The Summation Package Sigma: Underlying Principles and a Rhombus Tiling Application

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1 Discrete Mathematics ad Theoretical Computer Sciece 6, 2004, The Summatio Package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio Carste Scheider Research Istitute for Symbolic Computatio Johaes Kepler Uiversity Liz Alteberger Str. 69 A 4040 Liz, Austria Carste.Scheider@risc.ui-liz.ac.at received Aug 21, 2003, revised Apr 16, Aug 26, 2004, accepted Aug 31, We give a overview of how a huge class of multisum idetities ca be prove ad discovered with the summatio package Sigma implemeted i the computer algebra system Mathematica. Geeral priciples of symbolic summatio are discussed. We illustrate the usage of Sigma by showig how oe ca fid ad prove a multisum idetity that arose i the eumeratio of rhombus tiligs of a symmetric hexago. Whereas this idetity has bee derived alteratively with the help of highly ivolved trasformatios of special fuctios, our tools eable to fid ad prove this idetity completely automatically with the computer. Keywords: symbolic summatio, rhombus tiligs 1 Itroductio The overall object of this article is to give a itroductory overview of how a huge class of multisum idetities ca be prove ad discovered with the summatio package Sigma [Sch01], which is based o the computer algebra system Mathematica. The algebraic platform of Sigma is built o the costructive differece field theory of ΠΣ-fields [Kar81, Kar85, Bro00, Sch00, Sch01, Sch04b] that ot oly allows to simplify idefiite ad defiite sums of (q )hypergeometric terms, like [Gos78, Zei90, PS95a, PWZ96, PR97], but of ΠΣ-terms, i.e., ratioal terms of arbitrarily ested idefiite sums ad products. Due to the geerality of ΠΣ-terms, this opes up a ew class of symbolic summatio problems that caot be treated by the algorithms ad implemetatios [Weg97, Rie03] developed for (q )hypergeometric multisums or by those [CS98, Chy00] developed for holoomic ad -fiite terms. Supported by the Austria Academy of Scieces, the SFB-grat F1305 of the Austria FWF ad by grat P16613-N12 of the Austria FWF c 2004 Discrete Mathematics ad Theoretical Computer Sciece (DMTCS), Nacy, Frace

2 366 Carste Scheider I the first part of this article we shall discuss relevat techiques of symbolic summatio, ad we shall explai how these ideas ca be applied i the differece field settig of ΠΣ-fields ad i differece rig extesios like ( 1) ; see [Sch01, Sch04b]. More precisely, this meas that sequeces, that may cosist of ratioal terms of arbitrarily ested idefiite sums ad products, are traslated i a atural way ito the correspodig differece field/rig settig [Kar85, Sch04b], ad, by usig a very geeral algebraic machiery [Kar81, AP94, Bro00, Sch02b, Sch04a, Sch02a, Sch04c], the correspodig summatio priciples (telescopig, creative telescopig, solvig recurreces) are applied i this settig. This allows to carry over Zeilberger s paradigm from hypergeometric terms [PWZ96] to so-called ΠΣterms: give a defiite ested multisum, fid a recurrece ad, if possible, solve the recurrece i terms of simpler expressios tha the defiite sum itself. The the right combiatio of that solutios might give a closed form evaluatio of the defiite sum itself. The iterplay of these summatio techiques i the differece field settig ca be summarized with the defiite summatio spiral that is graphically illustrated i Figure 1. I the secod part of this article we shall demostrate how these summatio techiques withi Sigma eable the user to fid a alterative, completely automatic proof of a o-trivial multisum idetity that arises i [FK00]. I this article, M. Fulmek ad C. Krattethaler cout the umber of rhombus tiligs of a symmetric hexago with side legths N,M,N,N,M,N, with N ad M havig the same parity, which cotai a particular rhombus ext to the ceter of the hexago. Withi this coutig there arises the subproblem of fidig a closed form evaluatio S (1) +! (+3)! S(2) for the two sums S (1) := 1 ( 1) +k ( + k + 4)! k j=0 1 j+1 (k + 2)!(k + 3)!( k 1)!, S(2) := 1 ( 1) k ( + k + 4)!(1 ( 1) ( + 2)) (k + 1)(k + 2)! 2. ( k 1)! I order to achieve this, the authors i [FK00] derive closed form evaluatios for these sums, amely, S (1) = 1 + ( 1) ( ) + 2 (1 ( 1) ( + 2)) 2 k + 1 (1) ad ( S (2) = (1 ( 1) ( + 2)) 2 k + 1 ( 1) ( + 1)( + 2)( + 3) ), (2) which fially delivers the idetity S (1) +! ( + 3)! S(2) = ( 1) ( + 2) 2 (3) for all 0. However, the proof of this idetity, give i [FK00, Lemma 26], fills about four pages ivolvig highly complicated trasformatios of special fuctios. I [FK00], the authors were already aware that the sum idetity (3) could be prove with a prototype versio [Sch00] of our Sigma package [Sch01]. At that time, we were able to derive recurrece relatios for the two sums S (1) ad S (2). Afterwards, we combied those two recurreces usig gfu (see [SZ94], or [Mal96] for a Mathematica implemetatio) to oe recurrece of order 10 which cotais

3 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 367 S (1) +! (+3)! S(2) as a solutio. It is the a simple task to check that the right had side of idetity (3) is also a solutio of this combied recurrece. The fact that both sides of the equatio (3) agree with the first 10 iitial values fially shows the correctess of (3). However, at that time we were ot able to fid the explicit evaluatios (1) ad (2). This has bee chaged partially i [Sch00], where we could fid those evaluatios by assumig that the right had sides of (1) ad (2) deped o the harmoic umbers H = i=1 1 i. I this article we shall show that meawhile also the task of fidig the evaluatios (1) ad (2) ca be carried out with the summatio package Sigma, without ay guessig part, but oly with computer algebra methods. I other words, we preset a alterative proof of [FK00, Lemma 26] that ot oly shows the correctess of idetity (3), but also delivers the explicit evaluatios of the sums i (1) ad (2). Moreover we shall illustrate that the proof of [FK00, Lemma 26] becomes completely automatic, if oe uses Sigma, ad hece feasible without advaced kowledge of hypergeometric fuctios ad their trasformatios. I priciple, a reader may jump directly to the rhombus tilig applicatio i Sectio 3. At every algorithmic step there, a poiter to the appropriate subsectio of Sectio 2 is give where the ideas behid are outlied. 2 Symbolic Summatio i Differece Fields Symbolic summatio usually is divided ito two differet subbraches, amely idefiite ad defiite summatio. I cotrast to idefiite summatio, defiite summatio problems have closed form evaluatios oly for specifically chose summatio rages. For istace, the sum b k=a ( k) i geeral caot be simplified further, whereas for the specific bouds a = 0 ad b = that sum evaluates to 2. I the followig two subsectios we will explai i more details, how idefiite ad defiite summatio ca be treated with the summatio package Sigma [Sch01]. Moreover, additioal iformatio is give i ΠΣ-Remarks how the summatio problems are rephrased iterally i the differece field settig of ΠΣ-fields. Fially, we will summarize all these uderlyig differece field aspects i Subsectio Idefiite summatio Idefiite summatio deals with the problem of elimiatig summatio quatifiers without usig ay kowledge about the summatio rage. More precisely, followig [PS95b], we are iterested i the followig problem. Give a idefiite sum f (k) where f belogs to some ice domai of sequeces ad f (k) is idepedet of. Fid g(k) i the same class or some suitable extesio of it such that Alteratively, idefiite summatio asks for solvig f (k) = g(). Give f (k); fid g(k) such that holds withi a certai rage of k. Problem T: Telescopig. g(k + 1) g(k) = f (k) (4)

