Kam Hang Cheng, Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; e-mail: henry.cheng@family.ust.hk

Yik Man Chiang, Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; e-mail: machiang@ust.hk

Avery Ching, Department of Statistics, The University of Warwick, Coventry, CV4 7AL, United Kingdom. (Current address: Department of Mathematics, University of Dundee, Nethergate, Dundee DD1 4HN, Scotland, United Kingdom. e-mail: AChing001@dundee.ac.uk); e-mail: avery.ching@warwick.ac.uk

Abstract/Résumé:

This is a research announcement on a unifying study of generating functions of various sequences of special functions, using Bernstein’s theory of holonomic \(D\)-modules. Both new and well-known generating functions have been obtained in a systematic and algebraic way. New difference analogues of some special functions are also discovered. This announcement focuses on particular results about Hermite functions, Bessel functions and polynomials, Laguerre polynomials, and Gegenbauer polynomials.

Il s’agit d’une annonce de recherche sur une étude unificatrice des fonctions génératrices de diverses séquences de fonctions spéciales, en utilisant la théorie de Bernstein des \(D\)-modules holonomes. Des fonctions génératrices nouvelles et bien connues ont été obtenues de manière systématique et algébrique. De nouveaux analogues discrets de certaines fonctions spéciales sont également découverts. Cette annonce se concentre sur des résultats particuliers concernant les fonctions d’Hermite, les fonctions et polynômes de Bessel, les polynômes de Laguerre et les polynômes de Gegenbauer.

Keywords: D-modules, generating functions, holonomic systems of PDEs, special functions, transmutation formulae

AMS Subject Classification: Representations of entire functions by series and integrals, Monodromy; relations with differential equations and $D$-modules, Orthogonal polynomials and functions of hypergeometric type (Jacobi; Laguerre; Hermite; Askey scheme; etc.), None of the above; but in this section, Weyl theory and its generalizations, Algebraic aspects (differential-algebraic; hypertranscendence; group-theoretical), Commutators; derivations; elementary operators; etc. 30D10, 32S40, 33C45, 33E99, 34B20, 34M15, 47B47

PDF(click to download): $D$-module Approach to Special Functions and Generating Functions

Luc Vinet, FRSC ,Centre de Recherches Mathematiques, Universite de Montreal, P.O. Box 6128, Centre-ville Station, Montreal (Quebec), H3C 3J7, Canada and Centre de Recherches Mathematiques, Universite de Montreal, P.O. Box 6128, Centre-ville Station, Montreal (Quebec), H3C 3J7, Canada; e-mail: vinet@CRM.UMontreal.CA

Alexei Zhendanov, School of Mathematics, Renmin University of China, Beijing, 100872, China; e-mail: zhedanov@yahoo.com

Abstract/Résumé:

Eigenvalue problems on irreducible \(\mathfrak{su}(2)\) modules and their adjoints are considered in the Bargmann, Barut-Girardello and finite difference models. The biorthogonality relations that arise between the corresponding generating functions of the Krawtchouk polynomials are sorted out. A link with Padé approximation is made.

Des problèmes aux valeurs propres sur les modulesirréductibles de \(\mathfrak{su}(2)\) et leurs adjoints sont examinés dans les modèles de Bargmann, Barut–Girardello et aux différences finies. Les relations de biorthogonalité qui apparaissent entre les fonctions génératrices correspondantes des polynômes de Krawtchouk sont identifiées. Un lien avec l’approximation de Padé est fait.

Keywords: Krawtchouk polynomials, Pade approximation., biorthogonality, generating functions, su(2) models

AMS Subject Classification: Representations; algebraic theory (weights), Orthogonal polynomials and functions of hypergeometric type (Jacobi; Laguerre; Hermite; Askey scheme; etc.), Pad_¸ approximation 17B10, 33C45, 41A21

PDF(click to download): A Note on su(2) Models and the Biorthogonality of Generating Functions of Krawtchouk Polynomials