R. W. Gosper, Jr. (1978): Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75, 40--42.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a normal form for binomial series. For accomplishing this task quickly there is the rule SUMF. As an example we consider the Vandermonde sum. In[2] SUM[Binomial[N,l] Binomial[M,K l] fl,0,Infinityg] M ) N ) K l ) l ) l=0 and convert it into hypergeometric notation In[3]: SUMF K, N F ; 1 (1 K M) 2 1 1 K M K Out[3] 1) K 1.3. Summations for hypergeometric series. The package HYP includes 29 summation formulas in form of rules. All the available summations (with references) are listed and displayed in the manual. ....

....this task quickly there is the rule SUMF. As an example we consider the Vandermonde sum. In[2] SUM[Binomial[N,l] Binomial[M,K l] fl,0,Infinityg] M ) N ) K l ) l ) l=0 and convert it into hypergeometric notation In[3] SUMF K, N F ; 1 (1 K M) 2 1 1 K M K Out[3]= 1) K 1.3. Summations for hypergeometric series. The package HYP includes 29 summation formulas in form of rules. All the available summations (with references) are listed and displayed in the manual. Besides, there is the rule SListe which for a hypergeometric ....

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R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40--42.

....= f(x 1) for every f 2 K(x) Theorem 4 For every rational function F 2 K(x) there are polynomials a; b; c 2 K[x] such that (i) F = ii) a E b for all k 2 N , iii) a c and b Ec. For a proof, see [13] or [14] The original version of this theorem (without (iii) is due to Gosper [6]. Definition 7 (PNF) If a, b, c, F satisfy (i) and (ii) of Theorem 4, then (a; b; c) is a polynomial normal form or PNF of F . A PNF which satisfies (iii) of Theorem 4 is strict. Lemma 1 If (a; b; c) is a strict PNF of p=q where p; q 2 K[x] then a j p and b j q. 6 Proof: We have pbc = aqEc, ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40--42.

....which is reproduced in [6] The determinant evaluation in Theorem 1.10 does not seem to have appeared previously in the literature. The paper [7] contains our original proof, which is rather involved, but has its own appeal as it contains a nonautomatic ( application of Gosper s algorithm [13] (see also [14, Sec. 5.7] 21, Sec. II.5] Later we discovered that, in fact, there is a combinatorial argument which transforms the determinant in Theorem 1.10 into an instance of the determinant in Theorem 1.9, so that 10 these two determinant evaluations are actually equivalent. It is this ....

R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978), 40--42.

....of [2, p. 56 58] There the problem is to calculate the minimal polynomial satisfied by a product # 1 # k of k distinct branches of an algebraic function defined by P (z, #) 0. Gosper Petkovsek normal forms for rational functions. In many algorithms for symbolic summation (e.g. [15], 21] and [19] one has to solve a linear first order difference equation, the key equation. For example, Gosper s algorithm for hypergeometric indefinite summation [15] reduces the search for a hypergeometric solution f of a di#erence equation f(x 1) f(x) g(x) with hypergeometric right ....

....by P (z, #) 0. Gosper Petkovsek normal forms for rational functions. In many algorithms for symbolic summation (e.g. 15] 21] and [19] one has to solve a linear first order difference equation, the key equation. For example, Gosper s algorithm for hypergeometric indefinite summation [15] reduces the search for a hypergeometric solution f of a di#erence equation f(x 1) f(x) g(x) with hypergeometric right hand side g to the search of a polynomial solution of an auxiliary equation of a similar form. The key routine of this algorithm is the computation of the so called ....

R. W. Gosper, Jr. Decision procedure for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. U.S.A., 75(1):40--42, 1978.

....defined by P( 0. Gospev Petkovgek normal.forms .for rational.functions. In many algorithms for sym bolic summation (e.g. 15, 21 and [19] one has to solve a linear first order dif ference equation, he ke equation. For example, Gosper s algorkhm for hyperge ometric indefinite summation [15] reduces the search fbr a hpeweomeivic solution f of a diffbrence equation f(x 1) f(x) g(x) with hypergeometric right hand side g to the search of a polynomial solution of an auxiliary equation of a similar forB] The key routine of this algorithm is the computation of tile so called ....

R. W. Gosper, .Jr. Decision procedure for indefinite hypergeometric summation. Pvc. Nat. Aead. Sci. U.S.A., 75(1):40 42, 1978.

....the sum by Jacobi polynomials. Combinatorial sums and recurrences Consider the sums and (3) T = i By Lemma 4, calculating is equivalent to calculating these sums. We will focus on b(n, t) By Lemma 4, it corresponds to 1, y Using the Gosper Zeilberger . for z 2 W method [14, 15, 33] for generating recurrences from sums of binomial coefficients, we obtain the following lemma. LEMMA 13. 15] We have ( 2) 2 = 3. 4 )S , i (2. q 2)S , 5) Together with Lemma 4, this relates the amplitudes of I)l L) at timer, I l)lL) at timet l, and 1 2)lL) at time 2. We can ....

W. Gosper. Decision procedure for indefinite hypergeometric summation. In Proceedings of the National Academy of Sciences of U.S.A., volume 75, pages 40-42, 1978.