4 368 Carste Scheider The, give such a telescoper g(k) of f (k), oe derives by telescopig b f (k) = g(b + 1) g(a) (5) k=a if b a N 0. There are various algorithms that solve Problem T for ice domais of sequeces f (k), like [Gos78, PS95a] for hypergeometric terms, [PR97] for q hypergeometric terms, or [Chy00] for holoomic ad -fiite terms. I the summatio package Sigma the sequeces f (k) ad its telescoper g(k) are described i the algebraic settig of differece fields, more precisely of ΠΣ-fields [Kar81, Kar85], ad certai differece rigs; for more details see ΠΣ-Remark 1. This domai of sequeces essetially covers (q )hypergeometric terms, see [Sch04b], ad a importat subclass of holoomic ad -fiite terms that occurs frequetly i symbolic summatio. More geerally, our approach allows to formulate sequeces i terms of ratioal expressios cosistig of arbitrarily ested idefiite sums ad products that are out of scope of [Gos78, PR97, CS98, Chy00]. Without goig ito more details, we call all those sequeces f (k) ΠΣ-terms (i k) that ca be described i terms of ΠΣ-fields. Typical examples for ΠΣ-terms are for istace H k = k i=1 k k = i=1 1 i, H(2) ( 1 + H k ( 2k) ( ) k k 1 i 2, k! = i, i=1 ) ( ), H 2 k k k H(2) k, H k i=1 = k i=1 i + 1, i i j=1 (H2 j + j!) H 3 j + j!. (6) Oe of the crucial properties of a ΠΣ-term f (k) is that the sums ad products i the shifted versio f (k+1) ca be expressed by the sums ad products give i f (k), like H k+1 = H k + k+1 1 or (k + 1)! = (k + 1)k!. O the cotrary, sums like k H i i=1 are ot i the scope of ΠΣ-terms. (i+k) 4 ΠΣ-Remark 1. I the sequel a brief itroductio of ΠΣ-fields is give; further iformatio ca be foud i [Kar81, Kar85, Bro00, Sch01, Sch02b, Sch04b]. A differece field [Coh65], usually deoted by (F, σ), is othig else tha a field F together with a field automorphism σ : F F. Karr built up a differece field theory i a completely costructive maer that eables oe to describe a huge class of ested multisums. I short, the class of ΠΣ-fields cotais differece fields (F, σ) that ca be defied as follows. Basically F is costructed by a tower of fiite field extesios K = E 0 < E 1 < < E = F with costat field K, i.e., K = {σ(g) = g g F i } for all 0 i. Moreover the followig coditios for 1 i hold: E i := E i 1 (t i ) is a trascedetal extesio of E i 1 ad we either have σ(t i ) = a i t i (a product/π-extesio) or σ(t i ) = t i +a i (a sum extesio) for some a i E i 1 \{0}. I other words the class of ΠΣ-fields cotais differece fields such as F := K(t 1 )(t 2 )...(t ) where F is a field of ratioal fuctios over K. Moreover these trascedetal extesios allow to describe recursively defied ested sums ad products i ratioal terms. Besides such product ad sum extesios, a ΠΣ-field ca cotai more geeral extesios of the type σ(t i ) = α i t i + β i with α i,β i E \ {0} together with some techical side coditios that are described further, for istace, i [Kar81, Kar85, Bro00, Sch01, Sch02b, Sch04b]. Throughout this article all fields will have characteristic 0.

5 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 369 Clearly, ratioal fuctios as f (k) K(k) with the shift operator σ(k) = k +1 are cotaied i the class of ΠΣ-fields; also, most of the (q-)hypergeometric terms like f (k) = 2 k or f (k) = k! ca be rephrased i a ΠΣ-field (K(k)(h),σ) with σ(h) = 2h or σ(h) = (k + 1)h; for more details see [Sch04b]. I particular, all the terms give i (6) ca be formulated i ΠΣ-fields. O the other had, frequetly used objects like ( 1) k caot be formalized i ΠΣ-fields, sice we have the algebraic relatio (( 1) k ) 2 = 1. To overcome this problem, Sigma allows to hadle objects like α k, 1 α a th root of uity, i rig extesios of the type F[x] where (F,σ) is a ΠΣ-field with costat field K, α K, σ(x) = αx, ad x = 1. I particular, this meas that σ : F[x] F[x] is a rig automorphism, i.e., (F[x], σ) forms a differece rig, or a differece rig extesio of (F, σ). For more details see [Sch01, Sch04b]. For istace, with Sigma oe ca produce the right had sides of the idetities a ( ) ( ) ( ) 1 + ( 2k)H k = ( a)h a + 1, a 0, (7) k a a ( ) ( ) 2 ( ) ( a) ( 2k)H k = k 2 (1 + 2H a ), a 0; (8) a ote that the special case a = of theses idetities is treated i [PS03]; see also [DPSW04a, CD04, KR04]. We illustrate the usage of our package Sigma by discoverig ad provig idetity (7). First we start a Mathematica sessio by loadig the package I[1]:= << Sigma Sigma - A summatio package by Carste Scheider ad defiig the sum S(a) = mysum o the left had side of (7) as follows: c RISC-Liz I[2]:= mysum = SigmaSum[(1 + ( 2k)SigmaHNumber[k])SigmaBiomial[, k],{k, 0, a}] a ( ). Out[2]= (1 + ( 2 k + ) H j ) k Geerally, the fuctios SigmaSum ad SigmaProduct are used to defie ΠΣ-terms (i additio we allow summatio objects like ( 1) that ca be oly formulated i differece rig extesios). For this purpose there are also several other fuctios available, like SigmaHNumber, SigmaBiomial or SigmaPower to defie harmoic umbers, biomials or powers i terms of sums ad products which itself ca be coverted ito ΠΣ-fields or certai differece rig extesios. For istace, SigmaHNumber[k] produces the kth harmoic umber H k which alteratively could be described by SigmaSum[1/i,{i,1,k}]. The, by applyig the Sigma-fuctio SigmaReduce to mysum = S(a), we obtai the closed form evaluatio: I[3]:= SigmaReduce[mySum] ( ). Out[3]= 1 + ( a + ) H a a ΠΣ-Remark 2. Iterally, the Sigma-package proceeds as follows. 1. Costructio of the ΠΣ-field (F,σ): Take the ratioal fuctio field F := Q()(k)(b)(h) ad defie the field automorphism σ : F F

