Operational Methods in the Study of Sobolev-Jacobi Polynomials

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 cl...

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Bibliographic Details
Published in:Mathematics (Basel) 2019-01, Vol.7 (2), p.124
Main Authors: Behr, Nicolas, Dattoli, Giuseppe, Duchamp, Gérard, Penson, Silvia
Format: Article
Language:English
Subjects:
Calculus
Chemical reactions
Evolution
Gamma function
generalized normal-ordering
Hermite polynomials
Hypergeometric functions
Integrals
lacunary generating functions
Mathematical analysis
Mathematical Physics
Methods
Physics
Polynomials
Power
Quantum physics
Sobolev-Jacobi polynomials
umbral image techniques
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math7020124