D. Zeilberger. A holonomic systems approach to special function identities. J. Comput. Appl. Math., 32:321--368, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....sums can be found efficiently by means of Zeilberger s Creative Telescoping algorithm [21, 22, 19] A hypergeometric term is proper if it can be expressed as a product of a polynomial, several factorials of linear forms with integer coefficients, their reciprocals, and exponential functions. In [20] it is proved that proper hypergeometric terms are holonomic. Wilf and Zeilberger conjectured [19, p. 585] that a hypergeometric term is proper if and only if it is holonomic. Their conjecture concerns hypergeometric terms which depend on several discrete and continuous variables. We prove a ....

....is true of the factors of each (d Gamma 1) variate sequence T (n 1 ; n i Gamma1 ; k; n i 1 ; n d ) where k 2 N . Thus by Theorem 1, each factor of T (n) is holonomic, hence by Theorem 2, so is T (n) 2 For factorial terms, this result can be found in [17] and for proper terms in [20, 19, 14]. Wilf and Zeilberger conjectured [19, p. 585] that the converse of Theorem 3 holds as well. Taken literally, this is not true as we show in Example 6 of Section 3. However, we prove in Theorem 14 a slightly modified version of their conjecture, namely that over an algebraically closed field, ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990) 321--368.

....and a clear presentation of many such identities. this is due to tools like Superseeker , gfun and Mgfun [152, 24] and Rate (which is by far the most primitive of the three, but it is the most effective in this context) For hypergeometrics this is due to the WZ machinery (see [130, 190, 194, 195, 196]) And even if you should meet a case where the WZ machinery should exhaust your computer s capacity, then there are still computer algebra packages like HYP and HYPQ or HYPERG , which make you an expert hypergeometer, as these packages comprise large parts of the present hypergeometric ....

....value or coefficient in order to determine the multiplicative constant. As I explain in Appendix A, guessing can be largely automatized. It was already mentioned in the Introduction that proving binomial (hypergeometric) identities can be done by the computer, thanks to the WZ machinery [130, 190, 194, 195, 196] (see Footnote 5) Computing the degree bound is (in most cases) so easy that no computer is needed. You may use it if you want. It is only the determination of the multiplicative constant (item 5 above) by means of a special evaluation of the determinant or the evaluation of a special ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368. (p. 3, 15)

....like erf; Si; Bessel functions, hypergeometric functions, etc. are holonomic. The class of holonomic functions also admits several interesting algebraic properties which we recall in section 2. 1, and has recently been the object of intensive study in computer algebra and mathematics (e.g. [12, 8, 16]) The objective of this paper is to study holonomic functions from the exact numerical point of view: we require that all complex z numbers we compute with are e ective, i.e. for any rational 0 we can compute a Gaussian rational z 2 Q [i] with j z zj 6 . In this context, we are ....

D. Zeilberger. A holonomic systems approach to special functions identities. Journal of Comp. and Appl. Math., 32:321-368, 1990. 22

....such identities. ADVANCED DETERMINANT CALCULUS 3 this is due to tools like Superseeker 2 , gfun and Mgfun 3 [152, 24] and Rate 4 (which is by far the most primitive of the three, but it is the most effective in this context) For hypergeometrics this is due to the WZ machinery 5 (see [130, 190, 194, 195, 196]) And even if you should meet a case where the WZ machinery should exhaust your computer s capacity, then there are still computer algebra packages like HYP and HYPQ 6 , or HYPERG 7 , which make you an expert hypergeometer, as these packages comprise large parts of the present hypergeometric ....

....value or coefficient in order to determine the multiplicative constant. As I explain in Appendix A, guessing can be largely automatized. It was already mentioned in the Introduction that proving binomial (hypergeometric) identities can be done by the computer, thanks to the WZ machinery [130, 190, 194, 195, 196] (see Footnote 5) Computing the degree bound is (in most cases) so easy that no computer is needed. You may use it if you want. It is only the determination of the multiplicative constant (item 5 above) by means of a special evaluation of the determinant or the evaluation of a special ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368. (p. 3, 15)

....and presents it in a uniform way, with the added possibility of interactivity. We concentrate on special functions that are solutions of a linear di#erential or di#erence equation. These functions are called D finite or holonomic and many of their properties are algorithmically computable [9, 11, 8, 4, 5]. It turns out that this class covers about 60 of the functions described in [2] In other words, 60 of this reference can now be produced automatically with very little extra coding on top of standard computer algebra packages. Section 2 presents important properties of holonomic functions that ....