6 370 Carste Scheider by σ(c) = c for c Q(), σ(k) = k + 1, σ(b) = k 1 k+1 b ad σ(h) = h + k+1. Note that the k-shifts S ( k k) = ( ) k+1 = k ) k+1( k ad Sk H k = H k+1 = H k + k+1 1 are reflected by the actio of σ o b ad h. 2. Solvig the telescopig problem i (F,σ): Sigma [Sch02b] fids the solutio g = b(hk 1) for the telescopig equatio σ(g ) g = f with f = b(1 + ( 2k)h). This meas that g(k) = (kh k 1) ( k) is a telescoper for f (k) = (1 + ( 2k)H k ) ( k). Hece Sigma fids the telescoper g(k) = (kh k 1) ( k) ad the shifted versio g(k + 1) = ( k)hk ( k). The correctess of (4) for 0 k a is immediate ad therefore the closed form is verified. Similarly, oe obtais a closed form of the sum I[4]:= mysum = (3 + 2 k) ( 1) k k 1 + j. j=1 j (2 + j) ; by applyig it to the fuctio call SigmaReduce: I[5]:= SigmaReduce[mySum] 3 (1 + ) (2 + ) + 2 ( ) ( 1). + 4 (1 + ) (2 + ) 2 ( 1). 1+ι 1 ι 1 =1 ι 1 (2+ι 1 ) Out[5]= 4 (1 + ) (2 + ) If oe takes the shifted telescoper g(k + 1) of f (k) = (3 + 2k)( 1) k k j=1 i Out[5] with replaced by k, the proof of idetity 1+ j j(2+ j) to be the expressio (3 + 2k)( 1) k k j=1 1 + j j(2 + j) = ( )( 1) + ( + 2)( 1) 2( + 1)( + 2) i=1 i + 1 i(i + 2) (9) for 0 ca be carried out similarly to the proof of idetity (7) from above. I geeral, suppose that we are give a sum S(a,b) = b k=a f (k) with a ΠΣ-term f (k). If mysum = S(a,b) is defied with Sigma-fuctios as carried out i I[2], by typig i SigmaReduce[mySum] oe looks for a telescoper g(k) i terms of sums ad products that are give by the ΠΣ-term f (k). More precisely, first a ΠΣ-field is costructed i which the sums ad products occurrig i f (k) ca be expressed formally. Afterwards oe tries to solve the telescopig equatio i this ΠΣ-field. If such a g(k) ca be computed, by telescopig, see (5), the outermost summatio quatifier i the sum S(a,b) ca be elimiated. ΠΣ-Remark 3. More precisely, the followig differece field machiery is activated i Sigma; see also ΠΣ-Remark 2. First a cocrete ΠΣ-field (F,σ) is costructed for the ΠΣ-term f (k) i (4). I particular, this meas, oe has to defie a map which liks the give summatio objects, i.e., sequeces f (k), with elemets f, say, i the costructed ΠΣ-field; i other words, f F represets f (k); for more details Note that this sum simplificatio will play a importat role i Sectio 3.

7 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 371 see [Sch01, Chapter 2.5]. Give this traslatio machiery, it is decided costructively, if there exists a solutio g F for the telescopig problem σ(g ) g = f. (10) If oe fids such a g, oe costructs a sequece g(k) i terms of sums ad products for which (4) holds. This fially gives the evaluatio i (5). Based o Karr s differece field theory [Kar81], the traslatio betwee ΠΣ-terms ad correspodig ΠΣfields ca be carried out completely automatically for most istaces. Problematic cases ca be treated by buildig up the uderlyig ΠΣ-field maually; for more details see [Sch04b]. This user cotrolled costructio ca be achieved by callig SigmaReduce with the optio Tower {s 1 (k),...,s k (k)}, where s i (k) are ΠΣ-terms i k. This meas that Sigma first tries to costruct the ΠΣ-field for the term s 1 (k) ad the exteds the field i order to represet the remaiig s i (k) followig the iput order; fially, the ΠΣ-field is elarged with ecessary extesios i order to represet also f (k). Note that Sigma ca also treat idefiite summatio problems i terms of ( 1) k that ca be oly treated i differece rigs. For more details we refer to Subsectio 2.3. We wat to poit out that so far we oly dealt with idefiite summatio problems where the telescoper g(k) is searched i the domai give by the iput sequece f (k). But already for the slightly more geeral sum expressio I[6]:= mysum = (3 + 2 k) (x) k k 1 + j. j=1 j (2 + j) ; we would fail to fid such a telescoper i the groud field. Such kid of problems motivated us to geeralize the idefiite summatio approach to the followig refied versio [Sch04c]: with the Sigmapackage oe is able to decide costructively if certai classes of sum extesios provide simpler solutios. More precisely, give a ΠΣ-term f (k), Sigma ca search for a telescoper g(k) of f (k) that ot oly cosists of sums ad products give by f (k) but that ca cotai sum extesios with the followig property: they are ot more ested tha the give ΠΣ-term f (k) ad their summads are composed by ΠΣ-terms that occur i f (k). By settig the additioal optio SimplifyByExt Depth i the fuctio call SigmaReduce this refied algorithm ca be activated. I[7]:= res = SigmaReduce[mySum, SimplifyByExt Depth] x ( x + 2 x) x. 1+ι 1 ι 1 =1 ι 1 (2+ι 1 ) ι 1 =1 (1+ι 1) ( 3+x 2 ι 1 +2 x ι 1 ) x ι 1. ι 1 (2+ι 1 ) Out[7]= ( 1 + x) 2 (1+k)( 3+x 2k+2xk)x k I this example Sigma fids the additioal sum extesio E x () := k=1 k(2+k) that allows to fid the closed form evaluatio give i Out[7] with the same ested depth tha the summad itself. (If oe cosiders the special case x = 1, the sum E 1 (x) ca be simplified further to 3 2(3+3+2 )( 1) (+1)(+2) which fially gives (9).) Aalogously, this traslatio process ca be cotrolled i the Sigma-fuctios CreativeTelescopig, GeerateRecurrece ad SolveRecurrece that are explaied later.

