Christoph Koutschan

Advanced applications of the holonomic systems approach

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. Frédéric 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 Gröbner 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 Takayamas algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessels 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 Stembridges theorem. To make the underlying computations feasible we employed a new approach for finding creative telescoping operators.

A Fast Approach to Creative Telescoping

In this note we reinvestigate the task of computing creative telescoping relations in differential–difference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute several ideas of how to make an implementation of this approach reasonably fast and provide such an implementation. A selection of examples shows that it can be superior to existing methods by a large factor.

Proof of Ira Gessel's lattice path conjecture

We present a computer-aided, yet fully rigorous, proof of Ira Gessels tantalizingly simply stated conjecture that the number of ways of walking 2n steps in the region x + y ≥ 0,y ≥ 0 of the square lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals 16n(5/6)n(1/2)n(5/3)n(2)n.

Creative Telescoping for Holonomic Functions

The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts.

Proof of George Andrews’s and David Robbins’s q-TSPP conjecture

The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product formula has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture.

Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices

We study the face-centered cubic (fcc) lattice in up to six dimensions. In particular, we are concerned with lattice Green functions (LGFs) and return probabilities. Computer algebra techniques, such as the method of creative telescoping, are used for deriving an ODE for a given LGF. For the four- and five-dimensional fcc lattices, we give rigorous proofs of the ODEs that were conjectured by Guttmann and Broadhurst. Additionally, we find the ODE of the LGF of the six-dimensional fcc lattice, a result that was not believed to be achievable with current computer hardware.

Third order integrability conditions for homogeneous potentials of degree −1

We prove an integrability criterion of order 3 for a homogeneous potential of degree −1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r, θ) = r−1h(exp (iθ)) with h∈C[z],degh≤3. We then find all meromorphically integrable potentials of this form.

Regular languages and their generating functions: The inverse problem

The technique of determining a generating function for an unambiguous context-free language is known as the Schutzenberger methodology. For regular languages, Elena Barcucci et al. proposed an approach for inverting this methodology based on Soittolas theorem. This idea allows a combinatorial interpretation (by means of a regular language) of certain positive integer sequences that are defined by C-finite recurrences. In this paper we present a Maple implementation of this inverse methodology and describe various applications. We give a short introduction to the underlying theory, i.e., the question of deciding N-rationality. In addition, some aspects and problems concerning the implementation are discussed; some examples from combinatorics illustrate its applicability.

A generalized Apagodu-Zeilberger algorithm

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.

Fundamental laser modes in paraxial optics: from computer algebra and simulations to experimental observation

AbstractWe study multi-parameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagation-invariant laser modes in weakly varying media. A “smart” lens design and a similar effect of superfocusing of particle beams in a thin monocrystal film are also discussed. In the supplementary electronic material, we provide a computer algebra verification of the results presented here, and of some related mathematical tools that were stated without proofs in the literature. We also demonstrate how computer algebra can be used to derive some of the presented formulas automatically, which is highly desirable as the corresponding hand calculations are very tedious. In numerical simulations, some of the new solutions reveal quite exotic properties which deserve further investigation including an experimental observation.