Legendre-type Special Functions Defined by Fractional Order Rodrigues Formula

Abstract Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials of arbitrary fractional orders and their various applications utilizing Rodrigues matrix formulas. In this line of research, we use the fractional order of Rodrigues formula to provide further investigation on such Legendre polynomials from a different point of view. Some properties, such as hypergeometric representations, continuation properties, recurrence relations, and differential equations, are derived. Moreover, Laplace’s first integral form and orthogonality are obtained.

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

Mathematics ◽

10.3390/math9090984 ◽

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.

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A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

Abstract and Applied Analysis ◽

10.1155/2013/279681 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 60

Author(s):

Abdon Atangana ◽

Aydin Secer

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Fractional Derivatives ◽

Special Functions ◽

Advantages And Disadvantages ◽

Derivatives Of

The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

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Simple Recipe for Accurate Solution of Fractional Order Equations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4023009 ◽

2012 ◽

Vol 8 (3) ◽

Cited By ~ 1

Author(s):

Sambit Das ◽

Anindya Chatterjee

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Initial Conditions ◽

Special Functions ◽

Differential Algebraic Equations ◽

Accurate Solution ◽

Algebraic Equations ◽

Matlab Program ◽

Fractional Differential

Fractional order integrodifferential equations cannot be directly solved like ordinary differential equations. Numerical methods for such equations have additional algorithmic complexities. We present a particularly simple recipe for solving such equations using a Galerkin scheme developed in prior work. In particular, matrices needed for that method have here been precisely evaluated in closed form using special functions, and a small Matlab program is provided for the same. For equations where the highest order of the derivative is fractional, differential algebraic equations arise; however, it is demonstrated that there is a simple regularization scheme that works for these systems, such that accurate solutions can be easily obtained using standard solvers for stiff differential equations. Finally, the role of nonzero initial conditions is discussed in the context of the present approximation method.

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On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions

Fractal and Fractional ◽

10.3390/fractalfract4030033 ◽

2020 ◽

Vol 4 (3) ◽

pp. 33

Author(s):

Yudhveer Singh ◽

Vinod Gill ◽

Jagdev Singh ◽

Devendra Kumar ◽

Kottakkaran Sooppy Nisar

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Hypergeometric Function ◽

Fractional Order ◽

Integral Transform ◽

Special Functions ◽

Confluent Hypergeometric Function ◽

Complex Variables ◽

Several Complex Variables ◽

Volterra Type

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here.

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Generalized special functions in the description of fractional diffusive equations

Communications in Applied and Industrial Mathematics ◽

10.1515/caim-2019-0010 ◽

2019 ◽

Vol 10 (1) ◽

pp. 31-40 ◽

Cited By ~ 6

Author(s):

Clemente Cesarano

Keyword(s):

Heat Equation ◽

Fractional Order ◽

Numerical Solutions ◽

Special Functions ◽

Hermite Polynomials ◽

Generalized Equations ◽

Extended Form ◽

General Solutions

Abstract Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

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On the Agaciro Equation via the Scope of Green Function

Mathematical Problems in Engineering ◽

10.1155/2014/201796 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Abdon Atangana ◽

Innocent Rusagara

Keyword(s):

Exact Solution ◽

Green Function ◽

Fractional Order ◽

Mellin Transform ◽

Integral Transform ◽

Special Functions ◽

Order Derivative ◽

Third Order ◽

Fractional Order Derivative ◽

The Green Function

We have undertaken an investigation of a kind of third-order equation called Agaciro equation within the folder of both integer and fractional order derivative. In the way of deriving the general exact solution of this equation, we employed the philosophy of the Green function together with some integral transform operators and special functions including but not limited to the Laplace, Fourier, and Mellin transform. We presented some examples of exact solution of this class of third-order equations for integer and fractional order derivative. It is important to point out that the value of Agaciro equation can be extended to describe assorted phenomenon in sciences.

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On Generalisation of Polynomials in Complex Plane

Advances in Decision Sciences ◽

10.1155/2010/230184 ◽

2010 ◽

Vol 2010 ◽

pp. 1-9

Author(s):

Maslina Darus ◽

Rabha W. Ibrahim

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Complex Plane ◽

Special Functions ◽

Laguerre Polynomials ◽

Second Order ◽

Second Order Differential Equations

The generalised Bell and Laguerre polynomials of fractional-order in complex z-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.

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

Advances in Theoretical & Computational Physics ◽

10.33140/atcp.04.03.05 ◽

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.

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A fractional operator algorithm method for construction of solutions of fractional order differential equations

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-012-0020-5 ◽

2012 ◽

Vol 15 (2) ◽

Cited By ~ 5

Author(s):

Kanat Shinaliyev ◽

Batirkhan Turmetov ◽

Sabir Umarov

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Wide Class ◽

Special Functions ◽

New Method ◽

Fractional Operator ◽

Fractional Order Differential Equations

AbstractIn this paper a new method of construction of a solution for a wide class of fractional order differential equations is presented. This method is a generalization of the method of operator algorithms to fractional order differential equations. Along with this, new types of special functions are introduced and some of their properties are studied.

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