....qpochhammer. Other typical q hypergeometric terms are ratios of products of powers, q factorials, q binomial coefficients, and q shifted factorials that are integer linear in their arguments. 1 The ZEILBERG package (see [7] contains the hypergeometric versions. Those algorithms are described in [4], 11] 12] and [6] 1 2 ELEMENTARY Q FUNCTIONS 2 2 Elementary q Functions Our package supports the input of the following elementary q functions: ffl qpochhammer(a,q,infinity) a; q) 1 : 1 Y j=0 Gamma 1 Gamma a q j Delta ffl qpochhammer(a,q,k) a; q) k : 8 : Q k Gamma1 j=0 ....

Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 1978, 40--42. REFERENCES 16

....their arguments. The extensions of Gosper s and Zeilberger s algorithm mentioned in particular are valid for ratios of products of powers, factorials, # function terms, binomial coe#cients, and shifted factorials that are rational linear in their arguments. 2 Gosper Algorithm The Gosper algorithm [1] is a decision procedure, that decides by algebraic calculations whether or not a given hypergeometric term a k has a hypergeometric term antidi#erence g k , i.e. g k g k 1 = a k with rational g k g k 1 , and 1 The sum package contains also a partial implementation of the Gosper algorithm. ....

.... able to give a proof, instead, for the fact that H k does not possess a closed form evaluation: 6: gosper(1 k,k) Gosper algorithm: no closed form solution exists The following code gives the solution to a summation problem proposed in 4 REDUCE OPERATOR GOSPER 6 Gosper s original paper [1]. Let f k = k # j=1 (a b j c j 2 ) and g k = k # j=1 (e b j c j 2 ) Then a closed form solution for # k f k 1 g k is found by the definitions 7: operator ff,gg 8: let ff( k m) ff(k m 1) c (k m) 2 b (k m) a) when (fixp(m) and m 0) ff( k m) ....

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Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 1978, 40--42.

....base look up. Maple s sum procedure is a little dumb in first always trying Gosper s algorithm until this has definitely failed. 10.7. Gosper s algorithm Here we discuss Gosper s algorithm, of which the use in Maple was already shown by example in section 3. The fundamental reference is Gosper [9], see also Hayden Lamagna [10] Consider an indefinite summation S(k 2 ) Gamma S(k 1 ) k2 X k=k1 1 a k ; 10:7:1) where the summand a k is explicitly given and a k =a k Gamma1 is known to be rational in k. The indefinite sum S(k) is determined up to a constant term and also solves the ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40--42. 54 x11.2

....every f 2 K(x) Theorem 4 For every rational function F 2 K(x) there are polynomials a; b; c 2 K[x] such that (i) F = a b Delta Ec c , ii) a E k b for all k 2 N , iii) a c and b Ec. For a proof, see [6] or [7] The original version of this theorem (without (iii) is due to Gosper [3]. Definition 7 If a, b, c, F satisfy (i) and (ii) of Theorem 4, then (a; b; c) is a polynomial normal form or PNF of F . A PNF which satisfies (iii) of Theorem 4 is strict. Lemma 1 If (a; b; c) is a strict PNF of p=q where p; q 2 K[x] then a j p and b j q. Proof: We have pbc = aqEc, hence a j ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40--42.

.... 1) Gamma T1(n) is the summable part and T2 (n) the non summable part of T (n) 1 Introduction A sequence T (n) is a hypergeometric term (or simply a term) if the ratio T (n 1) T (n) is a rational function of n. We call this function the certificate of T . The well known Gosper s algorithm [4] solves the problem of indefinite hypergeometric summation: Given a hypergeometric term T (n) find another hypergeometric term T 1 (n) such that T (n) T 1 (n 1) Gamma T 1 (n) 1) provided that such a term exists. If it does, and if T (k) and T 1 (k) are defined for k = n 0 ; n 0 1; ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40--42.

....purpose. Generating functions (see Section 6) are one of the most common and powerful tools for proving identities. Here we only mention two recent developments that are of significance for both theoretical and practical reasons. One is Gosper s algorithm for indefinite hypergeometric summation [171, 175]. Given a sequence a 1 , a 2 , Gosper s algorithm determines whether the sequence of partial sums b n = n # k=1 a k , n = 1, 2, 3.3) has the property that b n b n 1 is a rational function of n, and if it is, it gives an explicit form for b n . We note that if b n b n 1 is a ....

....done by computer symbolic algebra systems such as Macsyma, Maple, and Mathematica. There are many widely available packages that can compute Taylor series expansions. Several can also compute certain types of limits, and some have implemented Gosper s indefinite hypergeometric summation algorithm [171]. They ease the burden of carrying out the necessary but uninteresting parts of asymptotic analysis. They are especially 156 useful in the exploratory part of research, when looking for identities, formulating conjectures, or searching for counterexamples. Much more powerful systems are being ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40--42.

....The extensions of Gosper s and Zeilberger s algorithm mentioned in particular are valid for ratios of products of powers, factorials, Gamma function terms, binomial coefficients, and shifted factorials that are rational linear in their arguments. 2 Gosper Algorithm The Gosper algorithm [1] is a decision procedure, that decides by algebraic calculations whether or not a given hypergeometric term a k has a hypergeometric term antidifference g k , i.e. g k Gamma g k Gamma1 = a k with rational g k =g k Gamma1 , and 1 The sum package contains also a partial implementation of the ....

.... able to give a proof, instead, for the fact that H k does not possess a closed form evaluation: 6: gosper(1 k,k) Gosper algorithm: no closed form solution exists The following code gives the solution to a summation problem proposed in 4 REDUCE OPERATOR GOSPER 6 Gosper s original paper [1]. Let f k = k Y j=1 (a b j c j 2 ) and g k = k Y j=1 (e b j c j 2 ) Then a closed form solution for X k f k Gamma1 g k is found by the definitions 7: operator ff,gg 8: let ff( k m) ff(k m 1) c (k m)2 b (k m) a) when (fixp(m) and m 0) ff( k m) ....

