Holonomic Functions Research Articles

Automatic computation of Stokes matrices

We describe a general procedure for computing Stokes matrices for solutions of linear differential equations with polynomial coefficients. The algorithms developed make an automation of the calculations possible, for a wide class of equations. We apply our techniques to some classical holonomic functions and also for some new special functions that are interesting in their own right: Ecalle’s accelerating functions.

Properties of q-holonomic functions

In a similar manner as in the papers by W. Koepf, D. Schmersau, Spaces of functions satisfying simple differential equations, Konrad-Zuse-Zentrum Berlin (ZIB), Technical Report TR 94-2 (1994) and Salvy, B., Zimmermann, P., GFUN: A package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software, (1994), pp. 163–177, where explicit algorithms for finding the differential equations satisfied by holonomic functions were given, in this paper we deal with the space of the q-holonomic functions which are the solutions of linear q-differential equations with polynomial coefficients. The sum, product and the composition with power functions of q-holonomic functions are also q-holonomic and the resulting q-differential equations can be computed algorithmically.

Efficient accelero-summation of holonomic functions

Let L ∈ K ( z ) [ ∂ ] be a linear differential operator, where K is the field of algebraic numbers. A holonomic function over K is a solution f to the equation L f = 0 . We will also assume that f admits initial conditions in K at a non-singular point z ∈ K . Given a broken-line path γ = z ⇝ z ′ between z and z ′ , which avoids the singularities of L and with vertices in K , we have shown in a previous paper [van der Hoeven, J., 1999. Fast evaluation of holonomic functions. Theoret. Comput. Sci. 210, 199–215] how to compute n digits of the analytic continuation of f along γ in time O ( n log 3 n log log n ) . In a second paper [van der Hoeven, J., 2001b. Fast evaluation of holonomic functions near and in singularities. J. Symbolic Comput. 31, 717–743], this result was generalized to the case when z ′ is allowed to be a regular singularity, in which case we compute the limit of f when we approach the singularity along γ . In the present paper, we treat the remaining case when the end-point of γ is an irregular singularity. In fact, we will solve the more general problem to compute “singular transition matrices” between non-standard points above a singularity and regular points in K near the singularity. These non-standard points correspond to the choice of “non-singular directions” in Écalle’s accelero-summation process. We will show that the entries of the singular transition matrices may be approximated up to n decimal digits in time O ( n log 4 n log log n ) . As a consequence, the entries of the Stokes matrices for L at each singularity may be approximated with the same time complexity.

Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite

While directed site-animals have been solved on several lattices, directed bond-animals remain unsolved on any nontrivial lattice. In this paper we demonstrate that the anisotropic generating function of directed bond-animals on the square lattice is fundamentally different from that of directed site-animals in that it is not differentiably finite. We also extend this result to directed bond-animals on hypercubic lattices. This indicates that directed bond-animals are unlikely to be solved by similar methods to those used in the solution of directed site-animals. It also implies that a solution cannot be conjectured using computer packages such as Gfun [A Maple package developed by B. Salvy, P. Zimmermann, E. Murray at INRIA, France, available from http://algo.inria.fr/libraries/ at time of submission; B. Salvy, P. Zimmermann, Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (2) (1994) 163–177] or differential approximants [A.J. Guttmann, Asymptotic analysis of coefficients, in: C. Domb, J. Lebowitz (Eds.), Phase Transit. Crit. Phenom., vol. 13, Academic Press, London, 1989, pp. 1–234, programs available from http://www.ms.unimelb.edu.au/~tonyg].

Fast Evaluation of Holonomic Functions Near and in Regular Singularities

A holonomic function is an analytic function, which satisfies a linear differential equationLf= 0 with polynomial coefficients. In particular, the elementary functions exp,log,sin, etc., and many special functions such as erf, Si, Bessel functions, etc., are holonomic functions. In a previous paper, we have given an asymptotically fast algorithm to evaluate a holonomic function f at a non-singular point z′ on the Riemann surface of f, up to any number of decimal digits while estimating the error. However, this algorithm becomes inefficient, when z′ approaches a singularity of f. In this paper, we obtain efficient algorithms for the evaluation of holonomic functions near and in singular points where the differential operator L is regular (or, slightly more generally, where L is quasi-regular—a concept to be introduced below).

An extension of Zeilberger's fast algorithm to general holonomic functions

We extend Zeilberger's fast algorithm for definite hypergeometric summation to non-hypergeometric holonomic sequences. The algorithm generalizes to the differential case and to q-calculus as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy.

Fast evaluation of holonomic functions

A holonomic function is an analytic function, which satisfies a linear differential equation with polynomial coefficients. In particular, the elementary functions exp, log, sin, etc. and many special functions like erf, Si, Bessel functions, etc. are holonomic functions. Given a holonomic function f (determined by the linear differential equation it satisfies and initial conditions in a non singular point z), we show how to perform arbitrary precision evaluations of f at a non singular point z′ on the Riemann surface of f, while estimating the error. Moreover, if the coefficients of the polynomials in the equation for f are algebraic numbers, then our algorithm is asymptotically very fast: if M( n) is the time needed to multiply two n digit numbers, then we need a time O( M( n log 2 n log log n)) to compute n digits of f( z′).

Computer Algebra Libraries for Combinatorial Structures

This paper introduces the framework of decomposable combinatorial structures and their traversal algorithms. A combinatorial type is decomposable if it admits a specification in terms of unions, products, sequences, sets, and cycles, either in the labelled or in the unlabelled context. Many properties of decomposable structures are decidable. Generating function equations, counting sequences, and random generation algorithms can be compiled from specifications. Asymptotic properties can be determined automatically for a reasonably large subclass. Maple libraries that implement such decision procedures are briefly surveyed (LUO, combstruct, equivalent). In addition, libraries for manipulating holonomic sequences and functions are presented (gfun, Mgfun).

Holonomic functions and their relation to linearly constrained languages

Dans cet article on considere la classe des langages avec des restrictions lineaires (LCL). Un langage L est dans la classe LCL si et seulement si il est l'ensemble de mots d'un langage algebrique non ambigu L' qui verifient des restrictions lineaires sur le nombre des occurrences des lettres. Nous montrons que a chaque langage dans LCL on peut associer une fonction generatrice holonome

An approach to the zero recognition problem by Buchberger algorithm

The class of “holonomic function” is considered. We present a quasi-algorithm that recognizes whether a holonomic function is zero or not. The algorithm consists of procedures which obtain differential operators that annihilate sums, products, and definite integrals with, respect to parameters of holonomic functions. A Weyl algebra-analog of Buchberger's algorithm is used. A “holonomic” approach to the zero recognition algorithm was initiated by D. Zeilberger (1990) who realized it by Sylvester's dyalitic elimination. Our algorithm uses Buchberger's algorithm to improve Zeilberger's algorithm.

On the regular holonomic character of theS-matrix and microlocal analysis of unitarity-type integrals

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.

Holonomic functions in a polycircle with nonnegative imaginary part

A well-known theorem of Nevanlinna on the representation of nonnegative measure of a function holomorphic in a circle and having nonnegative imaginary part is extended to functions of many complex variables, holomorphic in a polycircle and having there a nonnegative imaginary part.