificate if P
telescopers exist. Remark 6.4. The algorithm ReductionCT separates the
computation ofminimaltelescopers fromthat of certificates.
1. Finda kernelK andshell S ofT w.r.t. y s.t.T =SH
In applications where the certificates are irrelevant, we can
with K =σy(H)/H. drop step 4.2 and in step 4.3 compute ui and ri with ri ≡
2. ApplythemodifiedAbramov-PetkovˇsekreductiontoT r˜imodVK andσxi(ri−1)NH =∆y(uiH)+riH andthatR+
to get
ℓiriisaresidualformw.r.t.K,whereℓiisanindeterminate.
T =∆y(u0H)+r0H. (12) Tneheedrtaoticoanlcaullafutenctioij=n0uℓijucjaHn ibnetdhiesceanrdde.d, and we do not
If r =0, thenreturn (1,u H).
0 0
3. If the denominator of r is not integer-linear, return Remark 6.5. InPstead of applying the modified Abramov-
“No telescoper exists!”. 0 Petkovˇsek reduction to σx(ri−1)NH in step 4.1, it is also
possibletoapplythereductiontoσi(T),butourexperiments
4. Set N := σx(H)/H and R := ℓ0r0, where ℓ0 is an suggest that this variant takes consxiderably more time.
indeterminate.
For i=1,2,..., do Example 6.6. Let T = xy 3. Set f and g to be σx(T)/T
and σy(T)/T, respectively(cid:0). S(cid:1)ince g is shift-reduced w.r.t. y,
4.1. Viewσx(ri−1)NH asahypergeometrictermwith its kernel is equal to g itself, and the corresponding shell is
kernelK andshellσx(ri−1)N. Usingshellreduc- equal to 1. In step 4, we obtain σyi(T)≡(qi/v)H modUH,
tion w.r.t. K and polynomial reduction w.r.t. K, wherei=0,1,2, v=(y+1)3, H hasshiftquotient g w.r.t.y,
find u′i ∈ F and a res′idual form r˜i w.r.t. K q = 1(x+1)(x2−x+3y(y−x+1)+1), q =(x+1)3, and
s.t. σx(ri−1)NH =∆y(uiH)+r˜iH. 0 2 1
4.2. Set u˜i=σx(ui−1)N +u′i, so that q = (x+1)3 11x2−12xy+17x+20+12y+12y2 .
σxi(T)=∆y(u˜iH)+r˜iH. (13) B2y fin(dxin+g2a)n2 (cid:0)F-linear dependency among q ,q ,q , w(cid:1)e get
0 1 2
4.3. Follow the proof of Theorem 5.6 to compute ui
and ri in F(y)s.t. ri ≡r˜imodVK, L:=(x+2)2Sx2−(7x2+21x+16)Sx−8(x+1)2
σxi(T)=∆y(uiH)+riH, (14) is a telescoper of minimal order for T.
and that R + ℓiri is a residual form w.r.t. K, 7. IMPLEMENTATION AND TIMINGS
whereℓi is an indeterminate. We have implemented our algorithms in Maple. In order
4.4. UpdateR to beR+ℓiri. to get an idea about their efficiency, we compared their
4.5. Find ℓj ∈ F s.t. R = 0 by solving a linear runtime and memory requirements to the performance of
systeminℓ0,...,ℓi overF. Ifthereisanontrivial known algorithms. All timings are measured in seconds on
solution, return ij=0ℓjSxj, ij=0ℓjujH . a Linux computer with 388Gb RAM and twelve 2.80GHz
(cid:16) (cid:17) Dual core processors.
Theorem 6.3. Let T be a hPypergeometPric term over F(y). Forthefirst comparison, we considered univariatehyper-
If T has a telescoper, then the algorithm ReductionCT geometric terms of theform
terminates and returns a telescoper of minimalorder for T.
f(y) Γ(y−α)
T = ,
Proof. ByTheorem4.8,r =0impliesthat1isatelescoper g (y)g (y)Γ(y−β)
0 1 2
for T of minimal order.