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Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 1978, 40--42.

...., sumbasis, swapcol , swaprow , sylvester , toeplitz , trace, transpose , vandermonde , vecpotent , vectdim , vector , wronskian ] You can see which procedures are available now. As an example, we compute the determinant of the matrix # # 12a 3 456 789 # # by det( 1,2 a,3] 5,6,7] [9,10,11]] 16 16 a and the eigenvalues and eigenvectors for a =2: eigenvalues( 1,2,3] 5,6,7] 9,10,11] 0, 9 # 105, 9 # 105 eigenvectors( 1,2,3] 5,6,7] 9,10,11] 9 # 105, 1, # 11 2 1 2 # 105, 1, 7 2 1 2 # 105 # ] 9 # 105, 1, # 11 2 1 2 # 105, ....

...., vectdim , vector , wronskian ] You can see which procedures are available now. As an example, we compute the determinant of the matrix # # 12a 3 456 789 # # by det( 1,2 a,3] 5,6,7] 9,10,11] 16 16 a and the eigenvalues and eigenvectors for a =2: eigenvalues( 1,2,3] 5,6,7] [9,10,11]] 0, 9 # 105, 9 # 105 eigenvectors( 1,2,3] 5,6,7] 9,10,11] 9 # 105, 1, # 11 2 1 2 # 105, 1, 7 2 1 2 # 105 # ] 9 # 105, 1, # 11 2 1 2 # 105, 1, 7 2 1 2 # 105 # ] 0, 1, 1, 2, 1] Later we will show how important an efficient ....

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R. W. GOSPER JR., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978), pp. 40--42.

....of this important aspect. Maple knows to some extent about hypergeometric series, and, 1 Informatik, University of Erlangen Nurnberg, Erlangen, Germany. strehl informatik.uni erlangen.de 1 Binomial Sums besides always trying Gosper s algorithm for indefinite hypergeometric summation (see [1], 3] uses this knowledge in evaluating some types of hypergeometric sums, as can be seen from examples given in the next section. For a discussion of Maple s approach to hypergeometric series and sums (and what a hypergeometerist might else wish) I recommend the article [4] by T. Koornwinder. ....

William Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA, 75:40-42, 1978.

....; sumbasis; swapcol ; swaprow ; sylvester ; toeplitz ; trace; transpose; vandermonde; vecpotent ; vectdim; vector ; wronskian] You can see which procedures are available now. As an example, we compute the determinant of the matrix 0 1 2a 3 4 5 6 7 8 9 1 A by det( 1,2 a,3] 5,6,7] [9,10,11]] Gamma16 16 a and the eigenvalues and eigenvectors for a = 1: eigenvalues( 1,2,3] 5,6,7] 9,10,11] 0; 9 p 105; 9 Gamma p 105 2 eigenvectors( 1,2,3] 5,6,7] 9,10,11] 0; 1; f[1; Gamma2; 1]g] 9 p 105; 1; f 11 2 Gamma 1 2 p 105; 1; Gamma 7 2 1 2 p ....

....vector ; wronskian] You can see which procedures are available now. As an example, we compute the determinant of the matrix 0 1 2a 3 4 5 6 7 8 9 1 A by det( 1,2 a,3] 5,6,7] 9,10,11] Gamma16 16 a and the eigenvalues and eigenvectors for a = 1: eigenvalues( 1,2,3] 5,6,7] [9,10,11]] 0; 9 p 105; 9 Gamma p 105 2 eigenvectors( 1,2,3] 5,6,7] 9,10,11] 0; 1; f[1; Gamma2; 1]g] 9 p 105; 1; f 11 2 Gamma 1 2 p 105; 1; Gamma 7 2 1 2 p 105 g] 9 Gamma p 105; 1; f 11 2 1 2 p 105; 1; Gamma 7 2 Gamma 1 2 p 105 g] Later ....

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Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 1978, 40--42.

....series. For accomplishing this task quickly there is the rule SUMF. As an example we consider the Vandermonde sum. In[2] SUM[Binomial[N,l] Binomial[M,K l] fl,0,Infinityg] 1 ( n ( M ) N ) Out[2] i ( K l ) l ) l=0 and convert it into hypergeometric notation In[3]: SUMF 1 K, N F ; 1 (1 K M) 2 1 1 K M K Out[3] 1) K 1.3. Summations for hypergeometric series. The package HYP includes 29 summation formulas in form of rules. All the available summations (with references) are listed and displayed in the manual. ....

....SUMF. As an example we consider the Vandermonde sum. In[2] SUM[Binomial[N,l] Binomial[M,K l] fl,0,Infinityg] 1 ( n ( M ) N ) Out[2] i ( K l ) l ) l=0 and convert it into hypergeometric notation In[3] SUMF 1 K, N F ; 1 (1 K M) 2 1 1 K M K Out[3]= 1) K 1.3. Summations for hypergeometric series. The package HYP includes 29 summation formulas in form of rules. All the available summations (with references) are listed and displayed in the manual. Besides, there is the rule SListe which for a hypergeometric ....

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R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40--42.

