Computer Proofs of a New Family of Harmonic Number Identities

Transcription

1 Computer Proofs of a New Family of Harmoic Number Idetities Peter Paule 1 Research Istitute for Symbolic Computatio Johaes Kepler Uiversity Liz A 4040 Liz, Austria Peter.Paule@risc.ui-liz.ac.at Carste Scheider 2 Research Istitute for Symbolic Computatio Johaes Kepler Uiversity Liz A 4040 Liz, Austria Carste.Scheider@risc.ui-liz.ac.at Abstract I this paper we cosider five coectured harmoic umber idetities similar to those arisig i the cotext of supercogrueces for Apéry umbers. The geeral obect of this article is to discuss the possibility of automatig ot oly the proof but also the discovery of such formulas. As a specific applicatio we cosider two differet algorithmic methods to derive ad to prove the five coectured idetities. Oe is based o a extesio of Karr s summatio algorithm i differece fields. The other method combies a old idea of Newto which has bee exteded by Adrews) with Zeilberger s algorithm for defiite hypergeometric sums. 1 Itroductio For a positive iteger let H = deote the th harmoic 2 umber. It will be coveiet to defie H = 0 wheever is o-positive. The obect of this paper is the discussio of two ew algorithmic approaches which are used to prove the followig family of idetities for 1: 1 Partially supported by a ORCCA visitig professor appoitmet at the Symbolic Computatio Group, Uiversity of Waterloo, CANADA. 2 Supported by SFB-grat F1305 of the Austria FWF. Preprit submitted to Advaces i Applied Mathematics November 6, 2002

2 1 H + H ) = 1, 1) H + 2 H ) = 0, 2) H + 3 H ) = 1), 3) H + 4 H ) = 1), 4) H + 5 H ) = 1) ) It will be coveiet to rewrite the left sides of these idetities i the form where for α {1,..., 5}, R α) = R α) α ad S α) + S α) 6) ) α = α 2)H. 7) Biomial sums like that o the right side of 5) play a crucial role i Apéry s approach to prove the irratioality of ζ2) ad ζ3); see, for istace, the iformal report [vdp79]. I a attempt to prove certai supercogrueces for Apéry umbers which were coectured by Beukers, certai harmoic umber idetities popped up i [AO00,Ah02] see also the recet works of Morteso [Mor02a,Mor02b]). I particular, these formulas arise out of computatios ivolvig the p-adic gamma fuctio. This motivated S. Ahlgre to do a heuristic search i order to explore whether there are more harmoic umber idetities of a similar form. The result of this study was a family of coectured idetities, amely 1) 5) above. Util recetly there has bee o algorithm to derive defiite summatio idetities ivolvig harmoic umbers. For example, the solutio to bous problem 69 [GKP94, Chapt. 6], Fid a closed form for k=1 k 2 H +k, eds with the remark, It would be ice to automate the derivatio of formulas such as this. This situatio chaged due to work [Sch02a,Sch02b,Sch02c,Sch02d] of oe of the authors which exteds Karr s idefiite summatio algorithm 2

3 [Kar81,Kar85] Karr s algorithm is based o the theory of differece fields [Coh65]). Scheider exteds Karr s method to defiite summatio ad to solvig liear differece equatios with polyomial coefficiets ot oly of first but of arbitrary order. These developmets have bee implemeted i the form of the Mathematica package Sigma [Sch00], which we have used i all of our computatios for the examples below. Remark. Our emphasis i this article is o the problem of automatig the derivatio of such formulas as 1) 5). Cocerig computer assistace i provig such formulas there are other recet methods; see, e.g., Chyzak s geeralizatio of the Gosper-Zeilberger algorithm [CS98] or Wegschaider s package Multisum [Weg97] for simplifyig multiple hypergeometric sums. I Sectio 3 we will demostrate how idetities such as 1) 5) ca be proved ad foud with the Sigma package. We wat to emphasize that the uderlyig algebraic theory is quite complex but also very geeral. As a cosequece, the scope of applicatios of Sigma is much broader. Besides hyper- ad q-hypergeometric sums which could also ivolve harmoic umbers ad their q-aalogues, it ca also hadle summatio problems built by multiple ested sums of very geeral kid [Sch01]. Therefore it is atural to ask whether there is a more elemetary algorithmic approach for provig idetities like 1) 5). It turs out that this is ideed the case. I Sectio 2 we itroduce a ew algorithmic approach to prove defiite harmoic umber idetities such as 1) 5). The two buildig blocks of this approach are well-kow. Its algorithmic igrediet is Zeilberger s algorithm [Zei90,PWZ96] which is implemeted i all maor computer algebra systems. This is combied with a operator method for rewritig harmoic umbers i terms of biomial coefficiets which, as explaied below, traces back to Newto. I Sectio 4 we compare the methods of Sectio 2 ad Sectio 3, ad draw some coclusios. 2 A Algorithmic Versio of the Newto-Adrews Method Let L be the operator which evaluates fuctios fx) at x = 0, i.e., L fx) := f0). Let D be differetiatio with respect to x, i.e., D fx) := f x). It is a easy exercise to verify that for all itegers, x + LD = H. 8) This crucial observatio i may cases allows us to hadle harmoic umber idetities by reducig them to a hypergeometric problem, a techique ofte 3

