Extended Jacobi Functions via Riemann-Liouville Fractional Derivative

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.


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 ∈ N, − 1 ≤ < , and is the Riemann-Liouville fractional integral of with order − . Here, Γ denotes the classical gamma function. It is easy to see that the fractional derivative of the power function ( ) = is given by where ≥ −1, ≥ 0, and > 0. We know from [3] that if is a continuous function in [0, ] and has + 1 continuous derivatives in [0, ], 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 R into itself have derivatives of order , 0 < < 1, one has the chain derivative rule for fractional calculus: It is well known that the classical Gauss differential equation is given as follows: 2 Abstract and Applied Analysis 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 | − /(1 − )| < 1, | | < 1. Fujiwara [5] studied the polynomial ( , ) ( ; , , ) which is called the extended Jacobi polynomial (EJP) and defined by the Rodrigues formula × Γ ( + + 1) Γ ( + + 1) ) × ( ! ( + + 2 + 1) Γ ( + + + 1) ) −1 ) , , where , is the Kronecker delta and min{R( ), R( )} > −1; , ∈ N 0 := N ∪ {0}.

Extended Jacobi Functions (EJFs)
In this section, we define the extended Jacobi functions (EJFs) and obtain their some significant properties.
Definition 1. Assume that , > −1 and > 0. The extended Jacobi functions are defined to be as the following Rodrigues formula: where ] > 0 and ] / ] is the Riemann-Liouville fractional differentiation operator.

Theorem 2. The explicit form of the EJFs is given by
where , > −1 and > 0.
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 , > −1 and > 0.

Abstract and Applied Analysis 3
Proof. This formula can be proved similar to Theorem 2 by taking the following: Remark 4. If we get = 1, = −1, and = 1/2 in (18), then (18) is reduced to the explicit formula satisfied by the − Jacobi functions in [6].
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.
(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.

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