Gosper's Algorithm Research Articles

Gosper summability of rational multiples of hypergeometric terms

By the telescoping method, Sun has recently given some hypergeometric series whose sums are related to π. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term , we consider the Gosper summability of for being a rational function of k. We give an upper bound and a lower bound on the degree of the numerator of such that is Gosper summable. We also show that the denominator of can be read off from the Gosper representation of . Based on these results, we give a systematic method to construct series whose sums can be derived from the known ones. We also illustrate the corresponding super-congruences and the q-analogue of the approach.

Telescoping method, summation formulas, and inversion pairs

<p style='text-indent:20px;'>Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.</p>

Improved Abramov-Petkovšek's Reduction and Creative Telescoping for Hypergeometric Terms

The Abramov-Petkovysek reduction computes an additive decomposition of a hypergeometric term, which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We improve the Abramov- Petkovysek reduction so as to decompose a hypergeometric term as the sum of a summable term and a non-summable one. The improved reduction does not solve any auxiliary linear difference equation explicitly. It is also more efficient than the original reduction according to computational experiments. Based on this reduction, we design a new algorithm to compute minimal telescopers for bivariate hypergeometric terms. The new algorithm can avoid the costly computation of certificates.

Binomial options pricing has no closed-form solution

We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach. Our proof follows from Gosper's algorithm.

On the bottom summation

We consider summation of consecutive values (?(v), ?(v + 1), ..., ?(w) of a meromorphic function ?(z), where v, w ? ?. We assume that ?(z) satisfies a linear difference equation L(y) = 0 with polynomial coefficients, and that a summing operator for L exists (such an operator can be found--if it exists--by the Accurate Summation algorithm, or, alternatively, by Gosper's algorithm when ordL = 1). The notion of bottom summation which covers the case where ?(z) has poles in ? is introduced.

Parameter augmentation and the q-Gosper algorithm

We develop a method for deriving new basic hypergeometric identities from old ones by parameter augmentation. The main idea is to introduce a new parameter and use the q -Gosper algorithm to find out a suitable form of the summand. By this method, we recover some classical formulas on basic hypergeometric series and find extensions of the Rogers–Fine identity and Ramanujan’s ψ 1 1 summation formula. Moreover, we derive an identity for a ψ 3 3 summation.

Converging to Gosper's algorithm

Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm and Abramov's algorithm for finding rational solutions to linear difference equations with polynomial coefficients. Our approach easily extends to the q-analogues.

Elimination in Weyl Algebra and q-Identities

In this paper, the following are introduced briefly: the basic concept of q-proper-hypergeometric; an algorithmic proof theory for q-proper-hypergeometric identities; and elimination in the non-commutative Weyl algebra. We give an algorithm for proving the single-variable q-proper-hypergeometric identities that is based on Zeilberger’s approach and the elimination in Weyl algebra. Finally, we test several examples that have been proven by D. Zeilberger and H. Wilf using the WZ-pair method and Gosper algorithm.

Improved universal denominators

The paper presents an algorithm for improvement (degree reduction) of universal denominators, which are used, for example, for constructing rational solutions of linear differential and difference systems with polynomial coefficients. A variant of Zeilberger's algorithm is described; it uses construction of universal denominator instead of application of Gosper's algorithm.

A fast algorithm for proving terminating hypergeometric identities

An algorithm for proving terminating hypergeometric identities, and thus binomial coefficients identities, is presented. It is based upon Gosper's algorithm for indefinite hypergeometric summation. A MAPLE program implementing this algorithm succeeded in proving almost all known identities. Hitherto the proof of such identities was an exclusively human endeavor.

The Markov-WZ Method

Andrei Markov's 1890 method for convergence-acceleration of series bears an amazing resemblance to WZ theory, as was recently pointed out by M. Kondratieva and S. Sadov. But Markov did not have Gosper and Zeilberger's algorithms, and even if he did, he wouldn't have had a computer to run them on. Nevertheless, his beautiful ad-hoc method, when coupled with WZ theory and Gosper's algorithm, leads to a new class of identities and very fast convergence-acceleration formulas that can be applied to any infinite series of hypergeometric type.