Let r0 obtained from step 2 be of the form a0/b0+q0/v, where f ∈Z[y] of degree 20, gi =piσyλ(pi)σyµ(pi) with pi ∈
pwrhimeree wa0it,hb0K,v,∈q0F∈[y]W, dKe,gya(nad0)v<isdtehgeyd(be0n)o,mb0iniastosrtroofngKly. Zra[nyd]oomfdteegrrmees1o0f,tλh,isµt∈ypNe,faonrddiαff,eβre∈ntZc.hFoiocresaosfelµectainodnλof,
Table 1 compares the timings of Maple’s implementation
ByOre-Sato’stheorem[18,20]onhypergeometricterms,K
oftheclassical Abramov-Petkovˇsekreduction(AP)andour
is integer-linear and so is v. It follows that b is integer-
0
modified version (MAP). We apply the algorithms to T as
linear if and only if b v is. By Abramov’s criterion, T has
0
a telescoper if and only if the denominator of r is integer- well as to thesummableterms σy(T)−T.
0
For the second comparison, we considered bivariate hy-
linear. Thus, steps 2 and 3 are correct.
pergeometric termsof theform
It follows from (12) and σx(r0H) = σx(r0)NH that (13)
holds for i = 1. By Theorem 5.6, there exists a residual f(x,y) Γ(2αx+y)
form r1 w.r.t.K withr1≡r˜1modVK s.t.R+ℓ1r1 isagain T = g1(x+y)g2(2x+y) Γ(x+αy)
T σy(T)−T [4] S.A. Abramovand M. Petkovˇsek.Rational normal
(λ,µ) AP MAP AP MAP forms and minimal decompositions of hypergeometric
(0,0) 0.45 0.30 4.41 2.29 terms. J. Symbolic Comput., 33(5):521–543, 2002.
(5,5) 5.94 1.21 10.19 2.40
[5] G. Almkvistand D.Zeilberger. The method of
(5,10) 14.69 2.20 15.67 3.30
differentiatingundertheintegral sign. J. Symbolic
(10,10) 17.22 2.31 17.98 2.77
Comput.,10:571–591, 1990.
(10,20) 57.05 6.20 38.03 3.80
[6] M. Apagodu and D.Zeilberger. Multi-variable
(10,30) 316.51 15.73 74.55 3.73
Zeilberger and Almkvist-Zeilbergeralgorithms and the
(10,40) 514.84 32.64 134.29 3.99
sharpeningof Wilf-Zeilberger theory.Adv. in Appl.
Math., 37(2):139–152, 2006.
Table 1: Comparison of the Abramov-Petkovˇsek
[7] A.Bostan, S.Chen, F. Chyzak,and Z. Li. Complexity
reduction and the modified version for a collection of
of creative telescoping for bivariaterational functions.
non-summable terms T and summable terms σy(T)−T. InProc. of ISSAC ’10, pp.203–210, 2010. ACM.
[8] A.Bostan, S.Chen, F. Chyzak,Z.Li, and G. Xin.
Hermitereduction and creative telescoping for
with f ∈Z[x,y] of degree n, gi =piσzλ(pi)σzµ(pi) with pi ∈ hyperexponentialfunctions. InProc. of ISSAC’13,pp.
[z]ofdegreem,andα,λ,µ∈ . Foraselectionofrandom
Z N 77–84, 2013. ACM.
terms of this type for different choices of n,m,α,µ,λ,
[9] A.Bostan, P.Lairez, and B. Salvy.Creative
Table 2comparesthetimingsof Maple’s implementation of
telescoping for rational functions usingthe
Zeilberger’salgorithm(Z)andtwovariantsofthealgorithm
Griffith-Dwork method.In Proc. of ISSAC’13,pp.
ReductionCT from Section 6: For the column RCT we
1 93–100, 2013. ACM.
computedboththetelescoperandcertificate,andforRCT
2
[10] S.Chen. Some applications of differential-difference
we only compute the telescoper. The difference between
these two variants comes mainly from the time needed to algebra to creative telescoping. PhD thesis, E´cole
bring the rational function u in the certificate uH on a Polytechnique,Palaiseau, 2011.
common denominator. When it is acceptable to keep the [11] S.Chen and M. Kauers. Order-degreecurvesfor
certificateasanunnormalizedlinearcombinationofrational hypergeometriccreative telescoping. In Proc. of
functions, thetimings are virtually thesame as for RCT . ISSAC’12,pp.122–129, 2012. ACM.