Doron Zeilberger. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics, 32(3):321--368, 1990. AUTOMATICALLY GENERATED ENCYCLOPEDIA OF SPECIAL FUNCTIONS 5

....and presents it in a uniform way, with the added possibility of interactivity. We concentrate on special functions that are solutions of a linear differential or difference equation. These functions are called D finite or holonomic and many of their properties are algorithmically computable [9, 11, 8, 4, 5]. It turns out that this class covers about 60 of the functions described in [2] In other words, 60 of this reference can now be produced automatically with very little extra coding on top of standard computer algebra packages. Section 2 presents important properties of holonomic functions that ....

Doron Zeilberger. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics, 32(3):321--368, 1990. AUTOMATICALLY GENERATED ENCYCLOPEDIA OF SPECIAL FUNCTIONS 5

....b) Now we obtain (10.2.6) from (10.8.6) Zeilberger [14] presents a similar method for obtaining higher order recurrences. Just replace (10.8.3) by S(k) S(n 1; k) Gamma l X j=0 s j (n) S(n Gamma j; k) where the s j (n) are unknowns. By the theory of holonomic systems, Zeilberger [13] shows that in the generic case there is always a value of l for which the method will succeed. For explicitly given orthogonal polynomials and l = 1 this is a powerful method to get the three term recurrence relation for the orthogonal polynomials by computer algebra. There is one caveat. Even ....

D. Zeilberger, A holonomic systems approach to special functions identities, preprint.

....1 ; z 2 ; z r ) A sequence f n1;n2 ; n r is holonomic iff its generating function f(z 1 ; z 2 ; z r ) X n1;n2 ; n r fn1;n2 ; n r z n1 1 z n2 2 Delta Delta Delta z nr r is holonomic. The major closure theorem here is due to Stanley, Lipschitz, and Zeilberger [4, 5, 7, 8]. Theorem 6 (Holonomic Closure) Holonomic functions are closed under sums, Cauchy products, Hadamard products, diagonals, algebraic substitutions, integration, differentiation, direct and inverse Laplace transforms. Theorem 7 (Holonomic Asymptotics) A holonomic sequence f n is asymptotic to a ....

Zeilberger (Doron). -- A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics, vol. 32, 1990, pp. 321--368. 120

.... (x 2 Gamma 1)f 00 n (x) 2xf 0 n (x) Gamma n(1 n)f n (x) 0 : 3) Therefore they represent a holonomic system completely determined by the two holonomic equations, and the initial values f 0 (0) 1 ; f 1 (0) 0 ; f 0 0 (0) 0 ; f 0 1 (0) 1 : In recent work, Zeilberger [32] introduced holonomic systems (in a more general setting) and showed how by an elimination process the holonomic equations can be used to verify identities for holonomic systems. We will give a rigorous introduction to this approach in x 8. In [33] 34] Zeilberger published an algorithm which ....

....(x) Gamma( Gamma2m) Gamma(1=2 Gamma m Gamma n) M n;m (x) Gamma(2m) Gamma(1=2 m Gamma n) M n; Gammam (x) 1] 13.1. 32) W n;m (x) is represented as sum of products, the recurrence equation can be obtained from (4) and the recurrence equation of the Gamma function (see e.g. 25] [32], 17] 24] 2 We note that similarly, one can obtain holonomic recurrence equations for the Hahn type polynomials h ( n (x; N) see [22] x2.4) and p n (x; fi; fl; ffi) see [7] x 10.23) Note further that some of the above recurrence equations have appeared in the literature. Relation ....

[Article contains additional citation context not shown here]

Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32, 1990, 321-368.