....Mathematica implementations of these algorithms are described. Nontrivial examples are given in order to illustrate the usage of these packages which are available by email request to the first named author. 1. Introduction Gosper s algorithm for indefinite hypergeometric summation, see e.g. Gosper (1978) or Lafon (1983) or Graham, Knuth and Patashnik (1989) belongs to the standard methods implemented in most computer algebra systems. Exceptions are, for instance, the 2.xVersions of the Mathematica system where symbolic summation is done by different means. A brief discussion is given in section ....

Gosper, R.W. (1978). Decision procedures for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U.S.A. 75, 40--42.

....system Mathematica and developed a user interface that dispenses the user from working explicitly with difference fields. Instead, the user can handle all summation problems in terms of sums and products. This algorithm cannot only deal with series of hypergeometric terms, like Gosper s algorithm [Gos78, PS95], series with q hypergeometric terms, like [PR97] or holonomic series, like Chyzak s algorithm [CS98] but with series of terms where for example the harmonic numbers can appear in the denominator (see section 2.4) In some cases appropriate difference field extensions are necessary in order to ....

R. W. Gosper. Decision procedures for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. U.S.A., 75:40--42, 1978.

....such that R1y1 = 0. Divide L by R1 to obtain L = L1R1 . 3. Recursively call A on L1 . Let the output be fz2 ; z3 ; zmg. 4. For i = 2; 3; m construct solutions y i of R1y i = z i using (10) and (12) Return fy1 ; y2 ; ymg and stop. In step 4, Gosper s algorithm [Gos78] applied to (12) resp. its continuous analogue [Alm Zei90] applied to (10) may be able to eliminate some summation (resp. integration) signs from the final result. Theorem 5 Algorithm A returns a basis for the space Ker L A(k) Proof: Obviously algorithm A finds a factorization L = ....

R.W. Gosper, Jr. (1978): Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75, 40--42.

.... [12] and then looking up standard results in the q hypergeometric database containing summation and transformation formulas of r OE s basic hypergeometric series (see e.g. the appendices in Gasper and Rahman [6] or Slater [19] We will exploit the fact that the algorithms presented by Gosper [9] and Zeilberger [23] for indefinite and definite hypergeometric summation, respectively, can be after appropriate adaptations also applied in the q case. We will first investigate the underlying theoretical background of q analogues of these algorithms, then describe the author s ....

....X k=on p 1 F (n Gamma j; k) o 0: 44 2.2 The Mathematica Implementation In this subsection we shall introduce the author s Mathematica implementation of the q analogue of Zeilberger s algorithm. Nowadays Gosper s algorithm for indefinite hypergeometric summation (see, e.g. Gosper [9] or Graham, Knuth and Patashnik [10] is implemented in most computer algebra systems. Extensions to Zeilberger s algorithm have been done by Zeilberger [24] and Koornwinder [11] in Maple. A very powerful Mathematica version of Zeilberger s algorithm has been written by Paule and Schorn [17] ....

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R.W. Gosper, Decision procedures for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. U.S.A., 75 (1978), 40--42.

.... solutions of differential equations [Sin91] ffl finding (q )hypergeometric solutions of (q )difference equations [Pet92, Abr Pet95] ffl factoring linear differential and difference operators with rational coefficients [Bro Pet94] ffl indefinite hypergeometric summation (Gosper s algorithm) [Gos78], Supported in part by grant 95 01 01138a of the Russian Fund for Fundamental Research. y Research supported in part by CATHODE (ESPRIT WG 7213) z Supported in part by grant J2 6193 0101 94 of the Ministry of Science and Technology of Slovenia. ffl definite hypergeometric summation ....

R. W. Gosper, Jr. (1978): Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75, 40--42.

....for single so called hyperexponential integrals [AZ] there exist extremely fast algorithms, that exploit fully the fact that the summand is hypergeometric, as opposed to the holonomic approach, that uses only the fact that the summand is holonomic. These algorithms used an extension of Gosper s [Gos] algorithm for single sum indefinite hypergeometric summation, and of its continuous analog. We will show that these fast algorithms of [Z2] WZ1] WZ2] and [AZ] extend naturally to the most general situation of multisums integrals of special functions of hypergeometric type. The key ....

Gosper, R. W., Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978), 40--42.

....solutions. In this paper we present algorithm Hyper which can be used to find all hypergeometric solutions of (1.3) To give some motivation, we describe first an application of algorithm Hyper to definite hypergeometric summation. The problem of indefinite hypergeometric summation was solved by Gosper (1978) who discovered an algorithm for finding hypergeometric solutions of the non homogeneous first order recurrence a(n 1) Gamma a(n) h(n) where h(n) is a hypergeometric term. In other words, Gosper s algorithm decides whether the indefinite sum of a hypergeometric term is hypergeometric (apart ....

....as N 0 , we find the unknown coefficients ff i from N0 X k=t ff k X l k l b (k Gammat s 0 ) k Gammal = f t m Gammas 0 ; for t = N 0 ; N 0 Gamma 1; 0 ; which is a triangular system of linear equations. 3. A normal form for rational functions The following lemma appears in Gosper (1978), though without mentioning properties (AC) and (BC) Lemma 3.1. Let K be a field of characteristic zero and r(x) a non zero rational function over K. Then there exists a non zero constant Z 2 K and monic polynomials A(x) B(x) and C(x) over K such that r(x) Z A(x) B(x) C(x 1) C(x) ....

Gosper, R.W., Jr. (1978). Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad.

