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Order-Degree Curves for Hypergeometric Creative Telescoping ∗ † Shaoshi Chen Manuel Kauers DepartmentofMathematics RISC NCSU JohannesKeplerUniversity Raleigh,NC27695,USA 4040Linz,Austria [email protected] [email protected] 2 1 0 ABSTRACT of the polynomials appearing in the recurrence. These are 2 investigated in thepresent paper. Creativetelescopingappliedtoabivariateproperhypergeo- n Ideally, we would like to be able to determine for a given metrictermproduceslinearrecurrenceoperatorswithpoly- a sum all the pairs (r,d) such that the sum satisfies a linear nomial coefficients, called telescopers. We provide bounds J recurrence of order r with polynomial coefficients of degree forthedegreesofthepolynomialsappearingintheseopera- 0 atmostd. Thisisahardquestionwhichwedonotexpectto tors. Ourboundsareexpressedascurvesinthe(r,d)-plane 1 haveasimpleanswer. Theresultsgivenbelowcanbeviewed which assign to every order r a bound on the degree d of as answers to simplified variants of the problem. One sim- the telescopers. These curves are hyperbolas, which reflect ] plificationisthatwerestricttheattentiontotherecurrences C the phenomenon that higher order telescopers tend to have foundbycreativetelescoping[17],called“telescopers”inthe lower degree, and vice versa. S symbolic summation community (see Section 2 below for a s. Categories andSubject Descriptors definition). Thesecondsimplification isthatinsteadof try- c ing to characterize all the pairs (r,d), we confine ourselves [ I.1.2 [Computing Methodologies]: Symbolic and Alge- with sufficient conditions. braic Manipulation—Algorithms Ourmainresultsarethusformulaswhichprovidebounds 1 onthedegreedofthepolynomialcoefficientsinatelescoper, v General Terms depending on its order r. The formulas describe curves in 2 Algorithms the(r,d)-planewiththepropertythatforeveryintegerpoint 8 (r,d) above the curve, there is a telescoper of order r with 9 1 Keywords polynomial coefficients of degree at most d. As the curves . SymbolicSummation,CreativeTelescoping,DegreeBounds arehyperbolas,theyreflectthephenomenonthathigheror- 1 derrecurrenceequationsmayhavelowerdegreecoefficients. 0 1. INTRODUCTION This feature can be used to derive a complexity estimate 2 according to which, at least in theory, computing the mini- 1 Weconsidertheproblemoffindinglinearrecurrenceequa- mum order recurrence is more expensive than computing a : tions with polynomial coefficients satisfied by a given defi- v recurrencewithslightlyhigherorder(butdrasticallysmaller nite single sum over a proper hypergeometric term in two i polynomial coefficients). This phenomenon is analogous to X variables. This is one of the classical problems in symbolic thesituationinthedifferentialcase,whichwasfirstanalyzed summation. Zeilberger [16] showed that such a recurrence r by Bostan et al. [5]for algebraic functions, and recently for a always exists, andproposed thealgorithm nownamedafter integrals of hyperexponentialterms bytheauthors [6]. him for computing one [15, 17]. Also explicit bounds are Ouranalysisfornon-rationalproperhypergeometricinput known for the order of the recurrence satisfied by a given (Section 3) follows closely our analysis for the differential sum [14, 10, 4]. Little is known however about the degrees case [6]. It turns out that the summation case considered ∗ hereisslightly easier thanthedifferentialcaseinthatit re- SupportedbytheNationalScienceFoundation(NFS)grant quires fewer cases to distinguish and in that the resulting CCF-1017217. † degree estimation formula is much simpler than its differ- Supported by the Austrian Science Fund (FWF) grant ential analogue. For rational input (Section 4), we derive a Y464-N18. degreeestimationformulafollowing Le’salgorithmforcom- putingtelescopers of rational functions[2, 9]. 