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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 350182, 6 pages
Extended Jacobi Functions via Riemann-Liouville Fractional Derivative
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.
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.
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  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 , 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  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 .
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:
Theorem 5. The extended Jacobi functions have the following representation: where is the hypergeometric function defined in (7).
Corollary 7. The extended Jacobi functions (EJFs) hold the following hypergeometric function:
Corollary 9. The extended Jacobi functions can be presented as follows:
Corollary 11. The EJFs hold the following hypergeometric function:
Theorem 13. For the extended Jacobi functions, one has
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).
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 ). The fractional hypergeometric differential equation is defined as follows: where .
Definition 18 (see ). 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).
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