2
[12] S.Chen and M. Kauers. Tradingorder for degree in
(m,n,α,λ,µ) Z RCT1 RCT2 order creative telescoping. J. Symbolic Comput.,
(1,0,1,5,5) 17.12 5.00 1.80 4 47(8):968–995, 2012.
(1,0,2,5,5) 74.91 26.18 5.87 6 [13] S.Chen, M. Kauers, and C. Koutschan.A generalized
(1,0,3,5,5) 445.41 92.74 17.34 7 Apagodu-Zeilbergeralgorithm. In Proc. of ISSAC’14,
(1,8,3,5,5) 649.57 120.88 23.59 7 pp.107–114, 2014. ACM.
(2,0,1,5,10) 354.46 58.01 4.93 4 [14] F.ChyzakandB.Salvy.Non-commutativeelimination
(2,0,2,5,10) 576.31 363.25 53.15 6 in Ore algebras proves multivariateidentities. J.
(2,0,3,5,10) 2989.18 1076.50 197.75 7 Symbolic Comput., 26:187–227, 1998.
(2,3,3,5,10) 3074.08 1119.26 223.41 7
[15] M. C. Fasenmyer. Somegeneralized hypergeometric
(2,0,1,10,15) 2148.10 245.07 11.22 4
polynomials. Bull. Amer. Math. Soc., 53:806–812,
(2,0,2,10,15) 2036.96 1153.38 153.21 6
1947.
(2,0,3,10,15) 11240.90 3932.26 881.12 7
[16] R.W. Gosper, Jr. Decision procedurefor indefinite
(2,5,3,10,15) 10163.30 3954.47 990.60 7
hypergeometricsummation. Proc. Nat. Acad. Sci.
(3,0,1,5,10) 18946.80 407.06 43.01 6
U.S.A.,75(1):40–42, 1978.
(3,0,2,5,10) 46681.30 2040.21 465.88 8
[17] M. Mohammed and D.Zeilberger. Sharp upper
(3,0,3,5,10) 172939.00 5970.10 1949.71 9
boundsfor theorders of therecurrences outputby the
Zeilberger and q-Zeilberger algorithms. J. Symbolic
Table 2: Comparison of Zeilberger’s algorithm to Comput.,39(2):201–207, 2005.
reduction-based telescoping with and without construc- [18] O.Ore. Surla forme des fonctions hyperg´eom´etriques
tion of a certificate deplusieurs variables. J. Math. Pures Appl. (9),
9(4):311–326, 1930.
[19] M. Petkovˇsek,H.S. Wilf, and Doron Zeilberger.
A=B.A K Peters Ltd.,Wellesley, MA, 1996.
8. REFERENCES
[20] M. Sato. Theory of prehomogeneous vectorspaces
(algebraic part). Nagoya Math. J.,120:1–34, 1990.
[1] S.A. Abramov.The rational component of the
[21] D.Zeilberger. A fast algorithm for proving
solution of a first-orderlinear recurrence relation with
terminatinghypergeometric identities. Discrete Math.,
a rational right side. USSR Comp. Math. Math. Phys.,
80:207–211, 1990.
15(4):216–221, 1975.
[22] D.Zeilberger. A holonomic systemsapproach to
[2] S.A. Abramov.When does Zeilberger’s algorithm
special functions identities. J. Comput. Appl. Math.,
succeed? Adv. in Appl. Math., 30(3):424–441, 2003.
32:321–368, 1990.
[3] S.A. Abramovand M. Petkovˇsek.Minimal
[23] D.Zeilberger. The method of creative telescoping. J.
decomposition of indefinitehypergeometric sums. In
Symbolic Comput., 11:195–204, 1991.
Proc. of ISSAC’01,pp.7–14, 2001. ACM.