....) we can write L = L 2 Delta ( Gamma r) Let M = Lfl s ( Gamma 1=r) L 3 Delta ( Gamma 1) for some L 3 with order(L 3 ) order(L) Gamma 1. The non constant hypergeometric solutions of M can be obtained by computing the hypergeometric solutions of L 3 and applying Gosper s algorithm (c.f. [8] or [1] Multiplying the hypergeometric solutions of M by u gives the hypergeometric solutions of L. This process is known as reduction of order. It may speed up the algorithm in the previous section if a hypergeometric solution is found early in the computation, but it is not always an ....

Jr. Gosper, R. William. Decision procedure for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. U.S.A., 75(1):40--42, 1978.

....result of ideas of H. Bateman (see the introduction to [10] G. Andrews [1] and others, it is widely recognized that most of these are special cases of relatively few hypergeometric identities, and attention is now being turned to methods of systematizing these higher level relationships. Gosper [9] has shown how to find indefinite hypergeometric sums, where they exist, by quite a general procedure (see [11] In this paper we describe a general attack on definite hypergeometric, and other, sums, continuing the program started in [13 15] ffl The method can prove, in a unified way, ....

....theory of [13] that is based on I. N. Bernstein s theory of holonomic systems (see [3] we know that G(n; k) of CF exists, as do the the a i s. Hoping that indeed L = 1, we have to find G(n; k) of CF that satisfies (1) where F (n; k) is given. This is done by Gosper s beautiful algorithm [9], that decides for us when such a G exists, and when it does, finds it. It is amazing that we are lucky so often, and whenever we are, we get in addition to a proof of the original identity, some brand new identities, complete with proofs. Acknowledgments We would like to thank Shalosh B. Ekhad ....

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R. William Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 40-42.

.... and an integer a 2 Z, we want to compute the coefficients g0 ; gn 2 Z of the Taylor expansion g = X 0kn gkx k = E a f = f(x a) X 0in f i (x a) i ; 1) where E is the shift operator Ef = f(x 1) This is one ingredient of algorithms for symbolic summation (Abramov 1971, Gosper 1978, Paule 1995) and is a basic operation in many computer algebra systems (e.g. translate in Maple) Writing out (1) explicitly, for 0 k n we have gk = X kin i k f i a i Gammak : 2) An important special case is a = Sigma1. The following lemma says how the coefficient size of a ....

....of NTL. The conclusion is that in our computing environment method B is the best choice for small problems, and method E for large ones. 3 Solving linear first order difference equations Let F be a field of characteristic 0 (say F = Q) In many algorithms for symbolic summation (Abramov 1971, Gosper 1978, Paule 1995) a linear first order difference equation of the form a Delta Eu Gamma b Delta u = c; 3) 3 often called key equation, with given polynomials a; b; c 2 F [x] has to be solved for a polynomial u 2 F [x] For example, the algorithms of Gosper (1978) and Paule (1995) reduce the ....

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R. W. Gosper, Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U.S.A. 25 (1978), 40--42.

.... have been found by a computer program which is a q analogue of Zeilberger s fast algorithm for proving terminating hypergeometric identities, see Zeilberger [25] The program is written in Mathematica by Riese [20] The algorithmic backbone of the program is a q analogue of Gosper s algorithm [14], see also Graham, Knuth and Patashnik [15] based on recent work of the author [18] After completion the program will be available upon request. Riese s package is not the first implementation of a q analogue of Zeilberger s fast algorithm. There are Maple programs written by Zeilberger, cf. ....

R.W. Gosper, Decision procedures for indefinite hypergeometric summation,Proc.Natl.Acad. Sci. USA., 75, 40--42.

....(a; q) k : 8 : Q k Gamma1 j=0 (1 Gamma a q j ) if k 0 1 if k = 0 Q k j=1 (1 Gamma a q Gammaj ) Gamma1 if k 0 ffl qbrackets(k,q) q; k] q k Gamma 1 q Gamma 1 1 The ZEILBERG package (see [7] contains the hypergeometric versions. Those algorithms are described in [4], 11] 12] and [6] ffl qfactorial(k,q) k] q : q; q) k (1 Gamma q) k ffl qbinomial(n,k,q) n k q : q; q) n (q; q) k Delta (q; q) n Gammak Furthermore it is possible to use an abbreviation for the generalized q hypergeometric series (basic generalized hypergeometric ....

Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 1978, 40--42.

....valid for F (n; k) Gamma n k Delta 2 with m = l = 1. A modification [21] of the (fast) Zeilberger algorithm ( 41] see also [25] and [29] returns a holonomic recurrence equation valid for s(n) Zeilberger s algorithm is based on a decision procedure for indefinite summation due to Gosper [16]. In our example case, Zeilberger s algorithm finds the holonomic recurrence equation (1 n) s(n 1) 2(1 2n) s(n) for s(n) P k2ZZ Gamma n k Delta 2 = n P k=0 Gamma n k Delta 2 which fortunately has only two terms. Therefore, we are led to the representation s(n) n X ....

Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 1978, 40--42.

....is a rational function with respect to k. Typical hypergeometric terms are ratios of products of powers, factorials, Gamma function terms, binomial coefficients, and shifted factorials (Pochhammer symbols) that are integer linear in their arguments. 1 Gosper Algorithm The Gosper algorithm [1] is a decision procedure, that decides by algebraic calculations whether a given hypergeometric term a k has a hypergeometric term antidifference g k , i.e. g k Gamma g k Gamma1 = a k , and returns g k if the procedure is successful, in which case we call a k Gospersummable. Otherwise no ....