....) n = nfn 1 . Furthermore, the class of holonomic functions enjoys many closure properties: it is (algorithmically) stable under addition, multiplication, right composition with algebraic functions, di erentiation and integration, Hadamard product, etc. We refer to (Stanley, 1980; Lipshitz, 1989; Zeilberger, 1990) for more information on this subject. Holonomic functions are also available in some computer algebra systems (Salvy and Zimmermann, 1994) 3.4. Composition 3.4.1. Right composition with polynomials Let f = f 0 f p 1 z p 1 and g = g 1 z g q 1 z q 1 be polynomials, ....

Zeilberger, D. (1990). A holonomic systems approach to special functions identities. Journal of Comp.

....applications of this formula that are mostly related to some particular classes of numbers such as Catalan, Motzkin, Schroder and Stirling numbers. 365 The present technics can be considered as an alternative to existing methods such as the snake oil by Wilf [16] and the Zeilberger s algorithm [17]. 2 Riordan arrays and combinatorial sums Let ff k g k2N be a sequence of (real) numbers. The generating function f(t) of the sequence is defined as f(t) Gff k g = P 1 k=0 f k t k . The notation [t n ]f(t) denotes the coefficient of t n in the Taylor development of f(t) around t = 0. A ....

D. Zeilberger. A holonomic system approach to special functions identities. Journal of Computational and Applied Mathematics, 32:321--368, 1990. 375

....Today we already have computer programs for the automatic treatment of hypergeometric multisums. For instance, the WZ engine of of Wilf and Zeilberger [23] has been fine tuned by Wegschaider [21] A more general engine, based on Zeilberger s holonomic systems approach to special functions [24] has been designed by Chyzak [13] its underlying mechanism is elimination via Grobner bases methods for noncommutative operator algebras. These computer algebra packages are remarkably powerful for various applications. For instance, see Chyzak s computer proof [13] of Calkin s identity (1.3) or ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comp. Appl. Math. 32 (1990), 321--368.

....3 One can ask the same question about definite sums of the form a(n) X k F (n; k) 1.4) where summation ranges over all integers, and the two quotients F (n 1; k) F (n; k) and F (n; k 1) F (n; k) are rational functions of n and k. Using Bernstein s theory of holonomic functions, Zeilberger (1990) proved that every a of the form (1.4) satisfies a recurrence of the form (1.2) Zeilberger (1991) gives an algorithm which constructs such a recurrence. Wilf and Zeilberger (1992) generalize this to multiple and or q hypergeometric sums. Since algorithm Hyper can be used to obtain a basis for ....

Zeilberger, D. (1990). A holonomic systems approach to special functions identities. J. Comput. Appl.

....240z 2 Gamma 224z 80) h 00 (16z 5 Gamma 80z 4 168z 3 Gamma 184z 2 125z Gamma 45) h 0 (16z 4 Gamma 64z 3 136z 2 Gamma 144z 53) h = 0: The theory of holonomic functions and sequences allows automatic proof of a large class of identities. Zeilberger has given in [30] an algorithm to prove some combinatorial identities and some identities involving special functions. This algorithm works by searching for equations satisfied by each side of the identity to be proved or to disproved. Then, if these equations are compatible and sufficiently many initial ....

Zeilberger, D. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32 (1990), 321--368.

....1, see e.g. Andrews [6] which also contains a brief historical account. It is well known that number theoretic identities like these, or of similar type, can be deduced as limiting cases of q hypergeometric finite sum identities. Due to recent algorithmic breakthroughs, see for instance Zeilberger [24], or, Wilf and Zeilberger [23] proving these finite versions becomes more and more routine work that can be left to the computer. For instance, the computers Ekhad and Tre [12] delivered a purely verification proof of the following finite version, which was stated first in this form by Andrews ....

D. Zeilberger, A holonomic systems approach to special function identities, J. Comp. Appl. Math., 32 (1990), 321--368.

....1995) The Identification Problem for Transcendental Functions Wolfram Koepf koepf zib berlin.de Abstract In this article algorithmic methods are presented that have essentially been introduced into computer algebra within the last decade. The main ideas are due to Stanley [34] and Zeilberger [40] [43] Some of them had already been discovered in the last century (see e.g. 4] 5] but because of the complexity of the underlying algorithms have fallen into oblivion. The combination of these ideas leads to a solution of the identification problem for a large class of transcendental ....