A probabilistic model for the degree of the cancellation polynomial in Gosper's algorithm

Gosper's algorithm is a cornerstone of automated summation of hypergeometric series. Milenkovic and Compton in 2002 gave an analysis of the run time of Gosper's algorithm applied to a random input. The main part of this was an asymptotic analysis of the random degree of the cancellation polynomial c( k) under various stipulated laws for the input. Their methods use probabilistic transform techniques. These results may also be obtained via probabilistic methods. Limit laws of the type proved by Milenkovic and Compton will be shown to follow from a general functional central limit theorem. Advantages to this approach include applicability to a more general class of input distributions and transparancy, in the sense that a non-quantitative limit theorem may be proved with no computation. A disadvantage to probabilistic methods is that one may typically derive only the leading term behavior. This is due to the inherent limit on precision in the Central Limit Theorem.

A Non-Automatic (!) Application of Gosper's Algorithm Evaluates a Determinant from Tiling Enumeration

We evaluate the determinant det 1≤i,j≤n x (() - ()), which gives the number of lozenge tilings of a hexagon with cut off corners. A particularly interesting feature of this evaluation is that it requires the proof of a certain hypergeometric identity which we accomplish by using Gosper's algorithm in a nonautomatic fashion.

Algorithms forq -Hypergeometric Summation in Computer Algebra

This paper describes three algorithms for q -hypergeometric summation: • a multibasic analogue of Gosper’s algorithm, • the q - Zeilberger algorithm, and • an algorithm for finding q - hypergeometric solutions of linear recurrences together with their Maple implementations, which is relevant both to people being interested in symbolic computation and in q -series. For all these algorithms, the theoretical background is already known and has been described, so we give only short descriptions, and concentrate ourselves on introducing our corresponding Maple implementations by examples. Each section is closed with a description of the input/output specifications of the corresponding Maple command. We present applications to q -analogues of classical orthogonal polynomials. In particular, the connection coefficients between families of q -Askey–Wilson polynomials are computed. Mathematica implementations have been developed for most of these algorithms, whereas to our knowledge only Zeilberger’s algorithm has been implemented in Maple so far (Koornwinder, 1993 or Zeilberger, cf. Pe kov0sek et al., 1996). We made an effort to implement the algorithms as efficient as possible which in the q -Petkovšek case led us to an approach with equivalence classes. Hence, our implementation is considerably faster than other ones. Furthermore the q -Gosper algorithm has been generalized to also find formal power series solutions.

A Generalization of Gosper's Algorithm to Bibasic Hypergeometric Summation

An algebraically motivated generalization of Gosper's algorithm to indefinite bibasic hypergeometric summation is presented. In particular, it is shown how Paule's concept of greatest factorial factorization of polynomials can be extended to the bibasic case. It turns out that most of the bibasic hypergeometric summation identities from literature can be proved and even found this way. A Mathematica implementation of the algorithm is available from the author.

A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities

Based on Gosper's algorithm for indefinite hypergeometric summation, Zeilberger's algorithm for proving binomial coefficient identities constitutes a recent breakthrough in symbolic computation. Mathematica implementations of these algorithms are described. Nontrivial examples are given in order to illustrate the usage of these packages which are available by e-mail request to the first-named author.

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.

Greatest Factorial Factorization and Symbolic Summation

The greatest factorial factorization (GFF) of a polynomial provides an analogue to square-free factorization but with respect to integer shifts instead to multiplicities. We illustrate the fundamental role of that concept in the context of symbolic summation. Besides a detailed discussion of the basic GFF notions we present a new approach to the indefinite rational summation problem as well as to Gosper's algorithm for summing hypergeometric sequences.

The Summation of Rational Functions by an Extended Gosper Algorithm

The rational functions form the most elementary class of functions for which the problem of summation or antidifferencing is not straightforward. Gosper's algorithm finds the antidifference of a rational function only if that antidifference is itself a rational function. We present an algorithm based upon Gosper's method which finds the rational part of the antidifference and the purely transcendental summand, i.e., a summand whose antidifference can be expressed entirely as a sum of digamma functions and derivatives of digamma functions. This algorithm is analogous to Horowitz's improvement of the Hermite-Ostrogradski method for finding the antiderivative of a rational function. An earlier algorithm of Moenck is the analogue of the Hermite-Ostrogradski method itself. Our algorithm requires less work than Moenck's.

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].