.... 1) H k 1 Gamma 1) but, is able to give a proof, instead, for the fact that H k does not possess a closed form evaluation: 6: gosper(1 k,k) Gosper algorithm: no closed form solution exists The following code gives the solution to a summation problem proposed in Gosper s original paper [1]. Let f k = k Y j=1 (a b j c j 2 ) and g k = k Y j=1 (e b j c j 2 ) Then a closed form solution for X k f k Gamma1 g k is found by the definitions 7: operator ff,gg 8: let ff( k m) ff(k m 1) c (k m)2 b (k m) a) when (fixp(m) and m 0) ff( k m) ....

[Article contains additional citation context not shown here]

Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 1978, 40--42.

....described. Nontrivial examples are given in order to illustrate the usage of these packages. The algorithms are based on a new approach to q hypergeometric telescoping in which a new algebraic concept, q greatest factorial factorization (qGFF) plays a fundamental role. 1 Introduction Based on Gosper s [1978] algorithm for indefinite hypergeometric summation, Zeilberger s algorithm for proving definite hypergeometric summation and transformation formulae constitutes a recent breakthrough in symbolic computation. An excellent and detailed account of this theory can be found in the book of Petkovsek, ....

Gosper, R.W. [1978] Decision procedures for indefinite hypergeometric summation, Proc. Natl.

.... sums like n 2 Gamma n 2 = X 0k n k and q n Gamma 1 q Gamma 1 = X 0k n q k : For the rich class of hypergeometric expressions, which include products of rational functions, factorials, binomial coefficients, and exponentials, the summation problem has first been solved by Gosper [13]. Other contributions are due to Abramov [1] Moenck [18] Karr [15, 16] Zeilberger [29, 28, 30] Wilf Zeilberger [26, 27] Lisonek et al. 17] Pirastu Strehl [22] and Paule [19] Paule Strehl [20] give a unifying overview of more recent works. The method of Wilf and Zeilberger, which ....

....E: ae 7 Gamma ae(x 1) on F (x) and that Eg = oeg for some oe 2 F (x) i.e. g is a hypergeometric expression. For example, the Gamma function is hypergeometric, and (E Gamma) x) Gamma(x 1) x Gamma(x) shows that we have oe = x. Any rational function is hypergeometric. Gosper s [13] algorithm solves the hypergeometric summation problem: Is there another hypergeometric expression f such that Deltaf = Ef Gamma f = g, and if so, then compute one. He has also shown that if such an f exists, then f = g for some 2 F (x) i.e. f is a rational multiple of g and already ....

[Article contains additional citation context not shown here]

Gosper, Jr., R. W. Decision procedure for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. U. S. A. 25 (1978), 40--42.

....This encourages us to apply parallel methods to other symbolic computation problems in combinatorics. For instance, consider the case of indefinite summation of hypergeometric functions f , i.e. where Ef=f is a rational function in x. This problem can be solved by Gosper s algorithm (see Gosper (1978) and Lisonek et al. 1993) which is based on the solution of a linear system with polynomial entries. Also, Zeilberger s algorithm for definite hypergeometric summation (see Zeilberger (1991) would take advantage of an efficient parallel solver for systems of linear equations with symbolic ....

Gosper, R.W. (1978). Decision Procedure for Indefinite Hypergeometric Summation, Proc. Nat. Acad.

....2 F [y1 ; ym ] such that m )tn 1 = p2(q m )tn for all n, where q1 ; qm 2 F n f0g are the bases; ffl mixed hypergeometric, if there are polynomials p1 ; p2 2 F [x; y1 ; ym ] such that p1(n; q m )tn 1 = p2(n; q m )tn for all n. The well known Gosper s algorithm [8, 9] finds hypergeometric solutions fn of the nonhomogeneous first order recurrence fn 1 Gamma fn = tn where tn is a given hypergeometric sequence. Besides its obvious use for indefinite hypergeometric summation, it also plays a crucial role in the algorithms for definite hypergeometric summation, ....

....package gosper.m as MixedPoly form of rational functions is described in Section 5. After these preparations, we present in Section 6 an analogue of Gosper s algorithm for the mixed hypergeometric case. Our algorithm MixedGosper is a common generalization of the algorithms presented in [9, 24, 20]. When specialized to the bibasic case, it essentially agrees with the algorithm given in [20] However, looking at the case analysis in the computation of the multiplicities fl and ffi [20, pp. 7 8] it is not immediately clear how to extend that to the multibasic case. In Section 7 we provide ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40--42.

....from literature can be proved and even found this way. A Mathematica implementation of the algorithm is available from the author. AMS Subject Classification. Primary 33D65, 68Q40; Secondary 33D20. 1 Introduction Recently, Paule and Strehl [10] observed that the algorithm presented by Gosper [7] for indefinite hypergeometric summation extends quite naturally to the q hypergeometric case by introducing a q analogue of the canonical Gosper Petkovsek (GP) representation for rational functions. Based on the new algebraic concept of greatest factorial factorization (GFF) Paule [8] developed ....