4 used by G.E. Adrews i his work. I [AU85] oe fids the followig statemet: Richard Askey has poited out to us that ideed Issac Newto was the first to see that the partial sums of the harmoic series arise from differetiatio of a product [N60, p. 561]. We illustrate the method by a elemetary example, amely S) := H, 0. Usig 8) ad the the hypergeometric summatio idetity [GKP94, 5.9)] x + = 1 + ) x +, 9) 1 + x the give sum becomes S) = LD x + = LD x ) x +. 10) By applyig the product rule for differetiatio this simplifies further to S) = H +L ) x + L +L H 1 + x) 2 = H + H, 11) 1 + x which i tur becomes the well-kow fact [GKP94, 6.67)] H = + 1) H, 0. 12) I this particular example the give sum as well as the uderlyig hypergeometric summatio 9) are idefiite, but obviously the method exteds also to the defiite case. However, applyig the method i this classical fashio will always lead to the problem of simplifyig the hypergeometric sums which arise. Hece, from algorithmic poit of view, it is a atural step to lik the Newto-Adrews method with Zeilberger s paradigm of creative telescopig. How such a combiatio is tured ito a effective algorithm becomes trasparet i the proof of idetity 1). Proof of Idetity 1). It is coveiet to prove 1) i the equivalet form 2) H ) = 1 2, 0, 13) which is obtaied from 1) by geometric summatio ad by reversig the order of summatio i the sum H ). The left side of 13) is othig but S 1), ad we obtai from 8) that S 1) = LD t x) where t x) := x + 2) ). 14) 4

5 Applyig Zeilberger s algorithm we used Sigma) returs the recurrece relatio 2 + 1) t x) x ) t +1 x) + + 1) t +2 x) = 0, 0. 15) The ext step is to apply the differetiatio operator D to both sides of 15) which results i the mixed differetial-differece equatio 2+1) t x) t +1 x) x+3+3) t +1x)++1) t +2x) = 0, 0. 16) Fially we apply the operator L to both sides of 16) which gives 2 + 1) S 1) 3 + 3) S 1) ) S 1) +2 = t +1 0), 0. 17) Now it is a elemetary fact that for all 0, t 0) = 2) = 0, 18) which ca be also foud by Gosper s algorithm [Gos78]. Therefore, i order to fid the right side of 13) oe oly eeds to solve 2 S 1) 3 S 1) +1 + S 1) +2 = 0, 0, 19) with iitial coditios S 1) 0 = 0 ad S 1) 1 = 1, which agai ca be doe algorithmically. 2.1 The Newto-Adrews-Zeilberger Algorithm Before summarizig i the form of a algorithm descriptio, we recall that S is a hypergeometric sequece if there exists a ratioal fuctio rx) such that S +1 /S = r) for all sufficietly large. Similarly, a term f, ) is called hypergeometric i ad, if the quotiets f + 1, )/f, ) ad f, + 1)/f, ) are ratioal fuctios i ad. Newto-Adrews-Zeilberger Algorithm. Iput: a term f, ) which is hypergeometric i ad ; Output: a liear recurrece of type 22) or 24), respectively, for the sum S of the form S := H f, ) or S := f, )/H, respectively. 20) 5

