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.

scholarly journals Hermite-Fejér and Grünwald interpolation at generalized Laguerre zeros

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4855-4863
Author(s):  
Bonis De ◽  
David Kubayi
Keyword(s):  
Laguerre Polynomials ◽  
Function Spaces ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Generalized Laguerre Polynomials ◽  
Uniformly Convergent

We introduce special Hermite-Fej?r and Gr?nwald operators at the zeros of the generalized Laguerre polynomials. We will prove that these interpolation processes are uniformly convergent in suitable weighted function spaces.

scholarly journals HYPERGEOMETRIC ZETA FUNCTIONS

2010 ◽  
Vol 06 (01) ◽  
pp. 99-126 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Functional Inequality ◽  
Bernoulli Numbers ◽  
Classical Counterpart ◽  
New Family ◽  
Riemann Zeta

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

scholarly journals From special operators (1-Dx )n+α to Laguerre Polynomials

2021 ◽  
Vol 4 (3) ◽  
Keyword(s):  
Generating Functions ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Rodrigues Formula ◽  
Operator Calculus ◽  
Symbolic Formula ◽  
The Way

Laguerre polynomials Ln α (x) are shown to be the transforms of monomials by the special operators (1-Dx )n+α . From this their current properties such as Rodrigues formula, Lucas symbolic formula, orthogonality, generating functions, etc… are systematically obtained. This success opens the way for the study of special functions from special operators by the powerful operator calculus.

scholarly journals Expressions of the Laguerre polynomial and some other special functions in terms of the generalized Meijer $ G $-functions

2021 ◽  
Vol 6 (11) ◽  
pp. 11631-11641
Author(s):  
Syed Ali Haider Shah ◽  
◽  
Shahid Mubeen
Keyword(s):  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Exponential Functions ◽  
Confluent Hypergeometric Functions ◽  
Logarithmic Functions

<abstract><p>In this paper, we investigate the relation of generalized Meijer $ G $-functions with some other special functions. We prove the generalized form of Laguerre polynomials, product of Laguerre polynomials with exponential functions, logarithmic functions in terms of generalized Meijer $ G $-functions. The generalized confluent hypergeometric functions and generalized tricomi confluent hypergeometric functions are also expressed in terms of the generalized Meijer $ G $-functions.</p></abstract>

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.

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