Hypergeometric Terms Research Articles

Average Case Analysis of Gosper’s Algorithm for a Class of Urn Model Inputs

In this paper we perform an asymptotic average case analysis of some of the most important steps of Gosper’s algorithm for indefinite summation of hypergeometric terms. The space of input functions of the algorithm is described in terms of urn models, and the analysis is performed by using specialized probabilistic transform techniques. We analyze two different types of urn model classes: one in which the input functions are assumed to be rational, and another for which a certain function of two inputs is assumed to be rational. The first set of results shows that the asymptotic complexity of the algorithm is the same within each of the two classes. The second set of results indicates that the complexity of the algorithm scales differently for the two classes of models: one can observe the logarithmic versus square root type of difference that is also present in other combinatorial models.

Asymptotics of sums of hypergeometric terms

Asymptotics of sums of hypergeometric terms

Applicability of the [formula omitted]-analogue of Zeilberger’s algorithm

The applicability or terminating condition for the ordinary case of Zeilberger’s algorithm was recently obtained by Abramov. For the q -analogue, the question of whether a bivariate q -hypergeometric term has a q Z -pair remains open. Le has found a solution to this problem when the given bivariate q -hypergeometric term is a rational function in certain powers of q . We solve the problem for the general case by giving a characterization of bivariate q -hypergeometric terms for which the q -analogue of Zeilberger’s algorithm terminates. Moreover, we give an algorithm to determine whether a bivariate q -hypergeometric term has a q Z -pair.

Telescoping in the context of symbolic summation in Maple

This paper is an exposition of different methods for computing closed forms of definite sums. The focus is on recently-developed results on computing closed forms of definite sums of hypergeometric terms. A design and an implementation of a software package which incorporates these methods into the computer algebra system Maple are described in detail.

K-free recurrences of double hypergeometric terms

In their elegant and powerful approaches to automatically proving hypergeometric identities, Wilf and Zeilberger have taken a truly insightful advantage of holonomic hypergeometric terms. Moreover, they have shown that all proper hypergeometric terms are holonomic, and they have conjectured that the converse should hold as well. In this paper, we introduce a canonical representation of double hypergeometric terms, which we call the standard representation. We show that in the case of double hypergeometric terms, the Wilf–Zeilberger conjecture is true. We note that Abramov and Petkovšek [Preprint Series 39 (748) (2001)] has independently obtained the same result.

A Sister Celine type algorithm for definite summation and integration

Sister Celine's method [2] computes a recurrence for a sum or multisum Σ ∞ k = 0 f ( n, k ) where f ( n, k ) is a given hypergeometric term. It was generalized to the q -hypergeometric case and to integrals [3].

When does Zeilberger's algorithm succeed?

A terminating condition of the well-known Zeilberger's algorithm for a given hypergeometric term T( n, k) is presented. It is shown that the only information on T( n, k) that one needs in order to determine in advance whether this algorithm will succeed is the rational function T( n, k+1)/ T( n, k).

The calculation of average distance in mesh structures

When solving problems in such areas as image processing, computer vision, and computational geometry, within a parallel architecture, the mesh structure often stands out as a natural choice. In this paper, we analyze a scalable measurement of the average distance between two arbitrary but fixed processors in a network structure, which is useful in providing a more global characterization of its data transmission behavior. In particular, we provide a closed-form expression for the average distance between two processors in the case of a traditional mesh. For a more complex case, where a traditional mesh is augmented with some additional diagonal links, after providing a telling expression for the average distance between its two processors, we prove that this latter expression cannot be represented in a closed-form format, with respect to a fairly general class of “standard operations”, namely, the class of the hypergeometric terms. Besides suggesting another global measurement of the communication behavior for general computer networks, and deriving concrete results for some “popular” mesh structures, this paper provides both positive and negative results regarding the derivation of closed-form expressions for combinatorial quantities, thus, is also theoretically interesting. We also believe that some of the general techniques used in this paper should be applicable elsewhere when closed-form expression needs to be derived, or its existence is in question.

On the structure of multivariate hypergeometric terms

Wilf and Zeilberger conjectured in 1992 that a hypergeometric term is proper-hypergeometric if and only if it is holonomic. We prove a slightly modified version of this conjecture in the case of several discrete variables.

A criterion for the applicability of Zeilberger's algorithm to rational functions

We consider the applicability (or terminating condition) of the well-known Zeilberger's algorithm and give the complete solution to this problem for the case where the original hypergeometric term F( n, k) is a rational function. We specify a class of identities ∑ k=0 nF(n,k)=0,F(n,k)∈ C(n,k) , that cannot be proven by Zeilberger's algorithm. Additionally, we give examples showing that the set of hypergeometric terms on which Zeilberger's algorithm terminates is a proper subset of the set of all hypergeometric terms, but a super-set of the set of proper terms. Résumé Nous considérons l'applicabilité (ou la condition de terminaison) du célèbre algorithme de Zeilberger et nous donnons la solution complète de ce problème dans le cas où le terme hypergéométrique initial F( n, k) est une fonction rationnelle. Nous indiquons une classe d'identités ∑ k=0 nF(n,k)=0,F(n,k)∈ C(n,k) , qui ne peuvent être démontrées par l'algorithme de Zeilberger. De plus, nous donnons des exemples qui prouvent que l'ensemble des termes hypergéométriques pour lesquels l'algorithme de Zeilberger se termine est un sous-ensemble propre de l'ensemble de tous les termes hypergéométriques mais un super-ensemble de l'ensemble des termes propres.

Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms

We describe a multiplicative normal form for rational functions which exhibits the shift structure of the factors, and investigate its properties. On the basis of this form we propose an algorithm which, given a rational function R, extracts a rational part F from the product of consecutive values of R: ∏k=n0n−1R(k) =F(n)∏k=n0n−1V(k) where the numerator and denominator of the rational function V have minimal possible degrees. This gives a minimal multiplicative representation of the hypergeometric term ∏k=n0n−1R(k). We also present an algorithm which, given a hypergeometric term T(n), constructs hypergeometric terms T1(n) and T2(n) such that T(n) =ΔT1(n) +T2(n) and T2(n) is minimal in some sense. This solves the additive decomposition problem for indefinite sums of hypergeometric terms: ΔT1(n) is the “summable part", and T2(n) the “non-summable part" of T(n). In other words, we get a minimal additive decomposition of the hypergeometric term T(n).

Computing the minimal telescoper for sums of hypergeometric terms

Let T ( n, k ) be a hypergeometric term of n and k. We present in this paper an algorithm to construct the minimal telescoper for U ( n, k ) = ∑ m=b n -1 T ( m, k ), b ε ℤ, if it exists. We show a Maple implementation of this method and discuss the problem of finding closed forms of definite sums of U ( n, k ).

Some connection and linearization problems for polynomials in and beyond the Askey scheme

The connection problem is considered in a hypergeometric function framework for (i) the two most general families of polynomials belonging to the Askey scheme (Wilson and Racah), and (ii) some generalized Laguerre and Jacobi polynomials falling outside that scheme (Sister Celine, Cohen and Prabhakar–Jain), which are relevant to the study of quantum-mechanical systems and include as particular cases, the generalizations of the classical families with Sobolev-type orthogonality. In addition, using the same method three new linearization-like formulae for the Gegenbauer polynomials are also derived: a linearization formula that generalizes the m= n case of Dougall's formula, the analogue of the m= n case of Nielsen's inverse linearization formula for Hermite polynomials, and a connection formula for the squares. Closed analytical formulae for the corresponding connection and linearization coefficients are given in terms of hypergeometric functions of unit argument, which at times can be further simplified and expressed as single hypergeometric terms.

Multibasic and Mixed Hypergeometric Gosper-Type Algorithms

Gosper’s summation algorithm finds a hypergeometric closed form of an indefinite sum of hypergeometric terms, if such a closed form exists. We extend his algorithm to the case when the terms are simultaneously hypergeometric and multibasic hypergeometric. We also provide algorithms for finding polynomial as well as hypergeometric solutions of recurrences in the mixed case. We do not require the bases to be transcendental, but only that q1k1⋯qmkm≠1 unless k1=⋯=km= 0. Finally, we generalize the concept of greatest factorial factorization to the mixed hypergeometric case.

Algorithms for m-fold Hypergeometric Summation

Zeilberger's algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms F( n, k) is extended to certain nonhypergeometric terms. An expression F( n, k) is called hypergeometric term if both F( n+1, k)/ F( n, k) and F( n, k+1)/ F( n, k) are rational functions. Typical examples are ratios of products of exponentials, factorials, Γ function terms, binomial coefficients, and Pochhammer symbols that are integer-linear with respect to n and k in their arguments. We consider the more general case of such ratios that are rational-linear with respect to n and k in their arguments, and present an extended version of Zeilberger's algorithm for this case, using an extended version of Gosper's algorithm for indefinite summation. In a similar way the Wilf-Zeilberger method of rational function certification of integer-linear hypergeometric identities is extended to rational-linear hypergeometric identities. The given algorithms on definite summation apply to many cases in the literature to which neither the Zeilberger approach nor the Wilf-Zeilberger method is applicable. Examples of this type are given by theorems of Watson and Whipple, and a large list of identities ("Strange evaluations of hypergeometric series") that were studied by Gessel and Stanton. Finally we show how the algorithms can be used to generate new identities.

REDUCE package for the indefinite and definite summation

This article describes the REDUCE package ZEILBERG implemented by Gregor Stölting and the author which can be obtained from RedLib, accessible via anonymous ftp on ftp.zib-berlin.de in the directory pub/redlib/rules.The REDUCE package ZEILBERG is a careful implementation of the Gosper and Zeilberger algorithms for indefinite, and definite summation of hypergeometric terms, respectively. An expression a k is called a hypergeometric term (or closed form ), if a k /a k-1 is a rational function with respect to k. Typical hypergeometric terms are ratios of products of powers, factorials, Γ function terms, binomial coefficients, and shifted factorials (Pochhammer symbols) that are integer-linear in their arguments.The package covers further extensions of both Gosper's and Zeilberger's algorithm which in particular are valid for ratios of products of powers, factorials, Γ function terms, binomial coefficients, and shifted factorials that are rational-linear in their arguments.A similar MAPLE package is described elsewhere [2].

On the Normal Approximation to the Hypergeometric Distribution

In this paper a new normal approximation to a sum of hypergeometric terms is derived, which is a direct generalization of Feller's normal approximation to the binomial distribution [2]. For intervals that are asymmetric with respect to the mean, or when the distribution is skewed, the new approximation is a marked improvement over the classical procedure. The hypergeometric distribution is discussed in Section 2, along with the classical norming and the resulting approximation. Feller's remarkable normal approximation for the related binomial distribution is given in Section 3 with an indication of how it can be extended to cover the hypergeometric case. The result of such an extension is presented in Theorem 2 of Section 4. This theorem gives upper and lower bounds on the hypergeometric sum and hence provides a useful estimate of the relative error. Preliminary results to proving Theorem 2 are exhibited in Section 5. The proof follows in Section 6.