6 The algorithm ca be applied if Zeilberger s algorithm succeeds i fidig a recurrece for the sum t x) of the form x + 1 x + t x) := f, ) or t x) := f, ), respectively. 21) By 8) we have that S = LD t x). Cosequetly, by applyig to the t x)- recurrece successively the operators D ad L as described i the proof of idetity 1)), a recurrece for S ca be derived i the form d a d ) S +d + a d 1 ) S +d a 0 ) S = p i )t +i 0), 22) i=0 where the a l ) ad p i ) are polyomials i, ad where a d ) is o-zero. I additio, by Zeilberger s algorithm ad by differece equatio solvers like [Pet92] ad [vh99] we ca decide algorithmically see also [A02]) whether t 0) = f, ) 23) is a hypergeometric sequece i. If so, each t +i 0) is a ratioal fuctio multiple of t 0) ad therefore also σ := d i=0 p i )t +i 0). Cosequetly, the recurrece 22) simplifies to a d ) S +d + a d 1 ) S +d a 0 ) S = σ, 24) where σ is a hypergeometric sequece i. Applicatios. Suppose the Newto-Adrews-Zeilberger algorithm outputs a recurrece of the form 24). The differece equatio solvers like [Pet92] ad [vh99] ca be used to decide algorithmically whether S fids a closed form represetatio as a liear combiatio of hypergeometric terms. But eve if S does ot fid a closed form represetatio as a liear combiatio of hypergeometric terms, it might happe that for a give sequece R the sequece R + S does have such a represetatio, which is the case for the idetities 3) ad 4); see below. I geeral, suppose a liear recurrece for R is available i the form b e ) R +e + b e 1 ) R +e b 0 ) R = τ 25) where τ is a hypergeometric sequece ad the b l ) are polyomials i, ad b e ) is o-zero. The usig procedures from the packages [SZ94] or [Mal96], the recurreces 22) ad 24) ca be combied ito a sigle homogeeous 6

7 liear recurrece c h ) T +h + c h 1 ) T +h c 0 ) T = 0 26) where the c l ) are polyomials i with c h ) o-zero, which is satisfied by the sequece T := R + S. Fially by applyig differece equatio solvers like [Pet92] or [vh99] oe fids a closed form represetatio of R + S as a liear combiatio of hypergeometric terms. I priciple, there are possibilities to exted the Newto-Adrews-Zeilberger algorithm to the case where the summad of S ivolves products or quotiets of products) of harmoic umbers, but the oe has to cosider may extra coditios. Nevertheless, such methods could cotribute to possible extesios of computer algebra packages that rely oly o Zeilberger s algorithm. Due to the fact that Scheider s extesio of Karr s work described i Sectio 3 covers all these applicatios i a atural way, we refrai from presetig further details. Oly for comparig the two methods, we give short versios of the Newto-Adrews-Zeilberger derivatios of 2) 4). Cocerig idetity 5), we emphasize the well-kow fact that its right side is ot expressible as a hypergeometric term i, so the Newto-Adrews-Zeilberger algorithm caot derive this represetatio. However for the sake of completeess we will briefly describe how a variatio of this method ca be used to prove idetity 5). 2.2 Newto-Adrews-Zeilberger Proofs of 2) 5) Proof of Idetity 2). We use the well-kow Vadermode evaluatio ) 2 = 2 ) to rewrite 2) i the form 2) H The rewrite rule 8) gives that S 2) = LD t x) where t x) := ) 2 = 1 2, 1. 27) 2 x + 2) ) 2, 28) ad the Newto-Adrews-Zeilberger algorithm applied as i the proof of idetity 1) leads to the recurrece relatio )3 + 5) S 2) + 1) ) S 2) ) + 2)3 + 2) S 2) +2 = 3 + 5)4 + 1)t 0) )t +1 0), 1. 29) 7

8 Now it is a elemetary fact that for all 0, 2 t 0) = 2) = 0, 30) which ca be also foud by Gosper s algorithm [Gos78]. Therefore, i order to fid the right side of 27) oe oly eeds to solve )3 + 5) S 2) + 1) ) S 2) ) + 2)3 + 2) S 2) +2 = 0, 1, 31) with iitial coditios S 2) 1 = 1 ad S 2) 2 = 3, which agai ca be doe algorithmically by applyig differece equatio solvers like [Pet92] or [vh99]. Next we preset the Proof of Idetity 3). Accordig to 6), idetity 3) is of the form R 3) + S 3) = 1). 32) Now R 3) does ot have a represetatio as a hypergeometric term sice the Zeilberger output recurrece for R 3) is + 2) 2 R 3) ) R 3) ) 2 R 3) = 0, 33) which does ot have ay hypergeometric solutio. Nevertheless, sice the right side of 32) is hypergeometric, we ca apply the Newto-Adrews-Zeilberger algorithm to fid this evaluatio. With this procedure we fid 81 + ) ) ) S 3) ) 2 + ) ) S 3) ) ) S 3) ) 3 + ) ) S 3) ) 3 + ) ) ) S 3) +4 = 0 34) as the recurrece for S 3). As described above, we apply the package Geer- 8

