Orthogonal polynomials and recurrence equations, operator equations and factorization

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or q-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class - implemented in the computer algebra system Maple - are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

N–Dimensional Orthogonal Tile Sizing Problem

AMS subject classification: 68Q22, 90C90

Solution properties of the de Branges differential recurrence equation

In this 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums thas was published by Askey and Gasper in 1976. The de Branges functions Tn/k(t) are defined as the solutions of a system of differential recurrence equations with suitably given initial values. The essential fact used in the proof of the Bieberbach and Milin conjectures is the statement Tn/k(t)<=0. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Λn/k(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', Tn/k(t)=-kΛn/k(t), and the positivity results in both proofs Tn/k(t)<=0 are essentially the same. In this paper we study differential recurrence equations equivalent to de Branges' original ones and show that many solutions of these differential recurrence equations don't change sign so that the above inequality is not as surprising as expected. Furthermore, we present a multiparameterized hypergeometric family of solutions of the de Branges differential recurrence equations showing that solutions are not rare at all.

Corrections to the paraxial approximation solutions in transversely nonuniform refractive-index media

The coupled differential recurrence equations for the corrections to the paraxial approximation solutions in transversely nonuniform refractive-index media are established in terms of the perturbation method. All the corrections (including the longitudinal field corrections) to the paraxial approximation solutions are presented in the weak-guidance approximation. As a concrete application, the first-order longitudinal field correction and the second-order transverse field correction to the paraxial approximation of a Gaussian beam propagating in a transversely quadratic refractive index medium are analytically investigated. (C) 1999 Optical Society of America [S0740-3232(99)00310-5].

Nonharmonic transverse vibration of the H-bonded molecules and the THz spectra in ice and water

A nonlinear equation of motion is found for the dimer comprising two charged H2O molecules. The THz dielectric response to nonharmonic vibration of a nonrigid dipole, forming the hydrogen bond (HB), is found in the direction transverse to this bond. An explicit expression is derived for the autocorrelator that governs the spectrum generated by transverse vibration (TV) of such a dipole. This expression is obtained by analytical solution of the truncated set of recurrence equations. The far infrared (FIR) spectra of ice at the temperature - 7 degrees C are calculated. The wideband, in the wavenumber (frequency) v range 0... 100.0 cm(-1), spectra are obtained for liquid water at room temperature and for supercooled water at -5.6 degrees C. All spectra are represented in terms of the complex permittivity epsilon(v) and the absorption coefficient alpha(v). The obtained analytical formula for epsilon comprises the term epsilon(perpendicular to) pertinent to the studied TV mechanism with three additional terms Delta epsilon(q), Delta epsilon(mu), and epsilon(or) arising, respectively, from: elastic harmonic vibration of charged molecules along the H-bond; elastic reorientation of HB permanent dipoles; and rather free libration of permanent dipoles in 'defects' of water/ice structure. The suggested TV-dielectric relaxation mechanism allows us: (a) to remove the THz 'deficit' of loss epsilon" inherent in previous theoretical studies; (b) to explain the THz loss and absorption spectra in supercooled (SC) water; and (c) to describe, in agreement with the experiment, the low- and high-frequency tails of the two bands of ice H2O located in the range 10...300 cm(-1). Specific THz dielectric properties of SC water are ascribed to association of water molecules, revealed in our study by transverse vibration of HB charged molecules. (C) 2006 Published by Elsevier B.V.

Algorithmic computation of knot polynomials of secondary structure elements of proteins

The classification of protein structures is an important and still outstanding problem. The purpose of this paper is threefold. First, we utilize a relation between the Tutte and homfly polynomial to show that the Alexander-Conway polynomial can be algorithmically computed for a given planar graph. Second, as special cases of planar graphs, we use polymer graphs of protein structures. More precisely, we use three building blocks of the three-dimensional protein structure-alpha-helix, antiparallel beta-sheet, and parallel beta-sheet-and calculate, for their corresponding polymer graphs, the Tutte polynomials analytically by providing recurrence equations for all three secondary structure elements. Third, we present numerical results comparing the results from our analytical calculations with the numerical results of our algorithm-not only to test consistency, but also to demonstrate that all assigned polynomials are unique labels of the secondary structure elements. This paves the way for an automatic classification of protein structures.