....1 10 Gamma1000 . Therefore, the special (and common) fact about the exponential and factorial functions is that they both satisfy a differential or recurrence equation, respectively, that is homogeneous, linear, of order one, and has polynomial coefficients. We can generalize this observation [40]: A continuous function of one variable f(x) is holonomic, if it satisfies a homogeneous linear differential equation with polynomial coefficients. By linear algebra arguments, Stanley [34] showed that sums and products of holonomic functions and the composition with algebraic functions also form ....

[Article contains additional citation context not shown here]

Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32, 1990, 321--368.

.... 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 uses a variant of Gosper s algorithm as a subroutine, is able to produce routinely short proofs of ....

Zeilberger, D. A holonomic systems approach to special function identities. J. Comput. Appl. Math. 32 (1990), 321--368.

....tableaux ayant meme forme (voir (2) La forme des r ecurrences (1a et 2a) et certains aspects explicites des coefficients (1b, 1c, 2b, 2c) font l objet de ces conjectures dans le cas g en eral, et nous donnons encore plus de d etails (1c, 1e et 2c) pour le cas h impair. Un r esultat de Zeilberger [4] assure que de telles r ecurrences existent, mais sa d emonstration ne semble pas permettre de d eduire une forme explicite aussi pr ecise que (4) ou (6) On montre aussi que le comportement asymptotique des solutions des r ecurrences obtenues par le biais de nos conjectures est compatible avec ....

....of the numbers t (2) h (n) X n; l( h f 2 : 2) For the cases h = 2; 3; 4; 5 (see Regev [3] and Gouyou Beauchamps [1] nice explicit expressions have been given for the t h (n) s or their generating function. We also should mention at this point that Zeilberger has shown in [4] that the t h (n) s are P recursive, this is to say that they satisfy a recurrence of the form m X k=0 p k (n) t h (n Gamma k) 0 ; 3) for some polynomials p k (n) and some integer m. The same kind of result holds also for the t h (n) 2) s. However Zeilberger s argument gives no clear ....

D. Zeilberger, A Holonomic Systems Approach to Special Functions Identities, SIAM J. Math. Anal. -- 6 --

.... 2 2 d 2 3 d (z z ) y(z) z z ) y(z) 1 z) y(z) 1 dz 2 dz 2.1 Building up equations The class of holonomic functions and sequences enjoys nice closure properties. In particular we have the following theorem. Theorem 1 (Closure) [6, 13, 17, 22] (a) algebraic functions are holonomic; b) the sum of two holonomic functions is holonomic; c) the Cauchy product of two holonomic functions is holonomic (convolution of the sequences) d) the Hadamard product of two holonomic functions is holonomic (term wise product of the sequences) e) if ....

....direct and inverse Laplace Borel transform. Most of these closure properties are implemented in the gfun package. We now give a review of these, in the same order as the properties presented in the theorem. The algorithms described in this section can be found between the lines in [17] or [22]. algfuntodiffeq [Property (a) Given an irreducible polynomial P in two variables, this procedure computes a holonomic equation satisfied by any function y(z) solution of P (z; y(z) 0. The algorithm used by gfun was pointed out by Comtet [6] it can also be found in [4, pp. 276 278] ff) ....

Zeilberger, D. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32 (1990), 321--368.

....terms. So it comes as a surprise that Gosper s method can be used systematically and efficiently for problems of definite hypergeometric summation in a less direct way. It follows from the theory of holonomic systems, as put into action for our purposes by D. Zeilberger in the fundamental article [63], that (under certain restrictions, which are satisfied for the binomial sums we consider) a hypergeometric summation f(n) P k F (n; k) is P recursive or holonomic as a function of n, i.e. it satisfies a linear recurrence with polynomial coefficients (2) and (4) are typical examples with ....

