scholarly journals A note on degenerate generalized Laguerre polynomials and Lah numbers

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Hye Kyung Kim ◽  
Seong Ho Park
Keyword(s):  
Orthogonal Polynomials ◽  
Explicit Expression ◽  
Laguerre Polynomials ◽  
Type Formula ◽  
Generalized Laguerre Polynomials ◽  
Degenerate Version

AbstractThe aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an explicit expression, a Rodrigues type formula, and expressions for the derivatives. The novelty of the present paper is that it is the first paper on degenerate versions of orthogonal polynomials.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

SPATIOTEMPORAL STRONGLY NONLOCAL SPHERICAL LIGHT BULLETS

2013 ◽  
Vol 22 (01) ◽  
pp. 1350006
Author(s):  
WEI-PING ZHONG
Keyword(s):  
Numerical Simulations ◽  
Legendre Polynomials ◽  
Schrodinger Equation ◽  
Laguerre Polynomials ◽  
Three Dimensional ◽  
Integration Constant ◽  
Generalized Laguerre Polynomials ◽  
Existence And Stability ◽  
Light Bullets ◽  
Nonlocal Nonlinear

The general spherical beam solution of the three-dimensional (3D) spatiotemporal strongly nonlocal nonlinear (NN) Schrödinger equation in the form of light bullets is presented. The 3D spatiotemporal spherical beams are built by the products of generalized Laguerre polynomials and associated Legendre polynomials. By the choice of a specific integration constant, the spherical beam becomes an accessible soliton, which can exist in various forms. We confirm the existence and stability of these solutions by numerical simulations.

scholarly journals Meixner polynomials of the second kind and quantum algebras representing su(1, 1)

2009 ◽  
Vol 466 (2117) ◽  
pp. 1409-1428 ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Combinatorial Theory ◽  
Quantum Algebras ◽  
Combinatorial Approach ◽  
Meixner Polynomials ◽  
Link Type ◽  
Polynomial Coefficients ◽  
The Matrix ◽  
Math Phys

We show how Viennot’s combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges (Hodges & Sukumar 2007 Proc. R. Soc. A 463 , 2401–2414 ( doi:10.1098/rspa.2007.0001 ); Sukumar & Hodges 2007 Proc. R. Soc. A 463 , 2415–2427 ( doi:10.1098/rspa.2007.0003 )) on the matrix entries in powers of certain operators in a representation of su(1, 1). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequence of results by Koelink, Groenevelt and Van Der Jeugt (Van Der Jeugt 1997 J. Math. Phys. 38 , 2728–2740 ( doi:10.1063/1.531984 ); Koelink & Van Der Jeugt 1998 SIAM J. Math. Anal. 29 , 794–822 ( doi:10.1137/S003614109630673X ); Groenevelt & Koelink 2002 J. Phys. A 35 , 65–85 ( doi:10.1088/0305-4470/35/1/306 )), for the related operators, substitutions into essentially the same Laguerre polynomials and Meixner polynomials of the second kind may be used to express their eigenvectors. Our combinatorial approach explains and generalizes this ‘coincidence’.

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