9 atigfuctios.m with iput 33) ad 34) to obtai the recurrece 81 + ) ) )S 3) +2 + ) )S 3) )S 3) ) )S 3) ) ) )S 3) +4 = 0 35) for T := R 3) + S 3). Fially with the solvers [Pet92] or [vh99] oe fids that T = 1), which completes the proof of 3). Proof of Idetity 4) Sketch. Accordig to 6), idetity 4) is of the form 2 R 4) + S 4) = 1). 36) Agai, R 4) does ot have a represetatio as a hypergeometric term, so oe proceeds completely aalogously to the proof of 3). We refrai from givig the details; however, we metio the fact that despite obtaiig agai a order 4 recurrece for T := R 4) + S 4), the iteger coefficiets of the polyomials ivolved become quite large. Usig the Newto-Adrews-Zeilberger algorithm, ot oly ca we prove the idetities 1) 4), but we ca also fid the correspodig closed forms o their right sides. With the last idetity the situatio is slightly differet. Proof of Idetity 5) Sketch. Accordig to 6), idetity 5) is of the form R 5) + S 5) = A 37) where A = 1) ) is a sequece of Apéry umbers. Agai Zeilberger s algorithm ad the Newto- Adrews-Zeilberger algorithm deliver a recurrece for R 5) ad S 5), respectively. As described above, from these recurreces oe obtais a homogeeous liear recurrece for T := R 5) + S 5) which turs out to be of order 6 ad big eough to fill oe page). But this time the right side A is a defiite sum which does ot simplify to a hypergeometric term, so we are ot able to fid A as the solutio to this recurrece sice there is o algorithm available for this task so far. However, the task of provig idetity 5) ca be completed algorithmically, for istace, as follows. With Zeilberger s algorithm compute the recurrece + 2) 2 A ) A ) 2 A = 0. 39) 9

10 The usig procedures from the packages [SZ94] or [Mal96] with iput 39) ad the Newto-Adrews-Zeilberger recurrece for T := R 5) + S 5), oe computes a homogeeous liear recurrece for Q := R 5) + S 5) A which turs out to be of order 6. Fially, checkig that Q i = 0 for i from 1 to 6 completes the proof of 5). 3 Sigma: A Summatio Package for Discoverig ad Provig Karr developed a algorithm for idefiite summatio [Kar81,Kar85] based o the theory of differece fields [Coh65]. He itroduced so called ΠΣ-fields i which first order liear differece equatios ca be solved i full geerality. This algorithm deals ot oly with sums over hypergeometric terms, like Gosper s algorithm [Gos78,PP95], or over q-hypergeometric terms, like [PR97], but also with summatios over terms i which, for example, the harmoic umbers ca appear i the deomiator. Geerally speakig, Karr s algorithm is the summatio couterpart of Risch s algorithm [Ris70] for idefiite itegratio. Ispired by this algorithm, Scheider developed a sigificatly more geeral algorithmic summatio theory [Bro00,Sch02a,Sch02b,Sch02c,Sch02d] also based o differece field theory. I additio, Scheider implemeted his algorithms i the computer algebra system Mathematica. The correspodig summatio package Sigma also provides a user iterface that dispeses the user from workig explicitly with differece fields. Istead, the user ca hadle all summatio problems coveietly i terms of usual sum ad product expressios; see [Sch00,Sch01]. A importat aspect of Scheider s work is his extesio of Karr s origial method i such a way that defiite summatio problems ca be treated too. For example, i [Sch02a] it is show how the defiite summatio idetity H ) = 2 H 2 =1 1, 0. 40) 2 ca be derived automatically with the Sigma package. Note that idetity 40) expresses the first defiite summatio compoet H of S 1) as a liear combiatio of 2 times the idefiite sums H ad 1 =1, respectively. 2 10