D. Zeilberger. A holonomic systems approach to special function identities. J. Comput. Appl. Math., 32:321--368, 1990.

....approach (1990a) In the single sum case, due to the possible use of Gosper s summation algorithm, the corresponding procedures work very efficiently. In the multiple sum case, since nothing like Gosper s algorithm is available, one has to introduce for instance some elimination procedure. Zeilberger (1990a) described a method ( Sylvester s dialytic elimination ) which extends to the multiple sum case. As indicated in the same paper there is another possibility, i.e. to do the elimination by using non commutative Groebner bases methods. Computer Generated Proofs of a Binomial Double Sum Identity ....

....Only recently, Wilf and Zeilberger (1992) showed that every proper hypergeometric multisum integral identity, or q identity, with a fixed number of summations and or integration signs, indeed possesses a computer constructable proof. Despite proceeding along the lines of the holonomic paper of Zeilberger (1990a) the method, by which the authors succeed to prove a variety of interesting examples, does not involve elimination in the operator algebra. In connection with the third question, Paule (1992) found a way to apply Zeilberger s fast algorithm not only for proving (1.1) but also for finding the ....

Zeilberger, D. (1990a). A holonomic systems approach to special function identities. J. Computational and Applied Math. 32, 321--368.

.... 10 Gamma1000 . Therefore, the special (and common) fact about the exponential and factorial functions is that they both satisfy a differential or recurrence equation, respectively, that is homogeneous, linear, of order one, and has polynomial coefficients. We can generalize this observation [42]: A continuous function of one variable f(x) is holonomic, if it satisfies a homogeneous linear differential equation with polynomial coefficients; we call such a differential equation also holonomic. By linear algebra arguments, Stanley [36] showed that sums and products of holonomic functions ....

.... (2 alpha 2 beta 2 d 4 gamma 2 z 4 alpha z 2 beta z 3 d z 4 gamma z) V[1 alpha] 2 2 alpha (1 alpha) M ( 1 z) z V[2 alpha] 0 2 I am indebted to Jochem Fleischer who informed me about a misprint in formula (31) of [15] 4 Holonomic Systems of Several Variables In [42], Zeilberger considered the more general situation of functions F of several discrete and continuous variables. If we have d variables, and d (essentially independent) mixed homogeneous linear (partial) difference differential equations with polynomial coefficients in all variables are given for F ....

[Article contains additional citation context not shown here]

Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32, 321--368 (1990).

....algebra systems. Current interest in this algorithm is mainly due to the fact that it can also be used for definite hypergeometric summation (e.g. verifying binomial identities automatically , finding recurrence operators annihilating hypergeometric sums) in a non obvious and non trivial way (see Zeilberger (1990a) 1990b) Wilf and Zeilberger (1992) and the references given in the latter) One of the steps in Gosper s algorithm, crucial for its running time and memory requirement, is the determination of a degree bound for a possible polynomial solution of a certain difference equation the so called ....

....(6.3) and Sn = n X k=0 F n;k : 6.4) Remark: For an excellent account on how this recurrence is used to prove the irrationality of i(3) see (van der Poorten, 1979) Note that the double indexed sequence (F n;k ) n0;k0 is hypergeometric in both variables. Under slight side conditions (see e.g. Zeilberger (1990a) 1990b) or Wilf and Zeilberger (1992) for such sequences there exist a nonnegative integer d, polynomials c 0 (n) c d (n) being independent of k, and a double indexed sequence (G n;k ) n0;k0 , again hypergeometric in both variables, such that c 0 (n) F n;k c 1 (n) Fn 1;k : ....

Zeilberger, D. (1990a). A holonomic systems approach to special function identities. J. Comp. Appl.

....useful when it is possible to recognize whether two members of the class are identical or not. D. Zeilberger showed that a large set of combinatorial identities can be proved using properties of the class of P finite functions and sequences and the important subclass of holonomic functions [33]. A function is P finite when the set of its partial derivatives spans a finite dimensional vector space over the rational functions. Computationally, a P finite function is defined by a set of linear differential equations (linear relations between the partial derivatives) and a finite number ....

