47B47 — 1 articles found.

$D$-module Approach to Special Functions and Generating Functions

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (1) 2023, pp. 1–12
Vol.45 (1) 2023
Kam Hang Cheng; Yik Man Chiang; Avery Ching Details
(Received: 2022-08-31 , Revised: 2023-03-31 )
(Received: 2022-08-31 , Revised: 2023-03-31 )

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


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

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