بسم الله الرحمن الرحيم قَالُواْ سُبْحَانَكَ لاَ عِلْمَ لَنَا إِلاَّ مَا عَلَّمْتَنَا إِنَّكَ أَنتَ الْعَلِيمُ الْحَكِيمُ

of Some Orthogonal Polynomials A Study of Fractional Extension of Some Orthogonal Polynomials دراسة في الامتداد الكسري لبعض كثيرات الحدود المتعامدة

Eman Atiya Ibrahim Abed Elwahed Prepared by: Eman Atiya Ibrahim Abed Elwahed Supervised by : Prof. Dr.Tariq Omar Salim Prepared by: Eman Atiya Ibrahim Abed Elwahed Supervised by : Prof. Dr. Tariq Omar Salim Prepared by: Eman Atiya Ibrahim Abed Elwahed Supervised by : Prof. Dr. Tariq Omar Salim Prepared by: Eman Atiya Ibrahim Abed Elwahed Supervised by : Prof. Dr. Tariq Omar Salim

ABSTRACT Orthogonal polynomials are a type of special functions that has the property of orthogonality , Recently some research workers have extended the differential formula of some orthogonal polynomials by using fractional derivatives In this thesis we investigate this kind of orthogonal polynomials and the main properties. This thesis contains four chapters and the references list. Chapter one contains some basic and important concepts of fractional calculus which will be used in this thesis .Also, the definitions and main properties of some orthogonal polynomials have been mentioned .Also, some important rules and formulas have been stated.

  In chapter two,Jacobi function of fractional order have been studied,where several properties have been established and the orthogonality property is obtained. In chapter three, we establish some basic properties like derivative formulas, recurrence relations, continuous properties , and the orthogonal property of Laguerre functions of fractional order and it's generating function relation. In chapter four, we study some basic properties of Hermite functions of fractional order by using Mittag-Leffler function , and alsothe orthogonality property has been formulated.

Introduction and Preliminaries Chapter One Introduction and Preliminaries

Definition 1.1 : Laguerre polynomial of integer order n defined as , . The generalized Laguerre polynomial is defined as The orthogonality of Laguerre polynomial is written as :

Definition 1.2: The Jacobi polynomial of integer order n is defined as: The Rodrgue's type formula of Jacobi polynomial of integer order n is written as, . The Orthogonality of Jacobi polynomial can be written as:

Definition1.3 : Hermite Polynomials :of order n is defined as : . The Rodrigues type formula for can be expressed as . The Orthogonality of Hermite polynomial can be written as:

Fractional Jacobi Function Chapter two Fractional Jacobi Function

Definition 2.1: The fractional Jacobi function is defined as , Definition 2.2: The fractional Ultraspherical function is defined as Definition 2.3: The fractional order Legender function is defined as:

Definition 2. 3 : The fractional Chebichev functions of first and second kind are defined as:

Theorem 2.1: For the fractional Jacobi function defined in The following representation holds: Theorem 2.2: the fractional Jacobi function can be represented as Theorem 2.3:

The fractional Jacobi function satisfies the following property: Theorem 2.4: , The fractional Jacobi function satisfies the following property:

Theorem 2.5: Let be the fractional Jacobi function defined by (2.1) , then Theorem 2.6: where is Laguerre function of fractional order

Orthogonality Properties: By using Rodrigue's formula of fractional Jacobi function ,we can write the orthogonality property in the following theorem Theorem 2.7: The Orthogonality property of fractional Jacobi function of order α can is written as for

Corollary 2.1: The Orthogonality property of fractional Legender function of order α is written as Corollary 2.2: The Orthogonality property of fractional Chebichev function of first kind is written as

Corollary 2.6: The Orthogonality property of fractional Chebichev function of second kind is written as Corollary 2.7: The Orthogonality property of fractional Ultraspherical function of second kind is written as

Fractional Laguerre Functions Chapter Three Fractional Laguerre Functions

Definition3.1: The fractional Laguerre function of order α is defined as , . and the fractional laguerre function , of order –α is defined as:

Definition3.2: let , is defined by and the function by

Definition 3.3: let ,then .

Theorem 3.1: let , then , , and

, Theorem 3.2: let then, Corollary 3.1: Let then

Corollary 3.3: let . then . Continuation Properties : Theorem 3.3: If , then

Corollary 3.5 : Let . . , then Theorem 3.4: Let , then Corollary 3.6 : Let , then

Theorem 3.5 : The fractional Laguerre function is continuous as a function in . Theorem 3.6 : If then

Chapter four fractional hermite functions