8 372 Carste Scheider ΠΣ-Remark 4. I the differece field settig the followig problem is solved i Sigma. First a ΠΣfield (F,σ) is costructed i which the ΠΣ-term f (k) ca be represeted with f F. The it is decided costructively, if there exists a bigger ΠΣ-field (F(x 1,...,x e ),σ) with σ(x i ) x i F ad a g F(x 1,...,x e ) with σ(g ) g = f where g is ot more ested tha f itself. If Sigma fids such a g, it costructs a telescoper g(k) of f (k) i terms of additioal sums whose depth is ot larger tha the ΠΣ-term f (k) itself. For algorithmic details we refer to [Sch01, Sch04c]. Further examples, like I[8]:= mysum = H 2 k H(2) k I[9]:= SigmaReduce[mySum, SimplifyByExt Depth] Out[9]= 1 ( 6 H + 3 H 2 H 3 + ( 3 (1 + 2 ) 3 (1 + 2 ) H + 3 (1 + ) H 2 ) (2) 1 ) H + 3 ι 1 =1 ι 3 1 ad m k j I[10]:= mysum = k i=1 k=1h (H2 i + i!.) j=1 H 3 j + (j!.) 2 ; I[11]:= SigmaReduce[mySum, SimplifyByExt Depth] Out[11]= ( m + (1 + m) H m ) m ι 1 j=1 (H 2 j + j!.) ι 1 =1 H 3 ι 1 + ι 1! 2. m ι 1 =1 ( ι 1 + H ι1 ι 1 ) ι 1 j=1 (H 2 j + j!.) H 3 ι 1 + ι 1!. 2 show that these ew ideas sigificatly ehace the algorithmic tool box. 2.2 Defiite summatio ad the defiite summatio spiral I geeral, the problem of defiite summatio is harder tha idefiite summatio, sice i additio oe also has to take ito accout the summatio rage. Up to ow, all defiite summatio algorithms deal with such kid of problems by followig Zeilberger s paradigm [PWZ96]: give a defiite sum, fid a recurrece (with polyomial coefficiets) that cotais the defiite sum as a solutio. If oe ca guess a closed form evaluatio for a give defiite sum, oe may prove this idetity by showig that the cojectured right had side is also a solutio of the computed recurrece ad checkig that the first iitial values are the same. More geerally, oe also tries to fid solutios of a derived recurrece. Here the crucial poit is that the computed solutios should be of a simpler type tha the give defiite sum expressio. If oe succeeds i this, oe caot oly prove idetities but eve derive closed form evaluatios. Subsequetly we will work out the iterplay betwee those subproblems ad methods that ca be summarized with our defiite summatio spiral i Figure 1. Fially, a cocrete example will illustrate these aspects i Sectio 3.

9 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 373 defiite sum combiatio of solutios creative telescopig simplified solutios recurrece idefiite summatio solvig d Alembertia solutios Creative telescopig Fig. 1: The defiite summatio spiral. The first step i our defiite summatio spiral cosists of solvig the followig problem. Give a defiite sum S() := b k=a f (,k) (11) where a,b are of the form a = a 1 + a 2 ad b = b 1 + b 2 with a 1,b 1 Z ad a 2,b 2 idepedet of. Fid a recurrece of the form c 0 ()S() + + c d ()S( + d) = h(). (12) Most relevat summatio algorithms accomplish this task by solvig Problem CT or variatios of it. Problem CT: Creative Telescopig. Give f (,k) ad d N; fid c 0 (),...,c d (), free of k ad ot all zero, ad g(,k) such that holds withi a certai rage of ad k. g(,k + 1) g(,k) = c 0 () f (,k) + + c d () f ( + d,k) (13) The basic idea behid this is as follows. Suppose oe succeeds i computig such c i () ad g(,k) for give f (,k) ad d. The summig equatio (13) over k from a to b gives g(,b + 1) g(,a) = c 0 () b k=a f (,k) + + c d () b k=a f ( + d,k). (14)

10 374 Carste Scheider The with some mild extra coditios, oe ca express the sums b k=a f ( + i,k) i (13) i terms of S(+i). This implies a ot ecessarily homogeeous recurrece (12) for the defiite sum S(). A cocrete example i Remark 2 illustrates i details how this trasformatio from (13) to (12) ca be carried out. Summarizig, solvig Problem CT for a sequece f (,k) with a fixed d N eables oe to costruct a recurrece of order d that cotais the above defied sum S() as solutio. Note that d must be specifically chose for each attempt to solve Problem CT. Usually, oe first tries to solve Problem CT for d = 1, ad icremets d util oe fids a solutio. Origially, creative telescopig has bee itroduced i [Zei90] for hypergeometric terms f (, k) ad g(, k); for a Mathematica implemetatio see for istace [PS95a]. Various other approaches i more geeral settigs, like [PR97] for q hypergeometric terms, [CS98, Chy00] for holoomic ad -fiite terms, or [Weg97, Rie03] for (q )hypergeometric multisum terms follow this idea of creative telescopig or related paradigms. With the summatio package Sigma oe ca try to solve Problem CT for a give d N ad a ΠΣterm f (,k) i k, which also depeds o a extra parameter, if the followig property holds : also the shifted versios f ( + i,k) for 1 i d are ΠΣ-terms i k ad all those ΠΣ-terms ca be represeted i a commo ΠΣ-field. The, give such a d ad f (,k), oe ca search for a solutio of Problem CT, where g(,k) cosists of sums ad products that occur i f (,k). Due to the geerality of the iput class of ΠΣ-terms, this approach opes up the possibility to tackle various defiite summatio problems that caot be treated by the earlier approaches [PR97, CS98, Chy00, Weg97, Rie03]. ΠΣ-Remark 5. Give a ΠΣ-term f (,k) ad d N, creative telescopig is hadled i Sigma as follows. First a ΠΣ-field (F,σ) is costructed with costat field K(), trascedetal over K, i which the ΠΣterms f ( + i,k) i k ca be expressed by f i F for 0 i d. The oe decides costructively, if there exist c i () K(), ot all zero, ad a g F with σ(g ) g = c 0 () f c d () f d. (15) If oe succeeds i fidig such solutios c i () ad g, a ΠΣ-term g(,k) is costructed that gives a solutio for Problem CT. We wat to remark that with Sigma oe ca search for creative telescopig solutios also i algebraic differece rig extesios like ( 1). Suppose that we are give d N ad a defiite sum S() = b k=a f (,k) as i (11) where f ( + i,k) is a ΠΣ-term i k for 0 i d. The, if mysum = S() is defied with Sigma-fuctios as carried out i I[2], by typig i creasol = CreativeTelescopig[mySum,, RecOrder d] a set of creative telescopig solutios with (13) is searched where each foud solutio is ecoded i the form {c 0 (),c 1 (),...,c d (),g(,k)}. Moreover, by eterig TrasformToRecurrece[creaSol, mysum, ] Note that this property holds for almost all ΠΣ-terms f (,k) i k. I our implemetatio the trivial solutio {0,...,0,1} with 1 1 = 0 f (,k) f ( + d,k) is always icluded i the set of output solutios. There might be several o-trivial solutios, if d is chose too big.

