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 ....

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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.

.... 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.

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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.

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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 ....

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

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

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

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

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