Abstract and Applied Analysis

Volume 2013 (2013), Article ID 350182, 6 pages

http://dx.doi.org/10.1155/2013/350182

Research Article

Extended Jacobi Functions via Riemann-Liouville Fractional Derivative

Bayram Çekim and Esra Erkuş-Duman

Gazi University, Faculty of Science, Department of Mathematics, Teknikokullar, 06500 Ankara, Turkey

Received 19 January 2013; Accepted 3 April 2013

Academic Editor: Mohamed Kamal Aouf

Copyright © 2013 Bayram Çekim and Esra Erkuş-Duman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By means of the Riemann-Liouville fractional calculus, extended Jacobi functions are de…fined and some of their properties are obtained. Then, we compare some properties of the extended Jacobi functions extended Jacobi polynomials. Also, we derive fractional differential equation of generalized extended Jacobi functions.

1. Introduction

Fractional calculus is examined by many mathematicians such as Liouville, Riemann, and Caputo. In recent years, the theory of fractional calculus, integrals, and derivatives of fractional orders has become an active research area in mathematical analysis. This theory has also been applied for many fundamental areas such as biology, physics, electrochemistry, economics, probability theory, and statistics [1, 2].

In this paper, we use fractional calculus in the theory of special functions. More precisely, we study the extended Jacobi function via the Riemann-Liouville (fractional) operator. Furthermore, we define a generalized extended Jacobi function which is a solution of the fractional differential equation.

Throughout the paper, we consider the Riemann-Liouville fractional derivative of a function with order defined by

where , , and

is the Riemann-Liouville fractional integral of with order . Here, denotes the classical gamma function. It is easy to see that the fractional derivative of the power function is given by

where . We know from [3] that if is a continuous function in and has continuous derivatives in , then the fractional derivative of the product , that is, the Leibniz rule, is given as follows:

Furthermore, according to Jumarie [4], if both of the functions and from into itself have derivatives of order , one has the chain derivative rule for fractional calculus:

It is well known that the classical Gauss differential equation is given as follows:

As usual, (6) has a solution of the hypergeometric function defined by

where is the Pochhammer symbol

Furthermore, we have the following transformation for the hypergeometric functions

where .

Fujiwara [5] studied the polynomial which is called the extended Jacobi polynomial (EJP) and defined by the Rodrigues formula

where and . The EJPs are orthogonal over the interval with respect to the weight function . In fact, it is hold that

where is the Kronecker delta and .

2. Extended Jacobi Functions (EJFs)

In this section, we define the extended Jacobi functions (EJFs) and obtain their some significant properties.

Definition 1. Assume that and . The extended Jacobi functions are defined to be as the following Rodrigues formula:

where and is the Riemann-Liouville fractional differentiation operator.

Theorem 2. The explicit form of the EJFs is given by

where and .

Proof. If we use the Leibniz rule (4) in (12), then we have

It follows from the definition of fractional derivative that

From (15) and (14), we get that

Using the fact that

the proof is completed.

Corollary 3. The another explicit form of the EJFs is given by

where and .

Proof. This formula can be proved similar to Theorem 2 by taking the following:

Remark 4. If we get , and in (18), then (18) is reduced to the explicit formula satisfied by the Jacobi functions in [6].

Theorem 5. The extended Jacobi functions have the following representation:

where is the hypergeometric function defined in (7).

Proof. Writing instead of in (13) and using binomial expansion, we obtain

Using the following identity:

we have

Substituting (23) in (21), we get that

which is the desired result.

Remark 6. If we get , and in (20), then (20) is reduced to the hypergeometric function representation satisfying the Jacobi functions in [6].

Corollary 7. The extended Jacobi functions (EJFs) hold the following hypergeometric function:

Proof. Applying (9) to (20), the proof follows.

Remark 8. Taking , and in Corollary 7, we give the following hypergeometric function representation for the Jacobi functions in [6]:

Corollary 9. The extended Jacobi functions can be presented as follows:

Remark 10. Taking , and in Corollary 9, we give the following hypergeometric function representation for the Jacobi functions in [6]:

Corollary 11. The EJFs hold the following hypergeometric function:

Proof. Using (9) and (27), we complete the proof immediately.

Remark 12. Taking , and in Corollary 11, we give the following hypergeometric function representation for the Jacobi functions in [6]:

Theorem 13. For the extended Jacobi functions, one has

Proof. For the proof of (31), it is enough to use Corollaries 7 and 11, respectively.

Theorem 14. The extended Jacobi functions hold the differential equation of second-order

or

Proof. With the help of (6) and (7), the hypergeometric function satisfies

where Writing instead of in the last, we get

Thus, the extended Jacobi functions having the hypergeometric function (20) satisfy the above differential equation. Multiplying (32) by and rearranging, we have the second differential equation.

Theorem 15. The extended Jacobi functions satisfy the following properties:

(i), (ii), (iii), (iv), (v), (vi).

Proof. Consider the following.

(i) By (20), we have

(ii) By (13), we get

(iii) It is enough to take in (13).

(iv) It proof is enough to take in (27).

(v) Using (20) and differentiating with respect to , the result follows.

(vi) Repeating times operation in (v), we obtain (vi).

Corollary 16. As a consequence of Theorem 15(v) and (31), one has the following recurrence relations:

Now, we compare some properties of the extended Jacobi functions with extended Jacobi polynomials.

(i) Rodrigues Formula. Consider the following. Extended Jacobi functions:

Extended Jacobi polynomials:

(ii) Explicit Formula. Consider the following. Extended Jacobi functions:

Extended Jacobi polynomials:

(iii) Hypergeometric Representation. Consider the following. Extended Jacobi functions:

Extended Jacobi polynomials:

(iv) Value at . Consider the following. Extended Jacobi functions:

Extended Jacobi polynomials:

(v) Value at . Consider the following. Extended Jacobi Functions:

Extended Jacobi polynomials:

(vi) Differential Equation. Consider the following. Extended Jacobi functions:

Extended Jacobi polynomials:

(vii) Derivation. Consider the following. Extended Jacobi functions:

Extended Jacobi polynomials:

3. Generalized Extended Jacobi Functions

In this section, we define a fractional extended Jacobi differential equation and its solution which is the generalized extended Jacobi function.

Definition 17 (see [6]). The fractional hypergeometric differential equation is defined as follows:

where .

Definition 18 (see [6]). The fractional hypergeometric matrix function is defined by

where and

and the following property

holds for .

If we take instead of ,   instead of instead of , and in (53), we obtain the generalized extended Jacobi functions and theirs fractional differential equation.

Definition 19. The fractional extended Jacobi matrix differential equation is defined by

where .

Definition 20. The generalized extended Jacobi functions (GEJFs) are defined by

where

and the equation

holds for .

Theorem 21. The generalized extended Jacobi function is a solution of (57).

Proof. Using Definitions 17 and 18 and (5), the theorem can be proved.

References

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