11 3.1 Itroductory Example The defiite sum H is the secod compoet of the sum S 1). So, before turig to the other S α) we will first demostrate how oe ca derive for this sum a evaluatio similar to 40). We start the Mathematica sessio by loadig the package with I[1]:= << Sigma Sigma - A summatio package by Carste Scheider c RISC-Liz The we set up the summatio problem as follows: I[2]:= mysum = Out[2]= SigmaSum[ SigmaHNumber[]SigmaBiomial[, ], {, 0, }]. H Remark. The basic fuctios SigmaSum ad SigmaProduct are used to describe all ested sum ad product epressios that ca be formulated i ΠΣfields. To facilitate this task there are umerous other fuctios available, like SigmaHNumber, SigmaBiomial or SigmaPower. For istace, SigmaHNumber[] produces the th harmoic umber H which alteratively could be described by SigmaSum[1/k,{k,1,}]. Additioally, i order to eable the user to defie his/her ow obects that ca be formulated with ested sums ad products, various help fuctios are provided. I the first step we ask Sigma to compute a recurrece that is satisfied by mysum: I[3]:= rec = GeerateRecurrece[mySum] Out[3]= { 41 + ) SUM[] ) SUM[1 + ] 1 + )1 + ) SUM[2 + ] == 1 + } This meas that SUM[] = H ) =mysum) satisfies the output recurrece Out[3]. Remark. To compute such recurreces Zeilberger s creative telescopig [Zei90] has bee exteded from hypergeometric expressios to terms i ΠΣ-fields; for more iformatio see [Sch01]. Secodly, we try to fid solutios to this recurrece. I the give situatio it turs out that the algorithm does ot fid ay solutio i the uderlyig differece field F which has bee costructed iterally by the obects give i the recurrece rec. The Sigma package is desiged i such a way that whe it fails to fid a solutio to a recurrece withi a give differece field F, the 11

12 it also idicates that there is o sum extesio of F i which a solutio exists. Therefore we try to exted F by a appropriate product extesio. Fidig such product extesios is assisted by the fuctio FidProductExtesios which uses M. Petkovšek s package Hyper [Pet92,Pet94,PWZ96]. This package is able to fid all hypergeometric solutios of liear recurreces such as Out[3] ad has to be loaded first. I[4]:= << Hyper I[5]:= FidProductExtesios[rec[[1]], SUM[]] I use M. Petkovsek s package Hyper to fid product extesios! Out[5]= { 2 } i=1 This step was successful: the output tells us that if we exted the give differece field F by the ew elemet 2, the we will fid at least oe o-trivial solutio to Out[3]. But the Sigma package ca do much more. Namely, with the ext fuctio call we ca fid ot oly solutios i F2 ), but also solutios i all differece fields which exted F2 ) by ested sums built from the elemets of F2 ). I[6]:= recsol = SolveRecurrece[rec[[1]], SUM[], Out[6]= { {0, 2. }, { 0, 2. NestedSumExt, Tower {2. }] 2 + ι 1 } {, 1, 2. 1 }} 1 + ι 1 ) ι ι 1 )ι 1 2 ι 1. ι 1=2 I this example we have succeeded completely; the output describes two liear idepedet solutios of the homogeeous variatio of the recurrece Out[3], amely 2 ad 2 2+ι 1 ι1 =2 1+ι 1 ) ι 1, ad oe particular solutio of the ihomogeeous recurrece itself, amely 2 ι1 =2 ι 1=2 1 1+ι 1 ) ι 1. Remark. These kid of solutios are called d Alembertia solutios ad are itroduced i [AP94]; further results ca be foud i [HS99] ad [Sch01]. Fially, the closed form of mysum is that liear combiatio of the homogeeous solutios plus the ihomogeeous solutio which has exactly the same iitial values as mysum. This is also computed automatically: I[7]:= result = FidLiearCombiatio[recSol, mysum, 2, MiIitialValue 1] Out[7]= ι ι 1 )ι ι 1 )ι 1 2 ι 1. ι 1 =2 Note that we were oly able to fid this liear combiatio startig from 1. This closed form evaluatio of mysum for 1 ca be rewritte as follows. Applyig partial fractio decompositio to the summads gives ι 1 =2 2 + ι 1 = ι 1 )ι ι 1 ι 1 ad ι 1 )ι 1 2 = 1 ι ι 1 )2 1 ι 1 ι 1 2. ι 1 12

13 This motivates us to simplify Out[7] further by askig Sigma for a represetatio of the expressio result by the sums H ad 1 =1. This is doe by 2 the followig commad. I[8]:= SigmaReduce [ result,, Tower { 1 }] H k, 2. =1 Out[8]= H ) ) =1 Summarizig, with Sigma we foud that H = H )) 2 =1 2 holds for all 1; by ispectio we see that 41) holds for = 0 as well. 41) 3.2 Automatic Discovery of 1) ad 2) Combiig 40) ad 41) we obtai ) ) A H + B H ) = A H + B H = 1 B + 2 B + 2 A + B ) 1 ))) H. 2 =1 2 For the specific choice A = ad B = 2 this leads us immediately to 13) which, as poited out above, is equivalet to 1). Applyig Sigma as i Sectio 3.1 we ca fid automatically the followig two idetities which combie to ) 2 2 H = 2 H H 2 ), 0, 42) ) 2 H = H 2 H 2 ), 1, 43) A H + B H ) ) 2 = 1 4 B 2 2 A B ) 2 H H 2 )) 2. 13

