Holonomic function

Abstract: The holonomic systems approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for the algorithmic treatment of holonomic function identities. Frederic Chyzak later extended this framework by introducing the closely related notion of ∂-finite functions and by placing their manipulation on solid algorithmic grounds. For practical purposes it is convenient to take advantage of both concepts which is not too much of a restriction: The class of functions that are holonomic and ∂-finite contains many elementary functions (such as rational functions, algebraic functions, logarithms, exponentials, sine function, etc.) as well as a multitude of special functions (like classical orthogonal polynomials, elliptic integrals, Airy, Bessel, and Kelvin functions, etc.). In short, it is composed of functions that can be characterized by sufficiently many partial differential and difference equations, both linear and with polynomial coefficients. An important ingredient is the ability to execute closure properties algorithmically, for example addition, multiplication, and certain substitutions. But the central technique is called creative telescoping which allows to deal with summation and integration problems in a completely automatized fashion.Part of this thesis is our Mathematica package HolonomicFunctions in which the above mentioned algorithms are implemented, including more basic functionality such as noncommutative operator algebras, the computation of Grobner bases in them, and finding rational solutions of parameterized systems of linear differential or difference equations.Besides standard applications like proving special function identities, the focus of this thesis is on three advanced applications that are interesting in their own right as well as for their computational challenge. First, we contributed to translating Takayama's algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessel's lattice path conjecture. The computations that completed the proof were of a nontrivial size and have been performed with our software. Second, investigating basis functions in finite element methods, we were able to extend the existing algorithms in a way that allowed us to derive various relations which generated a considerable speed-up in the subsequent numerical simulations, in this case of the propagation of electromagnetic waves. The third application concerns a computer proof of the enumeration formula for totally symmetric plane partitions, also known as Stembridge's theorem. To make the underlying computations feasible we employed a new approach for finding creative telescoping operators.

Abstract: The Apagodu-Zeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary δ-finite functions. In analogy to the hypergeometric case, we introduce the notion of proper δ-finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.

Abstract: The previously proved results that every analytically renormalized Feynman integral is a regular holonomic function suggests that theS-matrix should be locally expressible as an infinite sum of regular holonomic functions A regularity propertyR is formulated that expresses the condition that theS-matrix be locally expressible near each physical pointp as a convergent sum of regular holonomic functions, with each term enjoying some of the regularity properties of a corresponding Feynman integral This propertyR holds at every physical pointp that has yet been analyzed by the methods of axiomatic field theory orS-matrix theory Some analyticity properties of unitarity-type integrals are then examined under the assumption that theS-matrix satisfies propertyR and a weak integrability condition These results rest heavily on some recently proved properties of regular holonomic functions