....natural to encompass both notions into a more general one. Our aim is to make effective operations on systems of linear operators constrained so that their solutions lie in a finite dimensional vector space. First approaches to the mixed differential difference case are due to D. Zeilberger [33] and N. Takayama [26] We use Ore polynomials and skew polynomial rings to also deal with q equa tions, and numerous other linear equations in pseudo derivatives. The solutions of these systems will be called finite. However, none of our algorithms deals with these finite solutions. Instead, ....

[Article contains additional citation context not shown here]

Zeilberger, D. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32 (1990), 321--368.

....integer d is the order of the holonomic recurrence. As in the case of holonomic functions the integer maxfdeg(p i (x) j0 i dg is called the degree of recurrence (1.3. 1) In literature holonomic sequences are sometimes called polynomially recursive, P recursive [GKP94] Sta80] or P finite [Zei90]. Let [Kn ] be the ring of polynomial sequences in K with addition and termwise (Hadamard) multiplication. Since [Kn ] has a unit element and no zero divisors, we can define (Kn ) the field of rational sequences, to be the set (Kn ) f p q j p = p(n) n0 ; q = q(n) n0 2 [Kn ]g= where ....

....C ff n (x) given in Example 1.3.1) Since this thesis concentrates on holonomic univariate functions, we omit exact definitions of multivariate holonomicity. Detailed descriptions and discussions were given, for example, by Chyzak [Chy94] Gessel [Ges90] Lipschitz [Lip89] and Zeilberger [Zei90]. Chapter 2 My Mathematica Package 2.1 Introduction In Chapter 1 we discussed some properties of holonomic sequences and their generating functions. A closer look at the proofs of these properties reveals the fact that these proofs contain algorithms to compute holonomic differential or ....

[Article contains additional citation context not shown here]

D. Zeilberger. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics, 32:321--368, 1990.

....implementation. Introduction Computer algebra consists in performing calculations on mathematical objects represented by a finite amount of information. A class of computer algebra objects is especially useful when it is possible to recognize whether two members of the class are identical or not. Zeilberger (1990b) showed that a large set of combinatorial identities can be proved using properties of the class of P finite functions and sequences and the important subclass of holonomic functions. A function is P finite when the set of its partial derivatives spans a finite dimensional vector space over ....

.... finiteness which generalizes P finiteness of functions and sequences. These Ore polynomials capture the properties of linear operators that are necessary to express our algorithms. The notion of finiteness makes it easy to describe mixed differential difference systems which were studied by Zeilberger (1990b) and Takayama (1989) and linear q equations which up to now have mainly been studied in the q hypergeometric case (equations of order 1) Our generalization makes it possible to have a general program working at the level of Ore polynomials. New types of systems of operators can be defined by ....

[Article contains additional citation context not shown here]

Zeilberger, D. (1990b). A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics, 32:321--368.

....1, see e.g. Andrews [6] which also contains a brief historical account. It is well known that number theoretic identities like these, or of similar type, can be deduced as limiting cases of q hypergeometric finite sum identities. Due to recent algorithmic breakthroughs, see for instance Zeilberger [24], or, Wilf and Zeilberger [23] proving these finite versions becomes more and more routine work that can be left to the computer. For instance, the computers Ekhad and Tre [12] delivered a purely verification proof of the following finite version, which was stated first in this form by Andrews ....

D. Zeilberger, A holonomic systems approach to special function identities, J. Comp. Appl. Math., 32 (1990), 321--368.

....Closure properties of holonomic functions and sequences are effectively computable. Consequently, identities between holonomic functions and sequences are decidable. Another part of gfun implements the numerous closure properties of holonomic functions and sequences (Lipshitz 1989; Stanley 1980; Zeilberger 1990). In particular, it is known that a sequence is holonomic (P recursive) if and only if its generating function is holonomic (D finite) the sum and product of two holonomic functions or sequences are holonomic; algebraic functions are holonomic; the composition of a holonomic function ....

....Libraries for Combinatorial Structures 15 diffeqtorec( y(z) f(n) f(n 2 3n 2)f(n) Gamma (7n 6 2n 2 )f(n 1) 5 4n n 2 )f(n 2) f(0) 1; f(1) 1=2g Finding such equations serves various purposes. 1. Identity proving To prove a combinatorial identity a = b, the technique promoted by Zeilberger (1990), Wilf Zeilberger (1992) consists in building up the equation satisfied by a Gamma b as exemplified above. The identity is then proved by checking sufficiently many initial conditions. Zeilberger published numerous examples of uses of this method, a detailed example based on gfun being given in ....