14 By choosig A = ad B = 2 we obtai 27) which is equivalet to equatio 2). Summarizig, by usig the package Sigma we ot oly succeeded i discoverig ad provig the first two idetities of the family 1) to 5), but derived additioally as a by-product the idetities 40),41), 42), ad 43). 3.3 Provig ad Fidig Idetities I the followig we cosider the idetities 3) 5). We abbreviate their left sides by T α) ; recallig 6) this meas that T α) := R α) +S α) for α {3, 4, 5}. We will use two differet approaches; oe direct ad oe more sophisticated. These are described i the two subsectios below. For each approach the geeral strategy will be the same; amely, we first compute recurreces for the give left sides T α). More precisely, i the first attempt we will compute these recurreces i the classical way; i.e. by creative telescopig as i the previous subsectio. I the secod attempt we compute recurreces i a more sophisticated maer, amely by itroducig additioal sum extesios. It is crucial that these extesios ca be foud automatically ad also that these extesios produce recurreces of smaller order tha the direct approach. It turs out that for the give idetities these smaller orders are eve miimal. I additio to provig the idetities, this fact eables us to fid the right had sides of 3) 5) without ay further computatios The Direct Approach As metioed above we first compute recurreces for the sums T α) {3, 4, 5}. for α I[9]:= mysum3 = 1 3 H ) H ) I[10]:= rec3 = GeerateRecurrece[mySum3]. ) 3 ) ; Out[10]= { 1 ) SUM[] ) SUM[1 + ] + 2 ) SUM[2 + ] == 0}. ) 4 ) I[11]:= mysum4 = 1 4 H ) H ) ; I[12]:= rec4 = GeerateRecurrece[mySum4] Out[12]= { ) ) SUM[] ) SUM[1 + ] ) ) SUM[2 + ] == 0 } 14

15 I[13]:= mysum5 = 1 5 H ) H ) I[14]:= rec5 = GeerateRecurrece[mySum5]. ) 5 ) ; Out[14]= { 1 + ) ) ) SUM[] 2 + ) ) SUM[1 + ] ) SUM[2 + ] ) ) SUM[3 + ] ) 4 + ) ) SUM[4 + ] == 0 } Oe ca easily verify that 1) is a solutio of recurrece rec3 ad that 1) 2 is a solutio of rec4. Checkig iitial values of both sequeces proves idetities 3) ad 4). Note that by applyig differece equatio solvers like [Pet92] ad [vh99] oe is eve able to fid the closed form solutios 1) ad 1) 2 automatically. Sice the right side of idetity 5) is a defiite sum, we have to proceed i a slightly differet way. Namely, we compute a recurrece that cotais all the solutios of rec5 ad the recurrece give i 39). Usig oe of the packages [SZ94] or [Mal96] it turs out that the resultig recurrece is agai rec5. Sice the right side A defied i 38) is a solutio of 39), the expressio T 5) A is a solutio of rec5. Cosequetly, checkig that the first four iitial values of T 5) A are 0 implies that T 5) A is zero for all 1, which completes the proof of idetity 5). Remark. A differet approach would be to combie T 5) A ito a sigle defiite sum expressio ad to compute its defiig liear recurrece by applyig the Sigma fuctio call GeerateRecurrece to it. Agai it turs out that the result is recurrece rec5. We wat to emphasize that both strategies oly prove idetity 5). They do ot fid its right side; this situatio will chage i the more sophisticated approach of Sectio Moreover, we remark that if oe applies the Sigma fuctio call GeerateRecurrece directly to the left side sums i 3) 5), it turs out that the computatios are much more ivolved ad the orders of the resultig recurreces i compariso to the orders of rec3 to rec5 are icreased by oe. This idicates that creative symmetrizig itroduced for hypergeometric sums i [Pau94] plays a essetial role also i the algorithmic 15