.... in the same way, we can prove the bibasic identity (cf. Gasper [3] ji ; q) n Gamma1 (1 Gamma ap 2k =b) b; p) n 1 k(n Gamma1) ffi n;0 ; by transforming it into the equivalent version (a; b; q) k (bp =a; p) n Gamma1 = ffi n;0 : In[7]: qTelescope[ 1 b a) 1 a qk pk) 1 b pk qk) 1)k qfac[a,q,k] qfac[b,q,k] qfac[b a p(k 1) p,n 1] pBinomial[n k,2] qfac[p,p,k] qfac[p,p,n k] qfac[a pn,q,k 1] qfac[b pn,q,k 1] k, 0, n] Out[7] 0, n = 0 4.3 Open Problems With the input specification described above we actually ....

[Article contains additional citation context not shown here]

R.W. Gosper, Decision procedures for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. U.S.A., 75 (1978), 40--42.

....: qm 2 F are constants called the bases; ffl multibasic and mixed hypergeometric (mmHS, for short) if there are polynomials p1 ; p2 2 F [x; y1 ; ym ] such that p1(n; q n 1 ; q n m )tn 1 = p2(n; q n 1 ; q n m )tn for all n. The celebrated Gosper s algorithm [7, 8] finds hypergeometric solutions fn of the inhomogeneous firstorder recurrence fn 1 Gamma fn = tn where tn is a given hypergeometric sequence. Besides its obvious use for indefinite hypergeometric summation, it also plays a crucial role in the algorithms for definite hypergeometric summation, ....

....integration of hyperexponential functions [4] basic [16, 15] and bibasic hypergeometric summation [13] We present in Section 6 an analogue of Gosper s algorithm for the multibasic and mixed hypergeometric case. Our algorithm m m Gosper 1 is a common generalization of algorithms presented in [8, 16, 13]. Sections 2 and 3 give the required algebraic and algorithmic preliminaries, while in Section 5 we develop the multibasic and mixed hypergeometric canonical form of rational functions. Although in Gosper s corresponding author 1 available as GosperSum in Mathematica package gosper.m at ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40--42.

.... 1 A sequence a 2 F will be called rational over F if there is a rational function r over F such that a n = r(n) A non zero sequence a 2 F will be called hypergeometric over F if there is a rational function r over F such that a n 1 =a n = r(n) 1) 1 The well known Gosper s algorithm [3] (see also [4] decides for a given hypergeometric sequence a n whether the first order recurrence s n 1 Gamma s n = a n (2) has a hypergeometric solution s n , and if so, finds one. In this note we generalize Gosper s algorithm to non homogeneous recurrences with rational coefficients of ....

....(see (7) below) Lemma 1 Let F be a field of characteristic zero, and r a rational function over F . Then there exist polynomials A, B, C over F such that r(n) A(n) B(n) C(n 1) C(n) 4) 2 and gcd(A(n) B(n k) 1 for all nonnegative integers k. For a (constructive) proof, see [3]. The algorithm to compute A, B, and C given r uses gcd and resultant computations on polynomials over F . We remark that if we additionally require that A; B; C are monic polynomials such that gcd(A(n) C(n) gcd(B(n) C(n 1) 1, then a factorization of the form r(n) Z A(n) B(n) C(n ....

[Article contains additional citation context not shown here]

R.W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1975) 40 -- 42.

No context found.

R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. USA 75 (1978), 40--42.

....its underlying mechanism finds a more transparent explanation than in the descriptions given so far. Of course, to a certain extent this judgement is subjective, so we invite the interested reader to form his her own by comparison to Gosper s original argumentation as described, for instance, in Gosper (1978) or in the book (Graham et al. 1989) In a subsection we briefly relate rational telescoping, as a special case of Gosper s algorithm, to Theorem 4.1. A sequence hf k i k0 is called hypergeometric over K if there exists a rational function ae 2 K(x) such that f k 1 =f k = ae(k) for all k 2 N. ....

....(E Gamma1 r) Delta W = V: 5.15) Proof. Because of gcd(q; r) 1 we have rjEU . Hence there exists W 2 K[x] such that U = E Gamma1 r) Delta W for which G(U; V; q; r) 0 reduces to (5.15) 2 It is the form (5. 15) in which the difference equation associated to a G form is to find in Gosper (1978) or in the book (Graham et al. 1989) Consider the problem of deciding constructively over K(x) the general, first order linear difference equation a Delta E Gamma b Delta = c with nonzero polynomials a; b; c 2 K[x] We conclude this section by the remark that following the derivation above, ....

Gosper, R.W. (1978). Decision procedures for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U.S.A. 75, 40--42.

.... (1994b) Rational Summation and Gosper Petkovsek Representation 3 ffl to show (from local considerations) how this information can be obtained from the Gosper Petkovsek representation of rational functions, which was originally invented in the context of indefinite hypergeometric summation, see Gosper (1978), Petkovsek (1992) and for which we give a purely combinatorial equivalent, Section 5] ffl to propose a new summation method for rational functions, based on (known) algorithms for producing the Gosper Petkovsek representation of rational functions, Section 6] In Section 5.1 we give a ....

....between the following alternatives: ffl Asking for a decision procedure for the existence of Delta Gamma1 ff 2 R, and giving an algorithm to construct such an element in the case of a positive answer. Different approaches and algorithms in this direction were presented in Abramov (1971) Gosper (1978), and Man (1993) 4 R. Pirastu and V. Strehl ffl Enlarging the domain of functions under consideration (e.g. by adding polygamma functions) so that at least every ff 2 R has an inverse w.r.t. Delta. Moenck s approach mentioned above goes into this direction. A general method in analogy to ....

[Article contains additional citation context not shown here]

Gosper, R. W. (1978), Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad.

.... solution of this problem, and also gives the software by means of which everyone can perform these sums sans peine (almost) 2 2 The methods The methods that have achieved the complete solution of this class of problems are the following: Sister Celine s method [1] Gosper s algorithm [3] . Zeilberger s algorithm ct ( creative telescoping ) 11] Wilf and Zeilberger s WZ method [9] Petkovsek s algorithm Hyper [6] Here is a brief description of the scope of each of these algorithms (full descriptions are in [8] Computer programs, in Maple or Mathematica versions, that ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40 -- 42.

....from literature can be proved and even found this way. A Mathematica implementation of the algorithm is available from the author. AMS Subject Classification. Primary 33D65, 68Q40; Secondary 33D20. 1 Introduction Recently, Paule and Strehl [10] observed that the algorithm presented by Gosper [7] for indefinite hypergeometric summation extends quite naturally to the q hypergeometric case by introducing a q analogue of the canonical Gosper Petkovsek (GP) representation for rational functions. Based on the new algebraic concept of greatest factorial factorization (GFF) Paule [8] developed ....

....n,0 , the electronic journal of combinatorics 3 (1996) #R19 15 by transforming it into the equivalent version # 1 b a # n # k=0 (1 ap k q k ) 1 bp k q k ) 1) k (a, b; q) k (bp k 1 a; p) n 1 (p; p) k (p; p) n k (ap n ,bp n ; q) k 1 p ( n k 2 ) # n,0 . In[7]: qTelescope[ 1 b a) 1 a q k p k) 1 b p k q k) 1) k qfac[a,q,k] qfac[b,q,k] qfac[b a p (k 1) p,n 1] p Binomial[n k,2] qfac[p,p,k] qfac[p,p,n k] qfac[a p n,q,k 1] qfac[b p n,q,k 1] k,0,n ] Out[7] 0, n = 0 4.3 Open Problems With the input specification described above we actually ....

[Article contains additional citation context not shown here]

R.W. Gosper, Decision procedures for indefinite hypergeometric summation,Proc.Natl. Acad.Sci.U.S.A.,75 (1978), 40--42.

.... solution of this problem, and also gives the software by means of which everyone can perform these sums sans peine (almost) 2 2 The methods The methods that have achieved the complete solution of this class of problems are the following: ffl Sister Celine s method [1] ffl Gosper s algorithm [3] ffl Zeilberger s algorithm ct ( creative telescoping ) 11] ffl Wilf and Zeilberger s WZ method [9] ffl Petkovsek s algorithm Hyper [6] Here is a brief description of the scope of each of these algorithms (full descriptions are in [8] Computer programs, in Maple or Mathematica versions, that ....

R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40 -- 42.

....all n, where q 1 ; q m 2 F n f0g are the bases; 4. mixed hypergeometric, if there are polynomials p 1 ; p 2 2 F [x; y 1 ; ym ] such that p 1 (n; q n 1 ; q n m )t n 1 = p 2 (n; q n 1 ; q n m )t n for all n. The well known Gosper s algorithm (Gosper, 1977; Gosper, 1978) finds hypergeometric solutions fn of the nonhomogeneous first order recurrence fn 1 Gamma fn = t n where t n is a given hypergeometric sequence. Besides its obvious use for indefinite hypergeometric summation, it also plays a crucial role in the algorithms for definite hypergeometric ....

....A mixed hypergeometric canonical form of rational functions is described in Section 5. After these preparations, we present in Section 6 an analogue of Gosper s algorithm for the mixed hypergeometric case. Our algorithm MixedGosper z is a common generalization of the algorithms presented in (Gosper, 1978; Zeilberger, 1990b; Riese, 1996) When specialized to the bibasic case, it essentially agrees with the algorithm given by Riese (1996) However, looking at his case analysis in the computation of multiplicities fl and ffi (Riese, 1996, pp. 7 8) it is not immediately clear how to extend that to ....

Gosper, R. W., Jr. (1978). Decision procedure for indefinite hypergeometric summation. Proc. Natl.

.... has been addressed earlier in Pirastu (1992) and Pirastu (1994) ffl to show (from local considerations) how this information can be obtained from the Gosper Petkovsek representation of rational functions, which was originally invented in the context of indefinite hypergeometric summation, see Gosper (1978), Petkovsek (1992) and for which we give a purely combinatorial equivalent, Section 5] ffl to propose a new summation method for rational functions, based on (known) algorithms for producing the Gosper Petkovsek representation of rational functions, Section 6] In Section 5.1 we give a ....

....ffl Asking for a decision procedure for the existence of Delta Gamma1 ff 2 R, and giving an algorithm to construct such an element in the case of a positive answer. Different 4 R. Pirastu and V. Strehl approaches and algorithms in this direction were presented in Abramov (1971) Gosper (1978), and Man (1993) ffl Enlarging the domain of functions under consideration (e.g. by adding polygamma functions) so that at least every ff 2 R has an inverse w.r.t. Delta. Moenck s approach mentioned above goes into this direction. A general method in analogy to Risch s integration method is ....

[Article contains additional citation context not shown here]

Gosper, R. W. (1978), Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad.

No context found.

R. W. Gosper, Jr. (1978): Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75, 40--42.

No context found.

Gosper Jr., R. W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75 (1978), 40--42.

No context found.

R. W. Gosper. Decision procedures for indefinite hypergeometric summation. Proc. Nat. Acad. Sci. U.S.A., 1978.

No context found.

Gosper Jr., R. W. (1978). Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 40--42.

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