A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen

Die q-Analysis ist eine spezielle Diskretisierung der Analysis auf einem Gitter, welches eine geometrische Folge darstellt, und findet insbesondere in der Quantenphysik eine breite Anwendung, ist aber auch in der Theorie der q-orthogonalen Polynome und speziellen Funktionen von großer Bedeutung. Die betrachteten mathematischen Objekte aus der q-Welt weisen meist eine recht komplizierte Struktur auf und es liegt daher nahe, sie mit Computeralgebrasystemen zu behandeln. In der vorliegenden Dissertation werden Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen vorgestellt. Alle Algorithmen sind in dem Maple-Package qFPS, welches integraler Bestandteil der Arbeit ist, implementiert. Nachdem in den ersten beiden Kapiteln Grundlagen geschaffen werden, werden im dritten Kapitel Algorithmen präsentiert, mit denen man zu einer q-holonomen Funktion q-holonome Rekursionsgleichungen durch Kenntnis derer q-Shifts aufstellen kann. Operationen mit q-holonomen Rekursionen werden ebenfalls behandelt. Im vierten Kapitel werden effiziente Methoden zur Bestimmung polynomialer, rationaler und q-hypergeometrischer Lösungen von q-holonomen Rekursionen beschrieben. Das fünfte Kapitel beschäftigt sich mit q-hypergeometrischen Potenzreihen bzgl. spezieller Polynombasen. Wir formulieren einen neuen Algorithmus, der zu einer q-holonomen Rekursionsgleichung einer q-hypergeometrischen Reihe mit nichttrivialem Entwicklungspunkt die entsprechende q-holonome Rekursionsgleichung für die Koeffizienten ermittelt. Ferner können wir einen neuen Algorithmus angeben, der umgekehrt zu einer q-holonomen Rekursionsgleichung für die Koeffizienten eine q-holonome Rekursionsgleichung der Reihe bestimmt und der nützlich ist, um q-holonome Rekursionen für bestimmte verallgemeinerte q-hypergeometrische Funktionen aufzustellen. Mit Formulierung des q-Taylorsatzes haben wir schließlich alle Zutaten zusammen, um das Hauptergebnis dieser Arbeit, das q-Analogon des FPS-Algorithmus zu erhalten. Wolfram Koepfs FPS-Algorithmus aus dem Jahre 1992 bestimmt zu einer gegebenen holonomen Funktion die entsprechende hypergeometrische Reihe. Wir erweitern den Algorithmus dahingehend, dass sogar Linearkombinationen q-hypergeometrischer Potenzreihen bestimmt werden können. ________________________________________________________________________________________________________________