16 treatmet of sums where, for istace, harmoic umbers are ivolved A More Sophisticated Approach: Recurreces with Sum Extesios Scheider s summatio theory provides a ew mechaism which fids certai sum extesios automatically. The details of this method are described i [Sch01, Sectio 4.4.3], so we restrict ourselves to brief descriptios of its applicatio to the idetities 3) 5). We shall see that the orders of the recurreces computed by this approach are sigificatly smaller tha those of rec3 to rec5. Idetity 3). For mysum3 resp. T 3) ) we are able to fid the followig recurrece of order 1 istead of order 2 as i Out[10]. I[15]:= rec3 = GeerateRecurrece[mySum3, SimplifyByExt DepthNumber] Out[15]= { 1 + ) SUM[] ) SUM[1 + ] == ι 1 ) ι 3 1.) 3 ) ι 3 1 } 1 + ι 1 ) 3 ι 1=1 I a secod step we ca show with Sigma that the sum o the right side is equal to for all 0. This shows that T 3) satisfies T 3) + T 3) +1 = 0 = 1). Ob- which allows us to read off the closed form represetatio T 3) viously this recurrece for T 3) is the miimal possible oe. Idetity 4). For mysum4 resp. T 4) ) we are able to fid the followig recurrece of order 1 istead of order 2 as i Out[12]. I[16]:= rec4 = GeerateRecurrece[mySum4, SimplifyByExt DepthNumber] Out[16]= ) ) SUM[] ) 2 SUM[1 + ] == ) ι 1 ) ι ι1) ) } 1 + ι 1 ) 4 ι 1=1 I a secod step we show with Sigma that the right side is equal to 0 for all 0. This proves that T 4) satisfies the recurrece ) T 4) ) T 4) +1 = 0; idetity 4) is a direct cosequece of this result. I particular, this recurrece has miimal order for 1) 2, therefore it is also the miimal recurrece 16

17 for T 4). Idetity 5). Fially, for mysum5 resp. T 5) ), we fid the followig recurrece of order 2 istead of order 4 as i Out[14]. I[17]:= rec5 = GeerateRecurrece[mySum5, SimplifyByExt DepthNumber] Out[17]= { 1 + ) ) SUM[]+ 2 + ) ) SUM[1 + ] ) 3 SUM[2 + ] == 1 + ) ) ι 1 ) ι ι1) 1 + ) ) ι 1=0 1 + ι 1 ) ) ) ι 1 ) ι ι1) } 1 + ι 1 ) ι 1 ) 5 I a secod step we show with Sigma that the right side is equal to 0 for all 0. Therefore we obtai the recurrece ι 1=0 1 + ) 2 T 5) ) T 5) ) 2 T 5) +2 = 0 for all 0. Observig that this is, up to a alteratig sig variatio, the well-kow recurrece 39) of the Apéry umbers eables us to guess the right side A i 5). The guess is verified by checkig the first two iitial values. Agai this recurrece is the miimal possible oe for T 5). Cosequetly, by idetifyig the output recurrece as the Apéry recurrece we have eve foud the right had side of idetity 5). 4 Coclusio Before we coclude with a ope problem we compare the two differet approaches of the previous sectios. I the Newto-Adrews-Zeilberger approach, usig 8) oe sets up a more geeral hypergeometric summatio problem that cotais the origial harmoic umber summatio. For this more geeral problem a Zeilberger recurrece is computed. I order to solve the origial harmoic umber summatio problem, this recurrece is specialized by differetiatio ad evaluatio. The geerality of this approach is also its computatioal bottleeck. More precisely, i may cases this asatz fids oly recurreces with a drastically higher recurrece order tha ecessary. For istace, for provig idetity 5) we have to compute a recurrece of order 6 istead of order 4 as i Out[14] or order 2 as i Out[17]. 17

18 Moreover, whe products or quotiets of several harmoic umbers appear i the summad, oe has to itroduce additioal variables i order to traslate the problem to the hypergeometric settig which reduces the efficiecy of the algorithm tremedously. Additioally, i this case, as poited out i Sectio 2, oe has to cosider may extra coditios. Depedig o their complexity, i geeral there is o guaratee that a desired recurrece for the give defiite summatio problem ca be derived by restrictig to hypergeometric tools oly. Nevertheless, i practice may problems are of the simple type 20); so the Newto-Adrews-Zeilberger approach could well serve as a useful extesio of ay implemetatio of Zeilberger s algorithm. The approach followed by the Sigma package is completely differet. Nested sum expressios, icludig summatios ivolvig harmoic umbers, are traslated i a atural way ito the correspodig differece field settig ad, by usig a very geeral algebraic machiery, the problem is solved there. Clearly, if oe restricts these geeral algorithms to the hypergeometric case, they caot compete i performace with the hypergeometric special purpose provers ad solvers. But, due to the richess of the uderlyig algebraic structure, the Sigma approach provides much more flexibility ad efficiecy whe dealig with defiite ested sum expressios. Here we wat to metio that with the Sigma package we ca go o to compute recurreces for the sums T α) as illustrated i Subsectio For istace, for α = 6 ad α = 7 we obtai recurreces that are quite out of scope for the aive hypergeometric approach, amely ad ) ) ) T 6) ) ) T 6) )3 T 6) +2 = 0 44) 1 + ) ) T 7) ) T 7) ) T 7) ) T 7) +3 = 0. 45) Usig the Sigma package, we computed recurreces for T α) up to α = 9. Remarkably, for 3 α 9, these recurreces are the same as the recurreces we computed with Sigma or Zeilberger s algorithm for the hypergeometric sum U α) := α 2), 46) also parameterized by α. We do ot kow whether the Zeilberger recurreces for U α) coicide with the miimal recurreces of the T α) for all α 3. Note that the sums T α) are highly o-trivial whereas it is easy to prove that the U α) evaluate to zero for all α, 0. 18

