Holonomic Functions Research Articles

Рациональные коэффициенты ортогональных разложений некоторых функций

Expansions of many elementary and special functions in series of orthogonal polynomials have coefficients known explicitly. However, almost always these coefficients are irrational. Thus any numerical method would give these coefficients approximately with computations in any arithmetic. This is also true for the spectral methods, which give effective approximations for holonomic functions. However, in some exceptional cases the coefficients of expansions obtained with spectral methods turn out to be rational, and are computed exactly in rational arithmetic. We consider these expansions in series of some classical orthogonal polynomials. We demonstrate that, in this way, an infinite number of linear forms can be obtained for some irrationalities, in particular, for the Euler constant.

Operations for D-Algebraic Functions

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass elliptic functions, and holonomic or D-finite functions are D-algebraic. These functions form a field, and are closed under composition, taking functional inverse, and derivation. We present implementation for each underlying operation. We also give a systematic way for computing an algebraic differential equation from a linear differential equation with D-finite function coefficients. Each command is a feature of our Maple package NLDE available at https://mathrepo.mis.mpg.de/DAlgebraicFunctions/NLDEpackage.

Immersion of nonlinear systems comprising holonomic functions

This paper proposes a model simplification method for nonlinear systems comprising holonomic functions. A holonomic function is a nonlinear function that can be associated with a set of partial differential equations called a Pfaffian system, which contains only rational functions in the variable. By considering the Pfaffian system satisfied by all the holonomic functions in the original nonlinear system, it can be extended to another nonlinear system comprising only rational functions in a new state. This conversion of nonlinear systems can be shown to be an immersion, which preserves the input-output map of the original nonlinear system.

On the Representation of Non-Holonomic Power Series

Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like tan(z), z/(exp(z)-1), sec(z), etc. are not holonomic, therefore their power series are inaccessible by non-pattern matching implementations like the current Maple convert/FormalPowerSeries up to Maple 2021. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of tan(z) and sec(z) the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series.
 By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations (Geddes et al. 1992) to characterize several non-holonomic power series. As a consequence of the defined normal relation, it turns out that our algorithm is able to detect identities between non-holonomic functions that were not accessible in the past. We discuss this algorithm and its implementation for Maple 2022.
 Our Maple and Maxima implementations are available under the FPS software which can be downloaded at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm.

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

On the algebraic dependence of holonomic functions

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular, if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.

An energy–momentum conserving scheme for geometrically exact shells with drilling DOFs

An energy–momentum conserving temporal integration scheme is presented for a recently proposed geometrically exact shell formulation with drilling degrees of freedom. The scheme is based on a novel idea of defining mixed discrete derivatives for holonomic constraint functions with displacements and rotations. By defining general discrete derivative expressions with unknown terms, the mixed discrete derivatives with second-order accuracy are constructed according to deformation modes to satisfy directionality and orthogonality properties simultaneously, thus preserving conservation laws of total energy and momenta. The analysis of shell structures is conducted using the weak form quadrature elements to ensure exact incorporation of constraints and conservation of total energy after discretization, as well as circumvent shear and membrane locking phenomena. Benchmark numerical examples are presented to demonstrate the validity of the present scheme.

Holonomic relations for modular functions and forms: First guess, then prove

One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and sequences. With particular attention to algorithmic aspects, we study various connections between these two worlds. Applications concern partition congruences, Fricke–Klein relations, irrationality proofs a la Beukers, or approximations to pi studied by Ramanujan and the Borweins. As a major ingredient to a “first guess, then prove” strategy, a new algorithm for proving differential equations for modular forms is introduced.

Resurgence and holonomy of the ϕ2k model in zero dimension

We describe the resurgence properties of some partition functions corresponding to field theories in dimension 0. We show that these functions satisfy linear differential equations with polynomial coefficients and then use elementary stability results for holonomic functions to prove resurgence properties, enhancing the previously known results on growth estimates for the formal series involved, which had been obtained through a delicate combinatorics.

Constructing reductions for creative telescoping

The class of reduction-based algorithms was introduced recently as a new approach towards creative telescoping. Starting with Hermite reduction of rational functions, various reductions have been introduced for increasingly large classes of holonomic functions. In this paper we show how to construct reductions for general holonomic functions, in the purely differential setting.

A sage implementation for DD-finite functions

We present here the Sage package dd_functions which provides many features to compute with DD-finite functions, a natural extension of the class of holonomic or D-finite functions. The package, focused on a functional approach, allows the user to compute closure properties, extract coefficient sequences and compute the composition (as formal power series) of functions treated in this package. All these operations reduce the problem to linear algebra computations where classical division-free algorithms are used.

A survey of $q$-holonomic functions

We give a survey of basic facts of q -holonomic functions of one or several variables, following Zeilberger and Sabbah. We provide detailed proofs and examples.

Towards an effective study of the algebraic parameter estimation problem

The paper aims at developing the first steps toward a symbolic computation approach to the algebraic parameter estimation problem defined by Fliess and Sira-Ramirez. In this paper, within the algebraic analysis approach, we first give a general formulation of the algebraic parameter estimation for signals which are defined by ordinary differential equations with polynomial coefficients such as the standard orthogonal polynomials (e.g., Chebyshev or Hermite, polynomials). Based on a result on holonomic functions, we show that the algebraic parameter estimation problem for a truncated expansion of a function into an orthogonal basis of L2 defined by orthogonal polynomials can be studied similarly. Then, using symbolic computation methods such as Gröbner basis techniques for (noncommutative) polynomial rings, we first show how to compute ordinary differential operators which annihilate a given polynomial and which contain only certain parameters in their coefficients. Then, we explain how to compute the intersection of the annihilator ideals of two polynomials and characterize the ordinary differential operators which annihilate a first polynomial but not a second one. These results, at the core of the algebraic parameter estimation, are implemented in the NonA package.