[Article contains additional citation context not shown here]

Zeilberger, D. (1990). A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32(3), 321--368.

....difference differential equations and finitely many initial conditions. The subject of investigation is the left ideal of operators (differential operators, shift operators, or q shift operators) that annihilates this function. The research was again stimulated by D. Zeilberger, who showed in [Zei90b] that the subclass of holonomic functions is closed w.r.t. definite summation. He also showed that one is able to compute, using elimination in the annihilation ideal of the function, a recurrence of the form (1.4) for any holonomic function. Zeilberger used Sylvester s dialytic elimination to ....

D. Zeilberger. A holonomic system approach to special functions identities. Journal of Computational and Applied Math., 32:321--368, 1990.

.... on f1 : ng that avoid the pattern It follows from the Robinson Schenstead algorithm and the hook length formula[3] that for any r, the number of permutations with no increasing subsequence of length r, is a certain binomial coefficient multisum, from which it follows immediately [8] that it is P recursive (holonomic) i.e. it satisfies a linear recurrence with polynomial coefficients in n) Typeset by A M S T E X A natural conjecture is: For any given finite set of patterns, PAT , the sequence aPAT (n) fi fi fi oe 2 Sn : oe has no occurrences of the given patterns fi ....

Doron Zeilberger, A Holonomic systems approach to special functions identities, J. of Computational and Applied Math. 32 (1990), 321-368.

....of Zeilberger s Creative Telescoping algorithm [13, 14, 11] A function T (n 1 ; n d ) is a proper hypergeometric term if it can be expressed as a product of a polynomial with several factorials of linear forms with integer coefficients, their reciprocals, and exponential functions. In [12] it is proved that proper hypergeometric terms are holonomic. Wilf and Zeilberger conjectured [11, p. 585] that a hypergeometric term is proper if and only if it is holonomic. Their conjecture concerns proper hypergeometric terms which depend on several discrete and continuous variables. We prove ....

....that each univariate sequence f i (k) f(i; k) and f j (n) f(n; j) satisfies a recurrence with polynomial coefficients. Thus by Theorem 1, each factor of T (n; k) is holonomic, hence by Theorem 2, so is T (n; k) 2 For factorial terms, this result can be found in [9] and for proper terms in [12, 11, 7]. Example 2 Consider the sequence T (n; k) 8 : n Gamma 2k) n 2k; 3 k ; n = 2k; Gamma1) n Gamma2k (2k Gamman Gamma1) n 2k: Then T (n 1; k) a (n Gamma 2k 1)T (n; k) and (n Gamma 2k) n Gamma 2k Gamma 1)T (n; k 1) a T (n; k) so T is hypergeometric. It is ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990) 321--368.

.... on f1 : ng that avoid the pattern It follows from the Robinson Schenstead algorithm and the hook length formula[3] that for any r, the number of permutations with no increasing subsequence of length r, is a certain binomial coefficient multisum, from which it follows immediately [8] that it is P recursive (holonomic) i.e. it satisfies a linear recurrence with polynomial coefficients in n) A natural conjecture is: For any given finite set of patterns, PAT , the sequence aPAT (n) fi fi fi fi n oe 2 Sn : oe has no occurrences of the given patterns o fi fi fi fi ....

Doron Zeilberger, A Holonomic systems approach to special functions identities, J. of Computational and Applied Math. 32 (1990), 321-368.

No context found.

Zeilberger, D. (to appear). A holonomic systems approach to special function identities. J. of Computational and Applied Math. Zippel, R. (1979) Probablistic algorithms for sparse polynomials. In: (Ng, ed.) Proceedings of Eurosam 79 , Lecture Notes in Computer Science 72, pp. 216--226. New York/Heidelberg/Berlin: SpringerVerlag.