2. PROPER HYPERGEOMETRIC TERMS Permission tomake digital orhardcopies ofall orpartofthis workfor AND CREATIVETELESCOPING personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare notmadeordistributedforprofitorcommercialadvantageandthatcopies Let be a field of characteristic zero and let (n,k) bearthisnoticeandthefullcitationonthefirstpage.Tocopyotherwise,to K K be the field of rational functions in n and k. We will be republish,topostonserversortoredistributetolists,requirespriorspecific considering extension fields E of (n,k) on which two iso- permissionand/orafee. K Copyright20XXACMX-XXXXX-XX-X/XX/XX...$10.00. morphismsSn andSk aredefinedwhichcommutewitheach other, leave every element of fixed, and act on n and 3. THE NON-RATIONALCASE K k via Sn(n) = n + 1, Sk(n) = n, Sn(k) = k, Sk(k) = We consider in this section the case where h cannot be k+1. A hypergeometric term is an element h of such an splitintoh=qh forq∈ (n,k)andanotherhypergeomet- 0 extension field E with Sn(h)/h ∈ K(n,k) and Sk(h)/h ∈ rictermh0 withSk(h0)/hK0 =1. Informally,thismeansthat (n,k). Proper hypergeometric terms are hypergeometric weexcludetermshwherey=1andeveryΓ-terminvolving K terms which can bewritten in theform kcanbecancelledagainstanotheronetosomerationalfunc- M ′ ′′ ′ ′′ tion. ThosetermsaretreatedseparatelyinSection4below. h=pxnyk Γ(amn+amk+am)Γ(bmn−bmk+bm), If h cannot be split as indicated, then also Ch cannot be mY=1Γ(umn+u′mk+u′m′)Γ(vmn−vm′ k+vm′′) split in this way, for any rational function C ∈ (n,k). In (1) particular, we can then not have Ch∈ and tKherefore we w′here p ∈′K[n,k],′x,y ∈ K, M ∈ N is fixed, am′′, a′′m′ , bm′′, always have Sk(Ch)−Ch6=0. This imKplies that whenever bvmm′′,∈um, uamn,dvmth,evmexaprreesnsioonnnsegxant,ivyeki,ntaengdersΓ,(.a.m.), brmef,erumto, wsuerehathveataLpaisirn(oLt,tChe)wzeirtohoLp(ehr)at=orS,ka(nCdhw)e−nCeehd, wnoetcwanorbrye K elements of E on which Sn and Sk act as suggested by the about this requirement any further. notation, e.g., The analysis in thepresent case is similar to that carried Sn(Γ(2n−k+1))=(2n−k+1)(2n−k+2)Γ(2n−k+1), outbyApagoduandZeilberger[10],whouseditforderiving aboundontheorderrofL,andsimilartoouranalysis[6]of 1 S (Γ(2n−k+1))= Γ(2n−k+1). thedifferentialcase. Themainideaistofollowstepbystep k 2n−k the execution of Zeilberger’s algorithm when applied to h. ′ Throughout the paper the symbols p, x, y, M, am, am, This eventually leads to a linear system of equations with ′′ a , ...will beused with the meaning theyhavein (1). For coefficientsin (n)which musthaveasolution wheneverit m K fittinglongformulasintothenarrowcolumnsofthislayout, isunderdetermined. Theconditionofhavingmorevariables we also use theabbreviations than equations in this linear system is the source of the ′ ′′ ′ ′′ estimate for choices (r,d) that lead toa solution. Am:=amn+amk+am, Bm :=bmn+bmk+bm, Um :=umn+u′mk+u′m′, Vm :=vmn+vm′ k+vm′′. 3.1 Zeilberger’s Algorithm Recall the main steps of Zeilberger’s algorithm: for some We follow the paradigm of creative telescoping. For a givenhypergeometrictermhasabove,wewanttodetermine choiceofr,itmakesanansatzL=ℓ0+ℓ1Sn+···+ℓrSnr with polynomials ℓ0,...,ℓr ∈ [n](free ofk,notallzero), anda undeterminedcoefficientsℓ0,...,ℓr,andthencallsGosper’s K algorithmonL(h). Gosper’salgorithm[7]proceedsbywrit- rationalfunctionC ∈ (n,k)(possiblyinvolvingk,possibly K ing zero), such that ℓ0h+ℓ1Sn(h)+···+ℓrSnr(h)=Sk(Ch)−Ch. Sk(L(h)) = Sk(P) Q L(h) P S (R) k In[nt]h[Sisnc]aissec,atlhleedoapteerlaetsocropLer:=forℓ0h,+anℓ1dStnh+e r·a·t·io+naℓlrSfunrnc∈- for some polynomials P,Q,R such that gcd(Q,Ski(R)) = 1 tKion C ∈ (n,k) is called a certificate for L (and h). The foralli∈ . Itturnsoutthattheundeterminedcoefficients numberrKiscalledtheorder ofL,andd:=maxri=0degnℓiis ℓ0,...,ℓr Nappear linearly in P and not at all in Q or R. called its degree. If h representsan actual sequencef(n,k), Next,thealgorithm searchesforapolynomialsolution Y of then a recurrence for the definite sum n f(n,k) can be theGosper equation k=0 obtained