MIMO Zero-Forcing Performance Evaluation Using the Holonomic Gradient Method

For multiple-input multiple-output (MIMO) spatial-multiplexing transmission, zero-forcing detection (ZF) is appealing because of its low complexity. Our recent MIMO ZF performance analysis for Rician--Rayleigh fading, which is relevant in heterogeneous networks, has yielded for the ZF outage probability and ergodic capacity infinite-series expressions. Because they arose from expanding the confluent hypergeometric function $ {_1\! F_1} (\cdot, \cdot, \sigma) $ around 0, they do not converge numerically at realistically-high Rician $ K $-factor values. Therefore, herein, we seek to take advantage of the fact that $ {_1\! F_1} (\cdot, \cdot, \sigma) $ satisfies a differential equation, i.e., it is a \textit{holonomic} function. Holonomic functions can be computed by the \textit{holonomic gradient method} (HGM), i.e., by numerically solving the satisfied differential equation. Thus, we first reveal that the moment generating function (m.g.f.) and probability density function (p.d.f.) of the ZF signal-to-noise ratio (SNR) are holonomic. Then, from the differential equation for $ {_1\! F_1} (\cdot, \cdot, \sigma) $, we deduce those satisfied by the SNR m.g.f. and p.d.f., and demonstrate that the HGM helps compute the p.d.f. accurately at practically-relevant values of $ K $. Finally, numerical integration of the SNR p.d.f. produced by HGM yields accurate ZF outage probability and ergodic capacity results.

Holonomic functions in mathematica

We present the Mathematica package HolonomicFunctions which provides a powerful framework for the automatic manipulation of multivariate holonomic functions, in the spirit of Zeilberger's holonomic systems approach. Its top-level functionalities are: converting a mathematical expression into a holonomic description, executing holonomic closure properties, and creative telescoping for general holonomic functions. To achieve these goals, many other, lower-level, functionalities had to be implemented which were not available in the Mathematica system: finding rational solutions of linear systems of (q-) difference / differential equations, noncommutative arithmetic in Ore algebras and computing Gröbner bases in such domains.

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Focusing on examples associated with holonomic functions, we try to bring new ideas on how to look at phase transitions, for which the critical manifolds are not points but curves depending on a spectral variable, or even fill higher dimensional submanifolds. Lattice statistical mechanics often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in the most general mathematical framework, be too complex, or simply could not be defined. In a learn-by-example approach, considering several Picard–Fuchs systems of two-variables ‘above’ Calabi–Yau ODEs, associated with double hypergeometric series, we show that D-finite (holonomic) functions are actually a good framework for finding properly the singular manifolds. The singular manifolds are found to be genus-zero curves. We then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics, the n-fold integrals χ(n), corresponding to the n-particle decomposition of the magnetic susceptibility of the anisotropic square Ising model. In this anisotropic case, we revisit a set of so-called Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find the first set of non-Nickelian singularities for χ(3) and χ(4), that also turns out to be rational or elliptic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model, or, equivalently, that they depend on the spectral parameter of the model. This has important consequences on the physical nature of the anisotropic χ(n)s which appear to be highly composite objects. We address, from a birational viewpoint, the emergence of families of elliptic curves, and that of Calabi–Yau manifolds on such problems. We also address the question of singularities of non-holonomic functions with a discussion on the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility χ.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities

A holonomic function is a differentiable or generalized function which satisfies a holonomic system of linear partial or ordinary differential equations with polynomial coefficients. The main purpose of this paper is to present algorithms for computing a holonomic system for the definite integral of a holonomic function with parameters over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference–differential system including parameters, then a holonomic difference–differential system for the integral can also be computed. In the algorithms, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.

On the average complexity for the verification of compatible sequences

The average complexity analysis for a formalism pertaining pairs of compatible sequences is presented. The analysis is done in two levels, so that an accurate estimate is achieved. The way of separating the candidate pairs into suitable classes of ternary sequences is interesting, allowing the use of fundamental tools of symbolic computation, such as holonomic functions and asymptotic analysis to derive an average complexity of O ( n n log n ) for sequences of length n.

Resolution of circuits from a physical point of view.

The resolution of electrical circuits and its acoustic homologous from a physical point of view is presented here. Based on the structural analogy with a mechanical system in equilibrium, it is possible to associate a Lagrangian depending on the generalized forces in a viscous medium with friction coefficients and with a set of holonomic constraint functions. By means of the Hamilton principle applied to this Lagrangian associated with a circuit, an equation of a function of a set of independent variables can be obtained, which turns out to be the condition for the resolution of a typical problem of linear programming. This minimized expression represents, in turn, the power dissipation. From all the possible values that could take a set of speed and pressure, the principle picks out one and only one (the true values) with the condition of producing the least power dissipation. All acoustic devices or systems that could be represented by an electric circuit may be solved by this method with the physical meaning described here.

A divergent generating function that can be summed and analysed analytically

We study a recurrence relation, originating in combinatorial problems, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. This is an explicit example, which we study detail, of a behaviour known to be typical for a large class of holonomic functions. We also express the solution using Bessel functions and Lommel polynomials.