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Mathematics 2019, 7(2), 124; https://doi.org/10.3390/math7020124

Operational Methods in the Study of Sobolev-Jacobi Polynomials

1
Institut de Recherche en Informatique Fondamentale (IRIF), Université Paris-Diderot, F-75013 Paris, France
2
ENEA—Frascati Research Center, Via Enrico Fermi 45, 00044 Rome, Italy
3
Laboratoire d’Informatique de Paris-Nord (LIPN), CNRS UMR 7030, Université Paris 13, Sorbonne Paris Cité, F-93430 Villetaneuse, France
4
Laboratoire de Physique Theorique de la Matière Condensée (LPTMC), CNRS UMR 7600, Sorbonne Universités, Université Pierre et Marie Curie, F-75005 Paris, France
*
Author to whom correspondence should be addressed.
Received: 3 November 2018 / Revised: 19 January 2019 / Accepted: 22 January 2019 / Published: 24 January 2019
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Abstract

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of Sobolev-Jacobi (SJ) polynomials explicitly. View Full-Text
Keywords: Sobolev-Jacobi polynomials; umbral image techniques; generalized normal-ordering; lacunary generating functions Sobolev-Jacobi polynomials; umbral image techniques; generalized normal-ordering; lacunary generating functions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Behr, N.; Dattoli, G.; Duchamp, G.H.E.; Licciardi, S.; Penson, K.A. Operational Methods in the Study of Sobolev-Jacobi Polynomials. Mathematics 2019, 7, 124.

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