11 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 375 oe obtais the resultig recurreces of the form (12) for the sum S() that oe ca compute from the creative telescopig solutios. All these steps ca be carried out i oe stroke by usig the Sigmafuctio call GeerateRecurrece[mySum,,RecOrder d]. As example we refer to the computatio steps I[13], I[14] ad I[22] i the Mathematica sessio that will be carried out i Sectio 3. Further examples ca be foud i [Sch01, PS03, DPSW04a, DPSW04b]. We wat to emphasize that for our iput class of ΠΣ-terms, i.e., idefiite ested sums ad products, we ca verify the correctess of the obtaied recurrece by the followig recipe: check that the computed telescopig equatio of Problem CT is correct for all k with a k b. The it suffices to verify that the ihomogeeous part h() i (12) is correctly determied. I Remark 2 we will illustrate with a cocrete example how these verificatio steps ca be carried out with the computer Solvig recurreces Suppose that we have derived a recurrece for a defiite sum, say S(k), of the type a m (k)s(k + m) + + a 0 (k)s(k) = b(k) (16) where the coefficiets a i (k) ad the ihomogeeous part b(k) are ΠΣ-terms; ote that exactly this type of recurreces ca be computed with the Sigma-fuctio call GeerateRecurrece. The ext step i Figure 1 asks for solvig the recurrece i terms of simpler expressios tha the defiite sum itself. The the right liear combiatio of those solutios might give the closed form evaluatio of the defiite sum itself. With the package Sigma there are various possibilities to achieve this task. The simplest strategy is to search for the solutios i the groud field give by the coefficiets ad the ihomogeeous part i (16). Namely, if a recurrece of the form (16) is iserted properly i the computer algebra system Mathematica, say i the variable rec like i I[15], usig the fuctio call SolveRecurrece[rec, S[k]] the user ca look for all solutios i terms of sums ad products give by the a i (k) ad b(k). The result of this fuctio call is of the form {{0,h 1 (k)},...,{0,h r (k)}} or {{0,h 1 (k)},...,{0,h r (k)},{1,g(k)}} (17) where {h 1 (k),...,h r (k)} gives a solutio set of the homogeeous versio of the recurrece ad g(k) gives a particular solutio of the recurrece itself. Cocrete applicatios ca be foud i the computatio steps I[16] ad I[24]. ΠΣ-Remark 6. Iterally, a ΠΣ-field (F,σ) is costructed i which the coefficiets a i (k) ad the ihomogeeous part b(k) ca be expressed by a i F ad b F. The i Sigma all solutios g F with a m σ m (g ) + + a 0 g = b (18) If the optio RecOrder d is omitted i the fuctio calls CreativeTelescopig or GeerateRecurrece, Sigma tries to solve Problem CT first for d = 1 ad the for d = 2,3,... util a solutio is foud; the termiatio is ot guarateed i this case.

12 376 Carste Scheider are searched. More precisely, a solutio set {h 1,...,h r} F, liearly idepedet over the costat field {c F σ(c) = c}, is computed for the homogeeous versio of (18). Moreover, a particular solutio g F for (18) is searched. The foud solutios are the reiterpreted i form of ΠΣ-terms h i (k),g(k) that give the solutios (17) for the origial recurrece. Note that the search of the solutios for (18) ca be also carried out i algebraic extesios like ( 1) k, i.e., the a i (k) ad b(k) may deped o ( 1) k. I may istaces the uderlyig differece field is too small i which the solutios S(k) are searched. Therefore, Sigma provides the possibility to exted the uderlyig solutio domai maually. Namely, by the fuctio call SolveRecurrece[rec,S[k],Tower {s 1,...,s e }] oe ca search for all solutios S(k) i terms of sums ad products occurrig i the a i (k) ad b(k) together with the additioal sums ad products give by s i (k); see also ΠΣ-Remark 3. The applicatio of this feature is demostrated i the computatio steps I[19], I[25], ad I[28]. However, the guessig of additioal ΠΣ-terms is a highly o-trivial task. I order to dispese the user from extedig the uderlyig differece field maually, the followig two possibilities should be applied. Fidig (q )hypergeometric solutios. Due to the pioeerig work [Pet92, vh98, APP98], oe has powerful solvers i had that allow to fid all solutios S(k) i (q )hypergeometric terms of a homogeeous recurrece with polyomial coefficiets i k or q k. These solvers perfectly complemet the summatio package Sigma. Fidig ested sum solutios ad d Alembertia solutios. With the fuctio call SolveRecurrece[rec,S[k],Tower {s 1,...,s e },NestedSumExt ] the user ca compute all ested sum solutios of a give recurrece rec of the form k 1 =0 b 1 (k 1 ) k 2 k 2 =0 k r 1 b 2 (k 2 ) k r =0 b r (k r ) (19) where the b i (k i ) are ΠΣ-terms i terms of sums ad products give by the s i ad by the a i (k) ad b(k) i the recurrece (16). Typical sum solutios ca be foud i Out[17] ad Out[26]. Remark 1. Iterally, those solutios ca be obtaied by factorizig its liear differece equatio as much as possible ito liear right factors over the give differece field or rig; the each factor correspods basically to oe idefiite summatio quatifier; see [AP94, Sch01]. A importat result is that the class of liearly ested sum solutios (19) over the give ΠΣ-terms cotais also all solutios that cosist of ratioal terms of arbitrarily ested sums over the give ΠΣ-terms; for the ratioal case see [HS99] ad for the geeral ΠΣ-field case see [Sch01]. Note that the class of sum solutios is cotaied i the class of d Alembertia solutio [AP94] which agai is icluded i the class of Liouvillia solutios [HS99]. A importat special case is the ratioal case, i.e., the coefficiets of the recurrece are i the field K(k) with the shift operator S(k) = k+1. The the d Alembertia solutios are of the type (19) where b i (k i ) are hypergeometric terms over K(k i ). Here the crucial observatio is that a hypergeometric term solutio of a recurrece gives also a liear right factor of a recurrece. Therefore, the applicatio of algorithms like [Pet92, vh98] might cotribute to a refied factorizatio of a give recurrece ito liear right factors, ad thus to further solutios of the recurrece; see [AP94, Sch01]. I combiatio with [AP94, Sch01]

13 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 377 ad maual extesios of the solutio domai (with the optio Tower), the user ca compute all those d Alembertia solutios with the summatio package Sigma; for further details ad illustrative examples we refer to [PS03, DPSW04b] Idefiite summatio Nested sum solutios ad d Alembertia solutios cosist of o-trivial ad highly ested idefiite sums of the form (19). If such solutios cotribute to the closed form evaluatio of the origial defiite sum expressio, i most istaces the foud evaluatio is ot simpler, but eve more complex, amely more ested. I order to overcome this problem, oe has to reduce those ested sums to expressios which are less ested tha the origially give defiite multisum. It turs out that all ested sum solutios ad may d Alembertia solutios ca be expressed i ΠΣ-fields or differece rig extesios like ( 1) k ; see [Sch04b]. I this case oe ca apply our idefiite summatio algorithms described i Sectio 2.1 i order to simplify those sum solutios ad d Alembertia solutios further. This simplificatio step is carried out, for istace, i I[18] ad I[27] Combiatio of solutios. Now assume that we maaged to compute a recurrece of order d for a defiite sum S() that holds for all 0, 0 a iteger, ad we foud a set of solutios of that recurrece that holds for all 0. More precisely, suppose that i a Mathematica sessio mysum stads for our defiite sum S() ad recsol for our set of solutios of the recurrece that is give i the form (17) with k replaced by. The with FidLiearCombiatio[recSol,mySum,d,MiIitialValue 0 ] the user ca try to fid a liear combiatio of the solutios of the homogeeous versio of the recurrece plus oe particular solutio of the ihomogeeous recurrece that evaluates to the same iitial values for { 0, 0 + 1,..., 0 + d 1} as the give defiite sum. If Sigma succeeds i fidig such a liear combiatio, this expressio equals S() for all 0. Note that Sigma might fail to fid this liear combiatio if a particular solutio or some solutios of the homogeeous versio of the recurrece are missig i recsol. 2.3 The Master Problem for symbolic summatio i differece fields The summatio problems sketched i the previous ΠΣ-Remarks ca be summarized by Problem PLDE: Solvig Parameterized Liear Differece Equatios. Give a ΠΣ-field (F,σ) with costat field K, a 0,...,a m F, ad f 0,..., f d F; fid all g F ad all c 0,...,c d K with a m σ m (g) + + a 0 g = c 0 f c d f d. Namely, specializig to d = 0 ad m = 1 with a 1 = 1 ad a 2 = 1, oe cosiders the telescopig problem (10) for idefiite summatio. Moreover, specializig to m = 1 with a 1 = 1 ad a 2 = 1, oe ca formulate the creative telescopig problem (15) if K = K () ad f i F stads for the ΠΣ-term f (+i,k) F i k for 0 i d. Furthermore, if oe sets d = 0, oe cosiders the problem to solve liear differece equatios (18) of order m.