Identifikation spezieller Funktionen, die durch Rodriguesformeln gegeben sind

Es ist allgemein bekannt, dass sich zwei gegebene Systeme spezieller Funktionen durch Angabe einer Rekursionsgleichung und entsprechend vieler Anfangswerte identifizieren lassen, denn computeralgebraisch betrachtet hat man damit eine Normalform vorliegen. Daher hat sich die interessante Forschungsfrage ergeben, Funktionensysteme zu identifizieren, die über ihre Rodriguesformel gegeben sind. Zieht man den in den 1990er Jahren gefundenen Zeilberger-Algorithmus für holonome Funktionenfamilien hinzu, kann die Rodriguesformel algorithmisch in eine Rekursionsgleichung überführt werden. Falls die Funktionenfamilie überdies hypergeometrisch ist, sogar laufzeiteffizient. Um den Zeilberger-Algorithmus überhaupt anwenden zu können, muss es gelingen, die Rodriguesformel in eine Summe umzuwandeln. Die vorliegende Arbeit beschreibt die Umwandlung einer Rodriguesformel in die genannte Normalform für den kontinuierlichen, den diskreten sowie den q-diskreten Fall vollständig. Das in Almkvist und Zeilberger (1990) angegebene Vorgehen im kontinuierlichen Fall, wo die in der Rodriguesformel auftauchende n-te Ableitung über die Cauchysche Integralformel in ein komplexes Integral überführt wird, zeigt sich im diskreten Fall nun dergestalt, dass die n-te Potenz des Vorwärtsdifferenzenoperators in eine Summenschreibweise überführt wird. Die Rekursionsgleichung aus dieser Summe zu generieren, ist dann mit dem diskreten Zeilberger-Algorithmus einfach. Im q-Fall wird dargestellt, wie Rekursionsgleichungen aus vier verschiedenen q-Rodriguesformeln gewonnen werden können, wobei zunächst die n-te Potenz der jeweiligen q-Operatoren in eine Summe überführt wird. Drei der vier Summenformeln waren bislang unbekannt. Sie wurden experimentell gefunden und per vollständiger Induktion bewiesen. Der q-Zeilberger-Algorithmus erzeugt anschließend aus diesen Summen die gewünschte Rekursionsgleichung. In der Praxis ist es sinnvoll, den schnellen Zeilberger-Algorithmus anzuwenden, der Rekursionsgleichungen für bestimmte Summen über hypergeometrische Terme ausgibt. Auf dieser Fassung des Algorithmus basierend wurden die Überlegungen in Maple realisiert. Es ist daher sinnvoll, dass alle hier aufgeführten Prozeduren, die aus kontinuierlichen, diskreten sowie q-diskreten Rodriguesformeln jeweils Rekursionsgleichungen erzeugen, an den hypergeometrischen Funktionenfamilien der klassischen orthogonalen Polynome, der klassischen diskreten orthogonalen Polynome und an der q-Hahn-Klasse des Askey-Wilson-Schemas vollständig getestet werden. Die Testergebnisse liegen tabellarisch vor. Ein bedeutendes Forschungsergebnis ist, dass mit der im q-Fall implementierten Prozedur zur Erzeugung einer Rekursionsgleichung aus der Rodriguesformel bewiesen werden konnte, dass die im Standardwerk von Koekoek/Lesky/Swarttouw(2010) angegebene Rodriguesformel der Stieltjes-Wigert-Polynome nicht korrekt ist. Die richtige Rodriguesformel wurde experimentell gefunden und mit den bereitgestellten Methoden bewiesen. Hervorzuheben bleibt, dass an Stelle von Rekursionsgleichungen analog Differential- bzw. Differenzengleichungen für die Identifikation erzeugt wurden. Wie gesagt gehört zu einer Normalform für eine holonome Funktionenfamilie die Angabe der Anfangswerte. Für den kontinuierlichen Fall wurden umfangreiche, in dieser Gestalt in der Literatur noch nie aufgeführte Anfangswertberechnungen vorgenommen. Im diskreten Fall musste für die Anfangswertberechnung zur Differenzengleichung der Petkovsek-van-Hoeij-Algorithmus hinzugezogen werden, um die hypergeometrischen Lösungen der resultierenden Rekursionsgleichungen zu bestimmen. Die Arbeit stellt zu Beginn den schnellen Zeilberger-Algorithmus in seiner kontinuierlichen, diskreten und q-diskreten Variante vor, der das Fundament für die weiteren Betrachtungen bildet. Dabei wird gebührend auf die Unterschiede zwischen q-Zeilberger-Algorithmus und diskretem Zeilberger-Algorithmus eingegangen. Bei der praktischen Umsetzung wird Bezug auf die in Maple umgesetzten Zeilberger-Implementationen aus Koepf(1998/2014) genommen. Die meisten der umgesetzten Prozeduren werden im Text dokumentiert. Somit wird ein vollständiges Paket an Algorithmen bereitgestellt, mit denen beispielsweise Formelsammlungen für hypergeometrische Funktionenfamilien überprüft werden können, deren Rodriguesformeln bekannt sind. Gleichzeitig kann in Zukunft für noch nicht erforschte hypergeometrische Funktionenklassen die beschreibende Rekursionsgleichung erzeugt werden, wenn die Rodriguesformel bekannt ist.