19 Aother ope problem is the questio of whether the sum T α) for all α 3 fids a represetatio i terms of a defiite hypergeometric sigle-sum. Refereces [AP94] S. A. Abramov ad M. Petkovšek. D Alembertia solutios of liear differetial ad differece equatios. I J. vo zur Gathe, editor, Proc. ISSAC 94, pages ACM Press, Baltimore, [A02] S. A. Abramov. Whe does Zeilberger s algorithm succeed? Adv. Appl. Math. to appear) [AO00] S. Ahlgre ad K. Oo. A Gaussia hypergeometric series evaluatio ad Apéry umber cogrueces. J. reie agew. Math., 518: , [Ah02] S. Ahlgre. Gaussia hypergeometric series ad combiatorial cogrueces. Symbolic computatio, umber theory, special fuctios, physics ad combiatorics, Dev. Math., Kluwer Acad. Publ., Dordrecht, 1 12, [AU85] G. E. Adrews ad K. Uchimura. Idetities i combiatorics IV: differetiatio ad harmoic umbers. Utilitas Mathematica, 28: , 1995) [Bro00] M. Brostei. O solutios of liear ordiary differece equatios i their coefficiet field. J. Symbolic Comput., 296): , Jue [CS98] F. Chyzak ad B. Salvy. No-commutative elimiatio i ore algebras proves multivariate idetities. J. Symbolic Comput., 262): , [Coh65] R. M. Coh. Differece Algebra. Itersciece Publishers, Joh Wiley & Sos, [Gos78] R. W. Gosper. Decisio procedures for idefiite hypergeometric summatio. Proc. Nat. Acad. Sci. U.S.A., 75:40 42, [GKP94] R. L. Graham, D. E. Kuth, ad O. Patashik. Cocrete Mathematics: a foudatio for computer sciece. Addiso-Wesley Publishig Compay, Amsterdam, 2d editio, [HS99] [vh99] P. A. Hedriks ad M. F. Siger. Solvig differece equatios i fiite terms. J. Symbolic Comput., 273): , M. va Hoei. Fiite sigularities ad hypergeometric solutios of liear recurrece equatios. J. Pure Appl. Algebra, ): , [Kar81] M. Karr. Summatio i fiite terms. J. ACM, 28: , [Kar85] M. Karr. Theory of summatio i fiite terms. J. Symbolic Comput., 1: ,

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21 [Sch02a] C. Scheider. Solvig parameterized liear differece equatios i ΠΣfields. Techical Report 02-03, RISC-Liz, J. Kepler Uiversity, July Submitted. Available at [Sch02b] C. Scheider. A collectio of deomiator bouds to solve parameterized liear differece equatios i ΠΣ-fields. Techical Report 02-04, RISC-Liz, J. Kepler Uiversity, July Submitted. Available at [Sch02c] C. Scheider. A collectio of degree bouds to solve parameterized liear differece equatios i ΠΣ-fields. Techical Report 02-05, RISC-Liz, J. Kepler Uiversity, July Submitted. Available at [Sch02d] C. Scheider. A uique represetatio of solutios of parameterized liear differece equatios i ΠΣ-fields. Techical Report 02-06, RISC-Liz, J. Kepler Uiversity, July Submitted. Available at [Weg97] K. Wegschaider. Computer geerated proofs of biomial multi-sum idetities. Diploma thesis, RISC Liz, Johaes Kepler Uiversity, May [Zei90] D. Zeilberger. A fast algorithm for provig termiatig hypergeometric idetities. Discrete Math., 802): ,