Three Recitations on Holonomic Systems and Hypergeometric Series
英文摘要：
A tutorial on what later became to be known as WZ theory, as well as a motivated account of the seminal Gosper algorithm.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
Holonomic Tools for Basic Hypergeometric Functions
With the exception of q-hypergeometric summation, the use of computer algebra packages implementing Zeilberger's "holonomic systems approach" in a broader mathematical sense is less common in the field of q-series and basic hypergeometric functions. A major objective of this article is to popularize the usage of such tools also in these domains. Concrete case studies showing software in action introduce to the basic techniques. An application highlight is a new computer-assisted proof of the celebrated Ismail-Zhang formula, an important q-analog of a classical expansion formula of plane waves …
cs.SCmath.CO
Polynomial reduction for holonomic sequences and applications in $π$-series and congruences
Polynomial reduction, designed first for hypergeometric terms, can be used to automatically prove and generate new hypergeometric identities from old ones. In this paper, we extend the reduction method to holonomic sequences. As applications, we describe an algorithmic way to prove and generate new multi-summation identities. Especially we present new families of $π$-series involving Domb numbers and Franel numbers.
math.COmath.NT
A variant of van Hoeij's algorithm to compute hypergeometric term solutions of holonomic recurrence equations
Linear homogeneous recurrence equations with polynomial coefficients are said to be holonomic. Such equations have been introduced in the last century for proving and discovering combinatorial and hypergeometric identities. Given a field K of characteristic zero, a term a(n) is called hypergeometric with respect to K, if the ratio a(n+1)/a(n) is a rational function over K. The solutions space of holonomic recurrence equations gained more interest in the 1990s from the well known Zeilberger's algorithm. In particular, algorithms computing the subspace of hypergeometric term solutions which cove…
cs.SCmath.CO
G-functions and multisum versus holonomic sequences
The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of $G$-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms. The proofs utilize the notion of a $G$-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by André.
math.AGmath.CO
Rank jumps in Codimension 2 A-Hypergeometric Systems
The holonomic rank of the A-hypergeometric system H_A(β) is shown to depend on the parameter vector βwhen the underlying toric ideal I_A is a non Cohen Macaulay codimension 2 toric ideal. The set of exceptional parameters is usually infinite.
math.AGmath.CO
Exceptional parameters for generic A-hypergeometric systems
The holonomic rank of an A-hypergeometric system $H_A(β)$ is conjectured to be independent of the parameter vector $β$ if and only if the toric ideal $I_A$ is Cohen Macaulay. We prove this conjecture in the case that $I_A$ is generic by explicitly constructing more than $\vol(A)$ many linearly independent hypergeometric functions for parameters $β$ coming from embedded primes of certain initial ideals of $I_A$.
math.AGmath.CO
更多相关论文收起更多论文
Multivariate Hypergeometric Terms
In this 1997 Ph.D. dissertation we prove a piecewise form of the discrete part of Wilf and Zeilberger's 1992 conjecture that a hypergeometric term is proper if and only if it is holonomic. We show that a holonomic hypergeometric term on $Z^n$ is piecewise proper and we show that without such a qualification the conjecture is false. We call a term piecewise proper if $Z^n$ can be expressed as the union of a finite number of polyhedral regions (the "pieces") and a set of measure zero (which we define to be a finite union of hyperplanes) such that the restriction of the term to each polyhedral re…
math.CO