14 378 Carste Scheider I [Kar81, Kar85], M. Karr developed a complete algorithm that solves Problem T i the geeral ΠΣ-field settig; oly some additioal properties are required for the costat field, that are worked out i [Sch04b]. I some sese, Karr s algorithm [Kar81] is the summatio couterpart to Risch s algorithm [Ris69, Ris70] for idefiite itegratio. I [Sch00, Sch01], it was observed for the first time that Karr s algorithm ot oly ca solve Problem T but also Problem CT i ΠΣ-fields. More precisely, Karr s algorithm ca solve Problem PLDE with m = 1. Aalogously to the fact that the exteded versio of Gosper s algorithm [Zei90] represets a sigificat geeralizatio to defiite hypergeometric summatio, with this observatio Karr s algorithm ca be viewed as a major step forward with respect to defiite summatio i geeral. Based o results i [Bro00], Karr s algorithm was streamlied i [Sch01, Sch02b] to a more compact ad efficiet algorithm. Moreover, i [Sch02b, Sch04a, Sch02a] together with results from [Bro00], this streamlied algorithm was geeralized to a method that eables the user to search for all solutios of Problem PLDE for a arbitrary order m. Although there are still ope problems i the resultig algorithms, oe fids evetually all the solutios for Problem PLDE by repeatig the computatio process ad icreasig step by step the rage i which the solutios may exist; these ideas are preseted i [Sch02b]. Furthermore we wat to emphasize that Sigma provides methods that eable the user to search for solutios of Problem PLDE i differece rig extesios, like ( 1) k, that cotai zero-divisors, like (1 ( 1) k )(1+( 1) k ) = 0; for more details see [Sch01]. Those ideas are partially eeded i the computatio steps I[5], I[18], I[19], I[25], ad I[28]. 3 A Rhombus Tilig Applicatio I the sequel we will prove the multisum idetities (1) ad (2) that arise i [FK00]. Followig our defiite summatio spiral i Figure 1, those idetities will ot oly be prove with our package Sigma, but we will also fid their right had sides. First we set up the summatio problem S (1) = mysum1 as carried out i I[2]. I[12]:= mysum1 = H k (3 + k + )!. ( 1) k. ( 1). k=1 (1 + k)! 2 ;. (2 + k) ( k + )!. Fidig a recurrece with creative telescopig Give this sum expressio, we are able to compute a recurrece relatio of order three by solvig the creative telescopig problem; see Problem CT. I[13]:= creasol1 = CreativeTelescopig[mySum1,, RecOrder 3] Out[13]= { {0,0,0,0,1}, { (2 + ) (3 + ) (4 + ) 2 (5 + ) (9 + 2 ), (3 + ) (4 + ) (5 + ) (9 + 2 ) ( ), (3 + ) (4 + ) (5 + ) (5 + 2 ) ( ), (3 + ) 2 (4 + ) (5 + ) 2 (5 + 2 ), ( 2 (1 + k) (5 + ) (5 + 2 ) (7 + 2 ) (9 + 2 ) (( 3 + k ) (4 + k + ) + k (3 + ) (4 + ) H k ) (3 + k + )!. ( 1) k. ( 1). ) / ( (1 k + ) (2 k + ) (3 k + ) (1 + k)!. 2 ( k + )!. )}}

15 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 379 Here the secod etry i the output of Out[13], say {c 0 (),c 1 (),c 2 (),c 3 (),g(,k)}, gives the solutio of Problem CT for d = 3 ad the summad f (,k) = H k( + k + 3)!( 1) k ( 1) (k + 2)(k + 1)! 2 ( k)! (20) of S (1) = k=1 f (,k). The, as described i Subsectio 2.2.1, we ca geerate from this result a recurrece for S (1) with the fuctio call I[14]:= TrasformToRecurrece[creaSol1, mysum1, ] Out[14]= { (1 + ) (2 + ) 2 (3 + ) (4 + ) (9 + 2 )!. SUM[] (1 + ) (2 + ) (3 + ) (9 + 2 ) ( )!. SUM[1 + ] (1 + ) (2 + ) (3 + ) (5 + 2 ) ( )!. SUM[2 + ]+ (1 + ) (2 + ) (3 + ) 2 (5 + ) (5 + 2 )!. SUM[3 + ] == 2 (5 + 2 ) (7 + 2 ) (9 + 2 ) (3 + )!. ( 1). } This meas that SUM[] = S (1) (=mysum1) satisfies the output recurrece Out[14]. We could also carry this out i oe step by the call GeerateRecurrece[mySum,,RecOrder 3] which just gives the same recurrece as i Out[14]. We wat to emphasize that the user ca verify the correctess of recurreces idepedetly of the steps of the algorithm, see the followig remark. Remark 2. With the c i () ad g(,k) give i Out[13] oe ca show that S (1) is a solutio of the recurrece Out[14] as follows. For (20) observe that f ( + i,k) = f (,k) f i where f 0 = 1, f 1 = k + 1 k, f ( k)( k) 2 = ( + 1 k)( + 2 k), f ( k)( k)( k) 3 = ( + 1 k)( + 2 k)( + 3 k). Moreover ote that the ΠΣ-term g(,k) shifted i k ca be rewritte as 2(k + 1)( + 5)( k)(2 + 5)(2 + 7)(2 + 9) g(,k + 1) = (k + 2)( + 2 k) (H k ( ) + k + 2) ( + k + 3)!( 1)k ( 1) (k + 1)! 2 ( k)! by usig the relatios H +1 = H ad ( 1)+1 = ( 1). The with these represetatios, we verify that (13) with d = 3 holds for all 0 k. First we check that there do ot occur ay poles durig the evaluatio i the chose represetatios of g(,k), g(,k + 1) ad f ( + i,k) for 0 i 3 withi the rage 0 k. The we substitute those specific terms i g(,k + 1) g(,k) (c 0 () f (,k) + + c 3 () f ( + 3,k)), brig these expressios over a commo deomiator, ad check symbolically that the polyomial expressio i the umerator vaishes. This shows the correctess of (13) for 0 k. Moreover, summig equatio (13) over k from 0 to gives c 0 () f (,k) + + c 3 () f ( + 3,k) = g(, + 1) g(,0).