....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 5. Current interest in Gosper s algorithm is mainly due to the fact, observed by Zeilberger (1990a, 1990b) that it also can be used in a non obvious and nontrivial way for definite hypergeometric summation. For instance, for verifying or finding binomial identities automatically , finding recurrence operators annihilating hypergeometric sums etc. A generalization of that approach can be ....

....(g n;k ) is hypergeometric in k (and n) Given the order d, the coefficient polynomials as well as the hypergeometric sequence g n;k are manufactured by Gosper s algorithm. In general, d is not known in advance but an upper bound can be computed. For proofs and more details of the method see e.g. Zeilberger (1990b) or Wilf Zeilberger (1992) For tutorials see Zeilberger (1993) a reprinted version is contained in this issue, or Wilf (1993) The latter also discusses the historical roots of the method referring to Sister Celine s technique, see e.g. Fasenmyer (1949) or to the work of Verbaeten, see e.g. ....

Zeilberger, D. (1990a). A holonomic systems approach to special function identities. J. Comp. Appl. Math. 32, 321--368.

No context found.

Zeilberger, D. A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32, 3 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very di#erent sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18]toeachof the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....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, virtually all known hypergeometric sum identities (and therefore legions of binomial coefficient identities too) It does this by means of certificates of proof, each of which consists of a pair of functions (F; G) a WZ pair ) that satisfy certain ....

....where F (n; k) and rhs(n) are of CF and rhs(n) 6= 0 is equivalent to one in which rhs(n) 1: simply divide through by rhs(n) Does every such F have a mate G Not always, but very often, and there is something weaker that is always guaranteed. In [14] it was shown, using the general theory of [13], that for every holonomic F (n; k) there exists a CF G(n; k) that is moreover of the form R(n; k)F (n; k) for some rational function R(n; k) such that for some L and polynomials (in n) a 0 (n) a 1 (n) aL (n) we have (8) a 0 (n)F (n; k) a 1 (n)F (n 1; k) Delta Delta Delta ....

[Article contains additional citation context not shown here]

Doron Zeilberger, A holonomic systems approach to special function identities (to appear).

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [14, 15, 16, 17], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [15, 17] to each of the three sums in (3.6) being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....la sommation hyperg eom etrique d efinie aux suites holonomes non hyperg eom etriques. L algorithme se g en eralise aussi au cas diff erentiel et du q calcul. Sa justification th eorique se fonde sur une description par op erateurs lin eaires et sur la th eorie de l holonomie. Introduction In [28], D. Zeilberger initiated an algorithmic treatment of special functions that led to efficient algorithms for summation and integration [21] In this approach, he considered a large class of functions and sequences that enjoys numerous closure properties, the class of holonomic functions. Simple ....

....equations. Hypergeometry does not imply holonomy, as exemplified by the sequence u given by u n;k = 1= n 2 k 2 ) see [25] To solve the elimination problem of determining an equation like (1) Zeilberger first gave a general but theoretical algorithm based on a skew Euclidean algorithm [28]. He This work was supported in part by the Long Term Research Project Alcom IT (#20244) of the European Union. himself called this algorithm the slow algorithm, and proposed his fast algorithm [27] for a restricted class of sequences: this algorithm is guaranteed to terminate on sequences ....

[Article contains additional citation context not shown here]

Zeilberger, D. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

....still factors nicely. The various hypergeometric calculations were done, with some patience, using the first author s Mathematica package HYP [8] For curiosity, we mention that, although at present it is quite hopeless to prove any of the identities in this paper by the recent algorithmic tools [13, 15, 16, 17, 18], these did, implicitly, have their place in this work. For example, the fact that the three seemingly very different sums in (3.6) can be combined into one single sum was discovered by applying the Gosper Zeilberger algorithm [13, 16, 18] to each of the three sums in (3.6) Being puzzled that one ....

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990), 321--368.

No context found.

D. Zeilberger. A holonomic systems approach to special function identities. J. Comput. Appl. Math., 32:321--368, 1990.

No context found.

Zeilberger,D. (1990a). A Holonomic systems approach to special functions identities. J. Comp. Appl.

No context found.

Zeilberger, Doron, A holonomic systems approach to special function identities, to appear. 4

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC