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

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

3. 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

4. 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.

5. 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.

6. Introduction and Preliminaries
Chapter One
Introduction and Preliminaries

7. 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 :

8. 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:

11. 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:

12. Fractional Jacobi Function
Chapter two
Fractional Jacobi Function

13. 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:

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

15. 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:

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

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

18. 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

19. 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

20. 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

21. Fractional Laguerre Functions
Chapter Three
Fractional Laguerre Functions

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

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

24. Definition 3.3:
let
,then .

25. Theorem 3.1:
let
, then ,
, and

26. ,
Theorem 3.2:
let
then,
Corollary 3.1:
Let
then

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

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

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

34. Chapter four fractional hermite functions