Synthesis of a systolic array genetic algorithm

The paper presents a design for a hardware genetic algorithm which uses a pipeline of systolic arrays. These arrays have been designed using systolic synthesis techniques which involve expressing the algorithm as a set of uniform recurrence relations. The final design divorces the fitness function evaluation from the hardware and can process chromosomes of different lengths, giving the design a generic quality. The paper demonstrates the design methodology by progressively re-writing a simple genetic algorithm, expressed in C code, into a form from which systolic structures can be deduced. This paper extends previous work by introducing a simplification to a previous systolic design for the genetic algorithm. The simplification results in the removal of 2N 2 + 4N cells and reduces the time complexity by 3N + 1 cycles.

Uniformization of affine dependence programs for parallel embedded system design

This paper is concerned with the uniformization of a system of afine recurrence equations. This transformation is used in the design (or compilation) of highly parallel embedded systems (VLSI systolic arrays, signal processing filters, etc.). In this paper, we present and implement an automatic system to achieve uniformization of systems of afine recurrence equations. We unify the results from many earlier papers, develop some theoretical extensions, and then propose effective uniformization algorithms. Our results can be used in any high level synthesis tool based on polyhedral representation of nested loop computations.

Uniformization tool for systolic array designs

The paper is concerned with the uniformization of a system of affine recurrence equations. This transformation is used in the design (or compilation) of highly parallel embedded systems (VLSI systolic arrays, signal processing filters, etc.). We present and implement an automatic system to achieve uniformization of systems of affine recurrence equations. We unify the results from many earlier papers, develop some theoretical extensions, and then propose effective uniformization algorithms. Our results can be used in any high level synthesis tool based on polyhedral representation of nested loop computations.

Towards an abstract domain for resource analysis of logic programs using sized types

We present a novel general resource analysis for logic programs based on sized types.Sized types are representations that incorporate structural (shape) information and allow expressing both lower and upper bounds on the size of a set of terms and their subterms at any position and depth. They also allow relating the sizes of terms and subterms occurring at different argument positions in logic predicates. Using these sized types, the resource analysis can infer both lower and upper bounds on the resources used by all the procedures in a program as functions on input term (and subterm) sizes, overcoming limitations of existing analyses and enhancing their precision. Our new resource analysis has been developed within the abstract interpretation framework, as an extension of the sized types abstract domain, and has been integrated into the Ciao preprocessor, CiaoPP. The abstract domain operations are integrated with the setting up and solving of recurrence equations for both, inferring size and resource usage functions. We show that the analysis is an improvement over the previous resource analysis present in CiaoPP and compares well in power to state of the art systems.

Resource usage analysis of logic programs via abstract interpretation using sized types

We present a novel general resource analysis for logic programs based on sized types. Sized types are representations that incorporate structural (shape) information and allow expressing both lower and upper bounds on the size of a set of terms and their subterms at any position and depth. They also allow relating the sizes of terms and subterms occurring at different argument positions in logic predicates. Using these sized types, the resource analysis can infer both lower and upper bounds on the resources used by all the procedures in a program as functions on input term (and subterm) sizes, overcoming limitations of existing resource analyses and enhancing their precision. Our new resource analysis has been developed within the abstract interpretation framework, as an extension of the sized types abstract domain, and has been integrated into the Ciao preprocessor, CiaoPP. The abstract domain operations are integrated with the setting up and solving of recurrence equations for inferring both size and resource usage functions. We show that the analysis is an improvement over the previous resource analysis present in CiaoPP and compares well in power to state of the art systems.

On Solutions of Holonomic Divided-Difference Equations on Non-Uniform Lattices

The main aim of this paper is the development of suitable bases (replacing the power basis x^n (n\in\IN_\le 0) which enable the direct series representation of orthogonal polynomial systems on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows to write solutions of arbitrary divided-difference equations in terms of series representations extending results given in [16] for the q-case. Furthermore it enables the representation of the Stieltjes function which can be used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in [5], see also [6].