from such a pair (L,C) as expPlained in the litera- P =QS (Y)−RY k ture on symbolic summation [12]. We shall not embark on the technical subtleties of this correspondence here but re- by making an ansatz Y = y0+y1k+y2k2+···+ysks for strict ourselves to analyzing of the set of all pairs (r,d) for some suitably chosen degree s, substituting the ansatz into which thereexists a telescoper of orderr and degree d. theequation,andcomparingpowersofkonbothsides. This The following notation will be used. leadstoalinearsysteminthevariablesℓ0,...,ℓr,y0,...,ys with coefficients in (n). Any solution of thissystem gives • For p∈ [n,k] and m∈ , let K K N risetoatelescoperLwiththecorrespondingcertificateC = pm :=p(p+1)(p+2)···(p+m−1) RY/P. Ifnosolution exists,theprocedureisrepeatedwith a greater value of r. with theconventionsp0=1 and p1 =p. For a hypergeometric term h and an operator L = ℓ + 0 • For p ∈ [n,k], deg p and deg p denote the degree ℓ1Sn+···+ℓrSnr, we have K n k ofpwithrespecttonork,respectively. degpwithout r Si(p) M AiamBibm any subscript denotesthe total degree of p. L(h)= ℓixi n m m h Xi=0 p mY=1UmiumVmivm • For z∈ , let z+ :=max{0,z}. R r M With this notation, we have ℓixiSni(p) Pi,m Sn(h) =xSn(p) M AammBmbm, = Xi=0 M mY=1 h h p mY=1UmumVmvm p UmrumVmrvm mY=1 Skh(h) =ySkp(p)mYM=1UAmuam′m′m((VBmm−−vbm′′m))vbm′′m. =(cid:18)Xi=r0ℓixiSni(p)mYM=1Pi,m(cid:19) M M ×xnyk Γ(Am)Γ(Bm) , µ= (am+bm−um−vm), mY=1Γ(Um+rum)Γ(Vm+rvm) mX=1 M M where ν =max (a′ +v′ ), (u′ +b′ ) . m m m m nmX=1 mX=1 o Pi,m =AimamBmibm(Um+ium)(r−i)um(Vm+ivm)(r−i)vm. We can write Notethattheseparametersareintegerswhichonlydepend S (L(h)) S (P) Q on h butnot on r or d. Except for µ, theyare all nonnega- k = k , tive. Notealsothatwehaveλ+µ≥0andϑ=λ+µ+≥|µ|. L(h) P S (R) k where Lemma 2. Let di:=degnℓi (i=0,...,r). Then r r M degP ≤δ+λr+max(di+iµ). P = ℓixiSni(p) Pi,m, i=0 Xi=0 mY=1 Furthermore, degkP ≤δ+ϑr. M Q=y Ama′m(Vm+rvm−vm′ )vm′ , Proof. It suffices to observethat mY=1 degPi,m ≤iam+ibm+(r−i)um+(r−i)vm M R= (Um+rum−u′m)u′mBmb′m. for all m = 1,...,M and all i = 0,...,r. For the degree mY=1 withrespecttok,observealsothatdegkℓi=0foralli. Dependingontheactualvaluesofthecoefficientsappearing We have some freedom in choosing the di. The choice inh,thisdecompositionmayormaynotsatisfytherequire- influencesthenumberof variables in theansatz ment gcd(Q,Ski(R)) = 1 for all i ∈ N. But even if it does r di not,itonlymeansthatwemayoverlooksomesolutions,but L= ℓi,jnjSni every solution we find still gives rise to a correct telescoper Xi=0Xj=0 and certificate. Since we are interested only in bounding aswellasthenumberofequations. Weprefertohavemany thesizeofthetelescopersofh,itissufficienttostudyunder variablesandfewequations. Forafixedtargetdegreed,the which circumstances theGosper equation maximum possible number of variables is (d+1)(r+1) by P =QSk(Y)−RY choosing d0 =d1 =···=dr =d. But this choice also leads to many equations. A better balance between number of with theabovechoice of P,Q,R has a solution. variables and number of equations is obtained by lowering 3.2 Counting Variablesand Equations some of the di with indices close to zero (if µ is negative) or with indices close to r (if µ is positive). Specifically, we ApagoduandZeilberger[10]proceedfromherebyanalyz- choose ing the linear system over (n) resulting from the Gosper K (ν+i−r)+|µ| if µ≥0 equation for a suitable choice of the degree of Y. They de- di:=d−(cid:26) (ν−i)+|µ| if µ<0. rive a bound on r but give no information on the degree d. Generalboundsforthedegreesofsolutionsoflinearsystems See [6, Ex. 11, Ex. 15.5 and the remarks after Thm. 14] withpolynomialcoefficientscouldbeapplied,buttheyturn for a detailed motivation of thecorresponding choice in the outtoovershootquitemuch. Inparticular,itseemsdifficult differentialcase. ThesupportoftheansatzforLlooksasin to capturethephenomenonthat increasing r may allow for thefollowingdiagram,whereeverytermnjSi isrepresented n decreasing d using such general bounds. by a bullet at position (i,j): We proceed differently. Instead of a coefficient compari- ν son with respect to powers of k leading to a linear system r r r r r r j=d over (n),weconsideracoefficientcomparisonwithrespect r r r r r rz }| { µ topoKwersofnandkleadingtoalinearsystemover . This r r r r r r r o. requiresustomakeachoicenotonlyforthedegreeoKfY ink r r r r r r r .. but also for the degree of Y in n as well as for the degrees r r r r r r r r voafrtihabelℓeis(aind=e0q,u.a.t.i,orn)siinnnth.isFsoyrsteexmpr,eistsiinsghetlhpefunlutmobaderopotf rr rr rr rr rr rr rr rr roµ r r r r r r r r r thefollowing definition. r r r r r r r r r j=0 Definition1. Foraproperhypergeometrictermhasin(1), i=0 i=r let With thischoicefor thedegrees di,thenumberof variables δ =degp, in theansatz for L is M M r ϑ=maxnmX=1(am+bm),mX=1(um+vm)o, Xi=0(di+1)=(d+1)(r+1)− 12|µ|ν(ν+1), M provided that d≥|µ|ν. The number of resulting equations λ= (um+vm), mX=1 is as follows. Lemma 3. If the di are chosen as above, then P contains Theorem 5. Let h be a proper hypergeometric term which at most cannot be written h = qh for some q ∈ (n,k) and a hy- 0 K pergeometric term h with S (h )/h = 1. Let δ,λ,µ,ν be 1 δ+ϑr+1 δ+2d+ϑr−2|µ|ν+2 0 k 0 0 2 as in Definition 1, let r≥ν and (cid:0) (cid:1)(cid:0) (cid:1) terms nikj. ϑν−1 r+ 1ν 2δ+|µ|+3−(1+|µ|)ν −1 d> 2 . Proof. If µ≥0, we have (cid:0) (cid:1) (cid:0) r−ν+1 (cid:1) di+iµ=d−(ν+i−r)+µ+iµ≤d−ν|µ|+rµ ThenthereexistsatelescoperLforhoforderranddegreed. for all i=0,...,r. Likewise, when µ<0, we have Proof. Asufficientconditionfortheexistenceofatelescoper of order r and degree d is that for some particular ansatz, di+iµ=d−(ν−i)+|µ|+iµ≤d−ν|µ| theequation for all i=0,...,r. Together with Lemma 2, it follows that P =QS (Y)−RY k degP ≤δ+(λ+µ+)r+d−ν|µ|=δ+ϑr+d−ν|µ| has a nontrivial solution. A sufficient condition for the ex- istenceofasolutionisthatthelinearsystemresultingfrom regardless of the sign of µ. We also have deg P ≤ δ+ϑr k coefficient comparison has more variables than equations. from Lemma2. Forthenumberoftermsnikj inP wehave For all d in question, we have d > ϑν ≥ |µ|ν. Therefore, degkP with theansatz described above, we have (1+degP−i)= 1(deg P+1)(2degP+2−deg P). Xi=0 2 k k (d+1)(r+1)− 21|µ|ν(ν+1) Plugging the estimates for degP and degkP into the right variables ℓi,j in P, hand side gives theexpression claimed in theLemma. 1 δ+ϑr+1−ν δ+2d+ϑr−2|µ|ν+2−ν 2 The support of P has a trapezoidal shape which is de- (cid:0) (cid:1)(cid:0) (cid:1) variables yi,j in Y, and termined by the total degree and the degree with respect to k: 1 δ+ϑr+1 δ+2d+ϑr−2|µ|ν+2 2 r (cid:0) (cid:1)(cid:0) (cid:1)  r r equations. Solving theinequality degP  rrr rrr rrr rr (d++112)(δr++ϑ1)r−+121|−µ|νν(νδ++12)d+ϑr−2|µ|ν+2−ν  rrr rrr rrr rrr uesntdime>ratt21eh(cid:0).eδ(cid:0)a+ssϑurm+pt1i(cid:1)o(cid:0)nδr+≥2(cid:1)d(cid:0)ν+foϑrrd−g2iv|µes|νt+he2c(cid:1)laimed de(cid:1)gree degkP 3.3 Examples and Consequences | {z } The next step is to choose the degrees for Y in n and k. ThisisdoneinsuchawaythatQSk(Y)−RY onlycontains Example 6. 1. For terms which are already expectedto occur in P, so that no Γ(2n+3k) additional equations will appear. h=(n2+k2+1) Γ(2n−k) Lemma 4. Let the di be chosen as before and suppose that we have δ = 2, ϑ = 2, µ = 0, ν = 4. Theorem 5 Y ∈ [n,k] is such that degY ≤ degP −ν and deg Y ≤ K k predicts a telescoper of order r and degree d whenever deg P −ν. Then P −(QS (Y)−RY) contains at most k k r≥4 and 21 δ+ϑr+1 δ+2d+ϑr−2|µ|ν+2 7r+5 d> . terms nikj.(cid:0) (cid:1)(cid:0) (cid:1) r−3 The left figure below shows the curve defined by the Proof. As for Lemma 3, using also max{degQ,degR} = right hand side (black) together with the region of all max{degkQ,degkR}=ν. points (r,d) for which we found telescopers of h with order r and degree d by direct calculation (gray). In Lemma 4 suggests theansatz thisexample,theestimateovershootsbyverylittleonly. s1 s2−i Y = yi,jkinj 2. The corresponding picture for Xi=0 Xj=0 Γ(2n+k)Γ(n−k+2) h= withs =deg P−ν ands =degP−ν,whichprovidesus Γ(2n−k)Γ(n+2k) 1 k 2 with isshownbelowontheright. Here,δ=0,ϑ=3,µ=0, 1 δ+ϑr+1−ν δ+2d+ϑr−2|µ|ν+2−ν ν = 3 and Theorem 5 predicts a telescoper of order r 2 and degree d whenever r≥3 and (cid:0) (cid:1)(cid:0) (cid:1) variables. We are now ready to formulate the main result 8r−1 of this section. Note that the inequality for d is a consid- d> . r−2 erably simpler formula thanthecorrespondingresult in the differential case (Thm. 14 in [6]). In this example, the estimate is less tight. d d Therefore, in order to compute a telescoper and its certifi- 40 40 cate most efficiently,we should minimize thecost function 30 30 C(r,d):= (ϑ+1)r+δ+2 3 δ+ϑr+d−(|µ|+1)ν+1 . (cid:0) (cid:1) (cid:0) (cid:1) Accordingtothefollowingtheorem,forasymptoticallylarge 20 20 inputitissignificantlybettertochooserslightlylargerthan theminimal possible value. 10 10 Theorem 8. Let h and λ,µ,ν be as in Theorem 5, τ ≥ 0 r 0 r max{ϑ,ν}, and suppose that κ ∈ is a constant such that 0 5 10 15 20 0 5 10 15 20 degree t solutions of a linear systRem with m variables and at most m equations over (n) can be computed with κm3t Thepoints(r,d)intheportionofthegrayregionwhichis K operations in . Then: belowtheblackcurverepresenttelescoperswherethecorre- K spondinglinearsystemresultingfromtheansatzconsidered 1. A telescoper oforder r=τ alongwithacorresponding in our proof is overdetermined but, for some strange rea- certificate can be computed using son, nevertheless nontrivially solvable. The small portions of white spacewhich lie abovethecurvesarenot in contra- κτ9+ 1(7−|µ|)κτ8+O(τ7) 2 diction with our theorem because they do not contain any operations in . pointswithintegercoordinates. (Thetheoremsaysthatev- K ery point (r,d) ∈ 2 above the curve belongs to the gray Z 2. If α > 1 is some constant and r is chosen such that region.) r = ατ +O(1), then a telescoper of order r and a Theorem5supplementstheboundgivenin[10]ontheor- corresponding certificate can be computed using deroftelescopersforahypergeometrictermbyanestimate forthedegreethattheseoperatorsmayhave. Inaddition,it α5 provides lower degree bounds for higher orders and admits κτ8+O(τ7) α−1 a bound on theleast possible degree for a telescoper. operations in . K Corollary 7. With the notation of Theorem 5, h admits a telescoper of order r=ν and degree In particular, a telescoper for h and a corresponding certificate can be computed in polynomial time. d= 1ν(2δ+2νϑ+|µ|−ν|µ|) 2 (cid:6) (cid:7) Proof. AccordingtoTheorem5,foreveryr≥τ thereexists as well as a telescoper of order a telescoper of orderr and degree d for any r= 1ν(1+2δ+2(ν−1)(ϑ−|µ|)) 2 (τ2−1)r+O(τ2) (cid:6) (cid:7) d>f(r):= . and degree d=ϑν. r−τ −1 Byassumption,suchatelescopercanbecomputedusingno Proof. Immediatebycheckingthatthetwochoicesforrand more than d satisfy theconditions stated in Theorem 5. C(r,d)=κ (τ +1)r+δ+2 3 δ+τr+d−(|µ|+1)τ +1 (cid:0) (cid:1) (cid:0) (cid:1) An accurate prediction for the degrees of the telescopers operationsin . Theclaimnowfollowsfromtheasymptotic K canalsobeusedforimprovingtheefficiencyofcreativetele- expansionsofC(τ,f(τ)+1)andC(ατ,f(ατ)+1)forτ →∞, scoping algorithms. Although most implementations today respectively. computethetelescoperwith minimumorder,it maybeless costly to compute a telescoper of slightly higher order. If Theleadingcoefficientinpart2isminimizedforα=5/4. we know in advance thedegrees d of the telescopers for ev- Thissuggeststhatwhenϑandνarelargeandapproximately eryorderr,wecanselectbeforethecomputationtheorderr equal, the order of the cheapest telescoper is about 20% whichminimizesthecomputationalcost. Ofcourse,thecost larger than theminimal expected order. dependson thealgorithm which is used. Itis not necessary (and not advisable) to follow the steps in the derivation of 4. THE RATIONALCASE Theorem5anddoacoefficientcomparisonover . Instead, K oneshouldfollow thecommon practice[8]of comparingco- Wenowturntothecasewherehcanbewrittenash=qh0 efficientsonlywithrespecttopowersofk andsolvealinear for some hypergeometric term h0 with Sk(h0)/h0 = 1. By system over (n). For nonminimal choices of r, this sys- the following transformation, we may assume without loss K tem will havea nullspace of dimension greater than one, of of generality h0 =1. which we need not compute a complete basis, but only a single vector with components of low degree. There are al- Lemma 9. Let h be a hypergeometric term and suppose gorithms known for computing such a vector using O(m3t) that h = qh0 for some q ∈ (n,k) and a hypergeometric K field operations when the system has at most m variables term h0 with Sk(h0)/h0 =1. Let a,b∈ [n,k] be such that K and equations and the solution has degree at most t [3, 13, Sn(h0)/h0 =a/b. Let L be a telescoper for q of order r and 5]. Inthesituationathand,wehavem=(r+1)+(δ+ϑr+1) degree d. Then there exists a telescoper for h of order r and variablesandasolutionofdegreet=δ+ϑr+d−(|µ|+1)ν+1. degree at most d+rmax{degna,degnb}. Proof. WriteL=ℓ0+ℓ1Sn+···+ℓrSnr andletC ∈ (n,k) Themainpart ofthecomputationalwork happensinthe K be a certificate for L and q, so L(q) = S (Cq)−Cq. For last two steps. It therefore appears sensible to assume in k i=0,...,r, let thefollowingdegreeanalysisthatwealreadyknowthedata ℓ˜i :=ℓiabSn(cid:16)ab(cid:17)···Sni−1(cid:16)ab(cid:17) utrah,teVh1de,re.g.tr.he,eaVnbs,oifnu1n,td.e.sr.mi,nsfstoefcromtmhsepoufdteteghdreeiirensdseotgferpneeu2sm,aaenrndadtcotoroeaffienxcdpierdnetesss- and L˜ :=ℓ˜0+ℓ˜1Sn+···+ℓ˜rSnr. Then nominator of h, say. L˜(qh0)=L(q)h0 =(Sk(Cq)−Cq)h0 =Sk(Cqh0)−Cqh0. 4.2 Counting Variablesand Equations Because of Also in the present case, the degree estimate is obtained Sk a = Sk(Sn(h0)) = Sn(Sk(h0)) = Sn(h0) = a, bceyrtbaainlalnincienagrstyhsetenmumovbeerr o.fTvahreialibnleeasrasnydsteemquwateiocnosnsoidfear (cid:16)b(cid:17) Sk(h0) Sk(h0) h0 b K originates from a particular way of executing steps 3 and 4 the operator L˜ is free of k. Thus, after clearing denomina- of the algorithm outlined above. tors, L˜ is a telescoper for h with coefficients of degree at most d+rmax{degna,degnb}. Theorem 10. Let u ∈ K[n] and let V1,...,Vs ∈ K[n][Sn] be operators of degree δi (i = 1...,s). Let fi = (ain+ From now on, we assume that h is at the same time a′ik+a′i′)−ei for some a′i′ ∈ , ai,a′i ∈ with a′i > 0 and a proper hypergeometric term and a rational function, or gcd(ai,a′i)=1, ei>0, suppoKse Z equivalently, that h is a rational function whose denomina- tor factors into integer-linear factors. Le [9] gives a precise ai − aj n+ a′i′ − a′j′ 6∈ description of the structure of telescopers in this case, and (cid:16)a′i a′j(cid:17) (cid:16)a′i a′j(cid:17) Z heproposesanalgorithmdifferentfromZeilberger’sforcom- pstuetpinsgofthheims .algOourirthdmeg,reseoewsetimstaatret ibsydberriivefledyfsoulmlowmianrgiztinhge efovrerayllri≥6=j siw=i1tha′ieian=deejv.erLyet h= u1 Psi=1Vi(fi). Then for themain steps of Le’s approach. P s ′ 4.1 Le’s Algorithm −r−1+ aiδi Given a rational proper hypergeometric term h, Le’s al- d> Xi=s1 +degnu gorithm computes atelescoper L for h as follows. r+1− a′ i Xi=1 1. Compute g ∈ (n,k) and polynomials p,q ∈ [n,k] with gcd(q,Si(Kq))=1 for all i∈ \{0} such tKhat there exists a telescoper L for h of order r and degree d. k Z p h=S (g)−g+ . k q Proof. According to Le’s algorithm, it suffices to find some L ∈ [n][Sn] and operators Ri ∈ (n)[Sn] with the prop- TifhLenisaantoepleesrcaotpoerrLforispa. tAelbersacmopoevr[f1o]rahndifPaanudleo[n1l1y] erty tKhat L(u1Vi)=Ri(Sna′i−1) forKall i. explain how to computeqsuch a decomposition. Denotebyρi theorderofVi. Writingd˜:=d−degnu,we makean ansatz L=L˜u with 2. Compute a polynomial u∈ [n], operators V1,...,Vs finormK[fni][=Sn(]a,inan+dar′iakti+onaa′i′l)−fuKenic(tiio=ns1,f.1.,..,.s.),sfuschofththaet L˜ = r d˜ ℓi,jnjSni Xi=0Xj=0 s p 1 q = uXi=0Vi(fi). sItotthhuastrLemhaainssdteogrceoensdtraunctdoLpeu1rVaito=rs RL˜Vii∈(i [=n][1S,n.]..w,ist)h. Such data always exists according to Lemma 5 in [9] L˜Vi = Ri(Sna′i −1). Since LV˜i has order r+Kρi and degree in combination with theassumption gcd(q,Ski(q))=1 d˜+δi, we consider ansatzes for the Ri of order r+ρi−a′i (aire∈chZos\e{n0s}u)c.hItthcaatna′ib>e f0u,rtehie>r a0s,sgucmd(eadi,tah′ia)t=th1efofri aannddddeeggrreeeed0˜.+Tδhi,enreswpeechtaivveelya,ltboegceatuhseerSna′i−1hasordera′i all i, and (cid:16)aa′ii − aaj′j(cid:17)n+(cid:16)aa′i′i′ − aa′j′j′(cid:17)6∈Z (r+1)(d˜+1)+Xi=s1(r+ρi−a′i+1)(d˜+δi+1) for all i6=j with ei=ej. variables in L˜ and the Ri, and comparing coefficients with 3. Fsuocrhit=hat1,S.na.′i.−,s1,icsoamrpiguhttedaivnisoopreorfaLtoir(u1LVii)∈. KIt(fnol)l[oSwns] Rreis(pSenac′it−to1)nleaandds tSonailnineaallrtshyesteremquwiritehd identities L˜Vi = from Le’s Lemma 4 that the operators Li with this s property are precisely the telescopers of the rational (r+ρi+1)(d˜+δi+1) functions Vi(fi). Xi=1 4. Compute a common left multiple L∈ [n][Sn] of the equations. Thissystemwillhaveanontrivialsolutionwhen- K operators L1,...,Ls. Then L is a telescoper for h. ever the number of variables exceeds the number of equations. Underthe assumption r≥ s a′,theinequality Again,itisaneasymattertospecializethegeneraldegree i=1 i s P boundtoadegreeestimate foralow ordertelescoper, orto (r+1)(d˜+1)+ (r+ρi−a′i+1)(d˜+δi+1) an orderestimate for a low degree telescoper. Xi=1 Corollary 12. With the notation of Theorem 10, h admits >Xi=s1(r+ρi+1)(d˜+δi+1) a tsie=l1es(δcoi−pe1r)ao′ifaosrdweerllra=saPtesil=e1scao′ipearndofdoergdreererd== desi=gn1auiδ+i is equivalent to Pand degree d=degnu. P d˜> −r−1+ si=1a′iδi. Proof. Clear by checking that the proposed choices for r r+1−Ps a′ and d are consistent with theboundsin Theorem 10. i=1 i P This completes theproof. Also like in the non-rational case, the bounds for the de- 4.3 Examples andConsequences grees of the telescopers can be used for deriving bounds on the computational cost for computingthem. In thepresent Example 11. 1. The rational function situation, let usassume for simplicity that thecost of steps (2n−3k)(3n−2k)2 1 and 2 of Le’s algorithm is negligible, or equivalently,that h= (n+k+2)(n+2k+1)(2n+k+1)(3n+k+1) theinputhisoftheform u1 si=0Vi(fi)withVi∈K[n][Sn]. We shall analyze the algoriPthm which carries out steps 3 can be written in the form h = u1 4i=1Vi(fi) where and4ofSection4.1inonestrokebymakinganansatzover ussuu=cchh(ntthh−aatt1δa)n1′1(==n+·a·3′2·)=(=2naδ4−′3 ==1)(613.,naT+′4h1e=)r(eP52fno,r+ea,n1dT),htthehoeereVfmii aa1rr0ee iKng(nt)heforrigahntorepmeriantdoerrsLo=fLℓ0u1V+iℓw1Sitnh+res·p··ec+tℓtroSSnrna,′ic−om1pauntd- equating their coefficients to zero. We assume, as before, predicts a telescoper of order r and degree d whenever thattheresultinglinearsystemissolvedusinganalgorithm r≥5 and whoseruntimeislinearintheoutputdegreeandcubicinthe 29−r d> +6. matrix size. Then thealgorithm requiresO(r3d)operations r−4 in . This curve together with the region of all points (r,d) K for which a telescoper of order r and degree d exists is Theorem 13. Let u ∈ [n], V1,...,Vs ∈ [n][Sn], and K K shown in the left figure below. f1,...,fs ∈ (n,k) be as in Theorem 10 and consider h= 2. The corresponding picture for the rational function u1 si=1Vi(fiK). Suppose that κ ∈ R is a constant such that degPree t solutions of a linear system with m variables and (n−k+1)2(2n−3k+5) at most m equations over (n) can be computed with κm3t h= K (n+k+3)(n+k+5)(n+2k+1)(2n+k+1)2 operations in . Assume δ1 = ··· = δs =: δ > 0 and ′ ′ K ′ ′ a =a =···=a =:a are fixed. Then: is shown in the figure below on the right. This input 1 2 s can be written in the form 1. A telescoper of order r=a′s can be computed using 1 4 a′4(δ−1)κs4+O(s3) h=Sk(g)−g+ Vi(fi) u Xi=1 operations in . withg= 10(n+3)2(n+4)(2n+2k+7) ,u=(3n+1)2(n− K 4)′2(n−2′)(n2−(n4)+2(n5+)(9n)(n++k9+),3)t(hne+kf+i4s)uch that a′1 =a′2 = 2. Irf=α α>a′1si+s sOo(m1)etchoennstaanttelaenscdoprerisocfhoorsdeenrsrucchanthbaet a3 =1, a4 =2, and the Vi such that δ1 =8, δ2 =δ3 = computed using δ =7. According to Theorem 10, we therefore expect 4 ar≥tel5esacnodper for h of order r and degree d whenever αα−31a′3(δ−1+(α−1)degnu)κs3+O(s2) 35−r d> +8. operations in . r−4 K In this example, the estimate is not as tight as in the In particular, a telescoper for h can be computed in polyno- previous one. mial time. d d 40 40 Proof. AccordingtothefirstestimatestatedinTheorem10, for every r ≥ a′s there exists a telescoper of order r and 30 30 degree d for any sa′δ−r−1 20 20 d>f(r):= r+1−sa′ +degnu. 10 10 Byassumption,suchatelescopercanbecomputedusingno more than C(r,d):=κr3d operations in . The claim now followsfromtheasymptoticexpansionsofKC(a′s,f(a′s)+1) 0 r 0 r ′ ′ 0 5 10 15 20 0 5 10 15 20 and C(αas,f(αas)+1) for s→∞, respectively. Whendeg u=0,theleadingcoefficientinpart2ismin- [7] William Gosper. Decision procedure for indefinite n imized for α=3/2. This suggests that when s is large and hypergeometricsummation. Proceedings of the ′ all the δi, and ai are approximately equal, the order of the National Academy of Sciences of the United States of cheapest operator exceeds the minimal expected order by America, 75:40–42, 1978. around 50%. [8] Christoph Koutschan.A fast approach to creative Itmustnotbeconcludedfromaliteralcomparisonofthe telescoping. Mathematics in Computer Science, exponentsinTheorems5and10thatLe’salgorithmisfaster 4(2–3):259–266, 2010. than Zeilberger’s, because τ in Theorem 5 and s in Theo- [9] HaQ.Le.A directalgorithm toconstruct theminimal rem 10 measure the size of the input differently. Neverthe- Z-pairsfor rational functions. Advances in Applied less, it is plausible to expect that Le’s algorithm is faster, Mathematics, 30(1–2):137–159, 2003. because it finds the telescopers without also computing a [10] Mohamud Mohammed and Doron Zeilberger. Sharp (potentially big) corresponding certificate. Our main point upperboundsfor theorders of therecurrencesoutput here is not a comparison of the two approaches, but rather bytheZeilberger and q-Zeilberger algorithms. Journal the observation that both of them admit a degree analysis of Symbolic Computation, 39(2):201–207, 2005. whichfitstothegeneralparadigmthatincreasingtheorder [11] Peter Paule. Greatest factorial factorization and cancauseadegreedropwhichissignificantenoughtoleave symbolic summation. Journal of Symbolic a trace in thecomputational complexity. Computation, 20:235–268, 1995. It can also be argued that the situations considered in [12] MarkoPetkovˇsek,HerbertWilf, andDoron Zeilberger. Theorems 8 and 13 are chosen somewhat arbitrarily (ϑand A=B.AK Peters, Ltd.,1997. ν growing while µ remains fixed; resp. s growing while all ′ [13] ArneStorjohann and Gilles Villard. Computing the the δi and ai remain fixed). Indeed, it would be wrong to rank and a small nullspace basis of a polynomial takethesetheoremsasanadvicewhichtelescopersaremost matrix.In Proceedings of ISSAC’05,pages 309–316, easily computed for a particular input at hand. Instead, in 2005. order to speed up an actual implementation, one should let [14] Herbert S.Wilf and Doron Zeilberger. An algorithmic theprogram calculate theoptimal choice for r from thede- proof theory for hypergeometric (ordinary and q) greeestimatesgivenTheorems5and10with theparticular multisum/integral identities. Inventiones parameters of theinput. Mathematicae, 108:575–633, 1992. Unfortunately, we are not able to illustrate the speedup obtained inthiswaybyan actualruntimecomparison for a [15] Doron Zeilberger. A fast algorithm for proving concrete example, because for examples which can be han- terminatinghypergeometric identities. 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