16 380 Carste Scheider The with S (1) +i = f ( + i,k) + i j=1 f ( + i, + j) (21) for i 0, the correctess of the recurrece rec with SUM[] = S (1) follows for all 0. Dividig the output recurrece i Out[14] by the o-zero factor ( + 3)( + 2)( + 1)! (for 0) gives the simplified versio: I[15]:= rec1 = (2 + ) (4 + ) (9 + 2 ) SUM[] (9 + 2 ) ( ) SUM[1 + ] (5 + 2 ) ( ) SUM[2 + ] + (3 + ) (5 + ) (5 + 2 ) SUM[3 + ] == 2 (5 + 2 ) (7 + 2 ) (9 + 2 ) ( 1). ; Solvig the recurrece with sum solutios (d Alembertia solutios) I the ext step we try to fid solutios of the recurrece rec1 give i Out[15]. To accomplish this task, Sigma provides the followig fuctio call; see Subsectio I[16]:= SolveRecurrece[rec1,SUM[]] Out[16]= {{0,1},{0,(2 + ) ( 1). }} Iterally Sigma costructs the uderlyig differece rig A = Q()[( 1) ] give by the objects i the recurrece ad afterwards tries to solve the recurrece formulated i this algebraic settig A. I this case Sigma fids two liearly idepedet solutios of the homogeeous versio of the recurrece, amely 1 ad ( + 2)( 1). Obviously, those solutios are ot sufficiet to describe the whole set of solutios of the give recurrece. Therefore we try to exted the uderlyig differece rig i form of sum solutios by settig i additio the optio NestedSumExt ; see Subsectio I[17]:= SolveRecurrece[rec1, SUM[], NestedSumExt, IdefiiteSummatio False] { Out[17]= {0,1},{0,(2 + ) ( 1). }, { 0, (3 + 2 ι 1 ) ( 1) ι ι 1 ( 1) 1. ι 2. }, ι 1 =0 ι 2 =1 ι 2 (2 + ι 2 ) { 1,2 (3 + 2 ι 1 ) ( 1) ι 1. ι 1 =0 ι ι 2 }} ι 2 =1 ι 2 (2 + ι 2 ) I this example Sigma succeeded completely sice it was able to compute three liearly idepedet solutios of the homogeeous versio of the recurrece ad oe particular solutio of the ihomogeeous recurrece itself. Simplifyig the solutios with idefiite summatio Now the essetial step is that those two sum solutios i Out[17] ca be simplified further with Sigma s idefiite summatio algorithm; see idetity (9). By default, i.e., omittig the optio IdefiiteSummatio False, those sum solutios are simplified immediately which results i: Remarks cocerig the optio IdefiiteSummatio False are give i the ext paragraph.

17 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 381 I[18]:= SolveRecurrece[rec1, SUM[], NestedSumExt ] { Out[18]= {0,1},{0,(2 + ) ( 1). }, { 0, 1 2 (1 + ) 1+ι 1 ι 1 =1 ι 1 (2+ι 1 ) }, 2 (1 + ) { ( ) ( 1). + 2 (1 + ) (2 + ) 2 ( 1). 1+ι 1 ι 1 =1 ι 1, 1 (2+ι 1 ) }} (1 + ) (2 + ) Lookig closer at this result, from the partial fractio decompositio of the summad ( i + 1 ) i=1 i(i + 2) = i=1 i + 1 i=1 i + 2 oe sees immediately that this sum ca be expressed i terms of the harmoic umbers H. This cosmetic chage of the solutio represetatio ca be also achieved by solvig the recurrece agai i the solutio domai exteded with H. I[19]:= recsol1 = SolveRecurrece[rec1,SUM[],Tower {H }] { Out[19]= {0,1}, { 0, (1 + ) (2 + ) H },{0,(2 + ) ( 1). }, (1 + ) (2 + ) { ( (1 + ) (2 + ) 2 H ) ( 1). }} 1, (1 + ) (2 + ) Remark 3. We wat to poit out that the correctess of the solutios i Out[19] for 0 (or of the represetatios Out[17] or Out[18] from above) ca be verified similarly as i Remark 2 by substitutig the solutios i the recurrece of Out[15] ad checkig equality for the resultig equatio. For istace, for the solutios give i Out[19], this ca be achieved by applyig the relatios H +1 = H ad ( 1)+1 = ( 1). Fidig a closed form evaluatio by combiig the solutios So far we computed a recurrece relatio of order 3 for the defiite sum S (1), that holds for all 0 (see Remark 2), ad foud solutios for that recurrece, that hold for all 0 (see Remark 3). Therefore a ca be obtaied by composig the particular liear combiatio of the homogeeous solutios plus the ihomogeeous solutio that matches the first three iitial values of S (1) for = 0,1,2; see Subsectio closed form of S (1) I[20]:= FidLiearCombiatio[recSol1, mysum1, 3, MiIitialValue 0] Out[20]= (1 + ) (2 + ) H + ( (1 + ) (2 + ) 2 H ) ( 1). (1 + ) (2 + ) This shows that S (1) = 5 3 2(1 + )(2 + )H + ( (1 + )(2 + ) 2 ) H ( 1), (1 + )(2 + ) or equivaletly (1), holds for all 0.

18 382 Carste Scheider I the same spirit we are able to fid a closed form evaluatio for the hypergeometric sum where S (2) 1 T := ( 1) k ( + k + 4)! (k + 1)(k + 2)! 2 ( k 1)! = (1 ( 1) ( + 2))T. More precisely, we first compute a recurrece for T = mysum2. I[21]:= mysum2 = (3 + k + )!. ( 1) k. k=1 k (1 + k)! 2 ;. ( k + )!. I[22]:= GeerateRecurrece[mySum2,, RecOrder 2] Out[22]= { (1 + ) (3 + ) (4 + ) (7 + 2 )!. SUM[] + 6 (1 + ) (3 + ) 2!. SUM[1 + ] (1 + ) (2 + ) (3 + ) (5 + 2 )!. SUM[2 + ] == 2 (5 + 2 ) (7 + 2 ) (4 + )!. } This meas that SUM[] = T (=mysum2) satisfies the output recurrece Out[22] for 0. Give the creative telescopig solutio, the verificatio of this recurrece relatio is immediate ad is omitted here. Note that this recurrece could have bee computed with ay other implemetatio that ca deal with creative telescopig for defiite hypergeometric sums [Zei90, PWZ96], like for istace [PS95a]. Dividig the output recurrece Out[22] by the o-zero term ( + 1)( + 3)! (for 0) gives the simplified versio: I[23]:= rec2 = (4 + ) (7 + 2 ) SUM[] 6 (3 + ) SUM[1 + ]+ (2 + ) (5 + 2 ) SUM[2 + ] == 2 (2 + ) (4 + ) (5 + 2 ) (7 + 2 ); Subsequetly we solve the recurrece rec2 give i I[23]. I the uderlyig algebraic settig of the recurrece we obtai the followig solutio I[24]:= SolveRecurrece[rec2,SUM[]] Out[24]= {{0,(1 + ) (2 + ) (3 + )}} which gives just a solutio of the homogeeous recurrece. Next, we ask for hypergeometric solutios of the homogeeous versio of the recurrece. For istace, the implemetatios [Pet92, vh98] give the additioal solutio ( 1). This gives the followig result. I[25]:= SolveRecurrece[rec2,SUM[],Tower {( 1). }] Out[25]= {{0,(1 + ) (2 + ) (3 + )},{0,( 1). }} Fially, we look for sum solutios of the recurrece ad get additioally a ihomogeeous solutio. I[26]:= SolveRecurrece[rec2,SUM[],Tower {( 1). }, NestedSumExt, IdefiiteSummatio False] Out[26]= { {0,(1 + ) (2 + ) (3 + )},{0,( 1). }, { 1,2 ( 1). ( ι1 + 9 ι ι 3 ) 1 ( 1) ι 1. ι 1 ι 1 =0 ι 2 =0 1 }} 1 + ι 2 Removig the optio IdefiiteSummatio False i the previous fuctio call, i.e., applyig i additio Sigma s idefiite summatio algorithm, leads to:

19 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 383 I[27]:= SolveRecurrece[rec2,SUM[],Tower {( 1). },NestedSumExt ] Out[27]= { {0,(1 + ) (2 + ) (3 + )},{0,( 1). }, { 1, }} + 2 (1 + ) (2 + ) (3 + ) ι 1 =0 1 + ι 1 I the ed, we just solve the recurrece agai i terms of H ad ( 1) which gives: I[28]:= recsol2 = SolveRecurrece[rec2,SUM[],Tower {H,( 1). }] Out[28]= { {0,(1 + ) (2 + ) (3 + )},{0,( 1). }, { 1, (1 + ) (2 + ) (3 + ) H }} The correctess of these solutios for 0 ca be verified as sketched i Remark 3. Combiig the solutios gives the closed form evaluatio of T = mysum2, amely I[29]:= FidLiearCombiatio[recSol2, mysum2, 2, MiIitialValue 0] Out[29]= (1 + ) (2 + ) (3 + ) H ( 1). which fially shows that (2) holds for all 0. 4 Coclusios I this survey article we illustrated how closed form evaluatios of a very geeral class of defiite multisums ca be discovered with the summatio package Sigma followig the defiite summatio spiral. As example, we derived ad proved the closed form evaluatios of S (1) ad S (2) from [FK00] purely algorithmically with computer algebra methods. For these computatios the user is completely dispesed from workig explicitly i differece fields or rigs; istead oe ca work coveietly i terms of usual sum ad product expressios. Ackowledgemets I would like to thak Christia Krattethaler for his valuable commets.

20 384 Carste Scheider 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, [APP98] [Bro00] [CD04] S. A. Abramov, P. Paule, ad M. Petkovšek. q-hypergeometric solutios of q-differece equatios. Discrete Math., 180(1-3):3 22, M. Brostei. O solutios of liear ordiary differece equatios i their coefficiet field. J. Symbolic Comput., 29(6): , Jue W. Chu ad L. De Doo. Hypergeometric series ad harmoic umber idetities. Preprit, [Chy00] F. Chyzak. A extesio of Zeilberger s fast algorithm to geeral holoomic fuctios. Discrete Math., 217: , [Coh65] R. M. Coh. Differece Algebra. Itersciece Publishers, Joh Wiley & Sos, [CS98] F. Chyzak ad B. Salvy. No-commutative elimiatio i ore algebras proves multivariate idetities. J. Symbolic Comput., 26(2): , [DPSW04a] K. Driver, H. Prodiger, C. Scheider, ad A. Weidema. Padé approximatios to the logarithm II: Idetities, recurreces, ad symbolic computatio. To appear i Ramauja Joural, [DPSW04b] K. Driver, H. Prodiger, C. Scheider, ad A. Weidema. Padé approximatios to the logarithm III: Alterative methods ad additioal results. To Appear i Ramauja Joural, [FK00] [Gos78] [HS99] M. Fulmek ad C. Krattethaler. The umber of rhombus tiligs of a symmetric hexago which cotais a fixed rhombus o the symmetric axis, II. Europea J. Combi., 21(5): , R. W. Gosper. Decisio procedures for idefiite hypergeometric summatio. Proc. Nat. Acad. Sci. U.S.A., 75:40 42, P. A. Hedriks ad M. F. Siger. Solvig differece equatios i fiite terms. J. Symbolic Comput., 27(3): , [Kar81] M. Karr. Summatio i fiite terms. J. ACM, 28: , [Kar85] M. Karr. Theory of summatio i fiite terms. J. Symbolic Comput., 1: , [KR04] [Mal96] C. Krattethaler ad T. Rivoal. Hypergéométrie et foctio zêta de Riema. Preprit, C. Malliger. Algorithmic maipulatios ad trasformatios of uivariate holoomic fuctios ad sequeces. Master s thesis, RISC, J. Kepler Uiversity, Liz, August 1996.

21 The package Sigma: Uderlyig Priciples ad a Rhombus Tilig Applicatio 385 [Pet92] [PR97] [PS95a] [PS95b] [PS03] M. Petkovšek. Hypergeometric solutios of liear recurreces with polyomial coefficiets. J. Symbolic Comput., 14(2-3): , P. Paule ad A. Riese. A Mathematica q-aalogue of Zeilberger s algorithm based o a algebraically motivated aproach to q-hypergeometric telescopig. I M. Ismail ad M. Rahma, editors, Special Fuctios, q-series ad Related Topics, volume 14, pages Fields Istitute Toroto, AMS, P. Paule ad M. Schor. A Mathematica versio of Zeilberger s algorithm for provig biomial coefficiet idetities. J. Symbolic Comput., 20(5-6): , P. Paule ad V. Strehl. Symbolic summatio - some recet developmets. I J. Fleischer et al., editor, Computer Algebra i Sciece ad Egieerig - Algorithms, Systems, ad Applicatios, pages World Scietific, Sigapore, P. Paule ad C. Scheider. Computer proofs of a ew family of harmoic umber idetities. Adv. i Appl. Math., 31(2): , [PWZ96] M. Petkovšek, H. S. Wilf, ad D. Zeilberger. A = B. A. K. Peters, Wellesley, MA, [Rie03] A. Riese. qmultisum - A package for provig q-hypergeometric multiple summatio idetities. J. Symbolic Comput., 35: , [Ris69] R. Risch. The problem of itegratio i fiite terms. Tras. Amer. Math. Soc., 139: , [Ris70] R. Risch. The solutio to the problem of itegratio i fiite terms. Bull. Amer. Math. Soc., 76: , [Sch00] C. Scheider. A implemetatio of Karr s summatio algorithm i Mathematica. Sém. Lothar. Combi., S43b:1 10, [Sch01] [Sch02a] [Sch02b] [Sch04a] [Sch04b] [Sch04c] C. Scheider. Symbolic summatio i differece fields. Techical Report 01-17, RISC-Liz, J. Kepler Uiversity, November PhD Thesis. C. Scheider. Degree bouds to fid polyomial solutios of parameterized liear differece equatios i ΠΣ-fields. SFB-Report 02-21, J. Kepler Uiversity, Liz, November C. Scheider. Solvig parameterized liear differece equatios i ΠΣ-fields. SFB-Report 02-19, J. Kepler Uiversity, Liz, November C. Scheider. A collectio of deomiator bouds to solve parameterized liear differece equatios i ΠΣ-extesios. To appear i SYNASC 2004, C. Scheider. Product represetatios i ΠΣ-fields. To appear i Aals of Combiatorics, C. Scheider. Symbolic summatio with sigle-ested sum extesios. Proc. ISSAC 04, pages , 2004.