Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 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 ErkuG-Duman
Gazi University, Faculty of Science, Department of Mathematics, Teknikokullar, 06500 Ankara, Turkey
Correspondence should be addressed to Esra Erkuş-Duman; eduman@gazi.edu.tr
Received 19 January 2013; Accepted 3 April 2013
Academic Editor: Mohamed Kamal Aouf
Copyright © 2013 B. Çekim and E. 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 defined 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 RiemannLiouville fractional derivative of a function 𝑓 with order 𝜇
defined by
𝐷𝜇 𝑓 (𝑡) := 𝐷𝑚 [𝐽𝑚−𝜇 𝑓 (𝑡)] ,
(1)
𝑡
1
∫ (𝑡 − 𝜏)𝑚−𝜇−1 𝑓 (𝜏) 𝑑𝜏
Γ (𝑚 − 𝜇) 0
𝐷𝜇 𝑡𝛼 =
Γ (𝛼 + 1) 𝛼−𝜇
𝑡 ,
Γ (𝛼 − 𝜇 + 1)
(2)
is the Riemann-Liouville fractional integral of 𝑓 with order
𝑚 − 𝜇. Here, Γ denotes the classical gamma function. It is
(3)
where 𝛼 ≥ −1, 𝜇 ≥ 0, and 𝑡 > 0. We know from [3]
that if 𝑓 is a continuous function in [0, 𝑡] and 𝜑 has 𝑛 + 1
continuous derivatives in [0, 𝑡], then the fractional derivative
of the product 𝜑𝑓, that is, the Leibniz rule, is given as follows:
∞
𝜇
𝐷𝜇 [𝜑 (𝑡) 𝑓 (𝑡)] = ∑ ( ) 𝜑(𝑘) (𝑡) 𝐷𝜇−𝑘 𝑓 (𝑡) .
𝑘
(4)
𝑘=0
Furthermore, according to Jumarie [4], if both of the functions 𝑓(𝑥) and 𝑥(𝑡) from R into itself have derivatives of order
𝛼, 0 < 𝛼 < 1, one has the chain derivative rule for fractional
calculus:
𝑑𝛼 𝑓 (𝑥) 𝑑𝑥 𝛼
𝑑𝛼
𝑓 (𝑥 (𝑡)) =
( ) .
𝛼
𝑑𝑡
𝑑𝑥𝛼
𝑑𝑡
where 𝑚 ∈ N, 𝑚 − 1 ≤ 𝜇 < 𝑚, and
𝐽𝑚−𝜇 𝑓 (𝑡) :=
easy to see that the fractional derivative of the power function
𝑓(𝑡) = 𝑡𝛼 is given by
(5)
It is well known that the classical Gauss differential equation
is given as follows:
𝑥 (1 − 𝑥) 𝑦󸀠󸀠 + [𝑐 − (𝑎 + 𝑏 + 1) 𝑥] 𝑦󸀠 − 𝑎𝑏𝑦 = 0.
(6)
2
Abstract and Applied Analysis
As usual, (6) has a solution of the hypergeometric function
defined by
∞
(𝑎)𝑘 (𝑏)𝑘 𝑘
𝑥 ;
(𝑐)𝑘 𝑘!
𝑘=0
𝐹 (𝑎, 𝑏; 𝑐; x) = ∑
|𝑥| < 1,
(7)
where (𝜆)𝑘 is the Pochhammer symbol
(𝜆)𝑘 = 𝜆 (𝜆 + 1) ⋅ ⋅ ⋅ (𝜆 + 𝑘 − 1) ,
Theorem 2. The explicit form of the EJFs is given by
∞
]+𝛼 ]+𝛽
)(
) (𝑥 − 𝑎)𝑘 (𝑥 − 𝑏)]−𝑘 ,
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = 𝑐] ∑ (
]−𝑘
𝑘
𝑘=0
(13)
where 𝛼, 𝛽 > −1 and 𝑐 > 0.
(𝜆)0 = 1.
(8)
Furthermore, we have the following transformation for the
hypergeometric functions
−𝑥
𝐹 (𝑎, 𝑏; 𝑐; 𝑥) = (1 − 𝑥) 𝐹 (𝑎, 𝑐 − 𝑏; 𝑐;
),
1−𝑥
−𝑎
Proof. If we use the Leibniz rule (4) in (12), then we have
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
=
(9)
∞
]
× ∑ ( ) {𝐷𝑥𝑘 [(𝑏 − 𝑥)𝛽+] ]} {𝐷𝑥]−𝑘 [(𝑥 − 𝑎)𝛼+] ]} .
𝑘
where | − 𝑥/(1 − 𝑥)| < 1, |𝑥| < 1.
Fujiwara [5] studied the polynomial 𝐹𝑛(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
which is called the extended Jacobi polynomial (EJP) and
defined by the Rodrigues formula
𝐹𝑛(𝛼,𝛽)
(−𝑐)]
(𝑥 − 𝑎)−𝛼 (𝑏 − 𝑥)−𝛽
Γ (] + 1)
𝑘=0
(14)
It follows from the definition of fractional derivative that
(−𝑐)𝑛
(𝑥 − 𝑎)−𝛼 (𝑏 − 𝑥)−𝛽
(𝑥; 𝑎, 𝑏, 𝑐) =
𝑛!
𝑑𝑛
× 𝑛 {(𝑥 − 𝑎)𝑛+𝛼 (𝑏 − 𝑥)𝑛+𝛽 } ,
𝑑𝑥
(10)
where 𝛼, 𝛽 > −1 and 𝑐 > 0. The EJPs 𝐹𝑛(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) are
orthogonal over the interval (𝑎, 𝑏) with respect to the weight
function 𝜔(𝑥; 𝑎, 𝑏) = (𝑥 − 𝑎)𝛼 (𝑏 − 𝑥)𝛽 . In fact, it is hold that
𝑏
𝐷𝑥]−𝑘 [(𝑥 − 𝑎)𝛼+] ]
∞
Γ (] + 𝛼 − 𝑟 + 1) 𝑘+𝛼−𝑟 (15)
]+𝛼 𝑟
.
= ∑(
𝑥
) 𝑎 (−1)𝑟
𝑟
Γ (𝑘 + 𝛼 − 𝑟 + 1)
𝑟=0
From (15) and (14), we get that
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) =
∫ (𝑥 − 𝑎)𝛼 (𝑏 − 𝑥)𝛽 𝐹𝑛(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) 𝐹𝑚(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) 𝑑𝑥
∞
Γ (𝛽 + ] + 1)
]
× ∑ ( ) (−1)𝑘
𝑘
Γ (𝛽 + ] − 𝑘 + 1)
𝑎
= ( (𝑐𝑚+𝑛 (−1)𝛼+𝛽+1 (𝑎 − 𝑏)𝑚+𝑛+𝛼+𝛽+1
𝑘=0
(11)
× (𝑏 − 𝑥)𝛽+(]−𝑘)
× Γ (𝛼 + 𝑛 + 1) Γ (𝛽 + 𝑛 + 1) )
∞
Γ (] + 𝛼 − 𝑟 + 1)
]+𝛼 𝑟
×∑(
) 𝑎 (−1)𝑟
𝑟
Γ
(𝑘 + 𝛼 − 𝑟 + 1)
𝑟=0
−1
× (𝑛! (𝛼 + 𝛽 + 2𝑛 + 1) Γ (𝛼 + 𝛽 + 𝑛 + 1) ) ) 𝛿𝑚,𝑛 ,
× 𝑥𝑘+𝛼−𝑟 .
where 𝛿𝑚,𝑛 is the Kronecker delta and min{R(𝛼), R(𝛽)} >
−1; 𝑚, 𝑛 ∈ N0 := N ∪ {0}.
In this section, we define the extended Jacobi functions (EJFs)
and obtain their some significant properties.
Definition 1. Assume that 𝛼, 𝛽 > −1 and 𝑐 > 0. The extended
Jacobi functions are defined to be as the following Rodrigues
formula:
]
(−𝑐)
(𝑥 − 𝑎)−𝛼 (𝑏 − 𝑥)−𝛽
Γ (] + 1)
𝑑]
× ] {(𝑥 − 𝑎)]+𝛼 (𝑏 − 𝑥)]+𝛽 } ,
𝑑𝑥
(16)
Using the fact that
2. Extended Jacobi Functions (EJFs)
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) =
(−𝑐)]
(𝑥 − 𝑎)−𝛼 (𝑏 − 𝑥)−𝛽
Γ (] + 1)
∞
Γ (𝛼 + 𝑘 + 1)
𝑎𝑟 (−1)𝑟 𝑥𝛼+𝑘−𝑟 = (𝑥 − 𝑎)𝛼+𝑘 ,
Γ
+
𝛼
−
𝑟
+
1)
𝑟!
(𝑘
𝑟=0
∑
(17)
the proof is completed.
Corollary 3. The another explicit form of the EJFs is given by
∞
(12)
where ] > 0 and 𝑑] /𝑑𝑥] is the Riemann-Liouville fractional
differentiation operator.
]+𝛼
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = 𝑐] ∑ (−1)𝑘 (
)
𝑘
𝑘=0
]+𝛽
×(
) (𝑥 − 𝑎)]−𝑘 (𝑥 − 𝑏)𝑘 ,
]−𝑘
where 𝛼, 𝛽 > −1 and 𝑐 > 0.
(18)
Abstract and Applied Analysis
3
Proof. This formula can be proved similar to Theorem 2 by
taking the following:
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
=
(19)
∞
∞
Remark 4. If we get 𝑎 = 1, 𝑏 = −1, and 𝑐 = 1/2 in (18),
then (18) is reduced to the explicit formula satisfied by the 𝑔−
Jacobi functions in [6].
Theorem 5. The extended Jacobi functions have the following
representation:
]
× 𝐹 (−], 𝛼 + 𝛽 + ] + 1; 𝛼 + 1;
𝑎−𝑥
),
𝑎−𝑏
(20)
where 𝐹 is the hypergeometric function defined in (7).
Proof. Writing (𝑥 − 𝑎) + (𝑎 − 𝑏) instead of (𝑥 − 𝑏) in (13) and
using binomial expansion, we obtain
∞
]+𝛼 ]+𝛽
= 𝑐] ∑ (
)(
) (𝑥 − 𝑎)𝑘 (𝑥 − 𝑎 + 𝑎 − 𝑏)]−𝑘
]−𝑘
𝑘
Corollary 7. The extended Jacobi functions (EJFs) hold the
following hypergeometric function:
]+𝛼
)
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = 𝑐] (𝑎 − 𝑏)] (
]
]+𝛼 ]+𝛽
=𝑐 ∑(
)(
) (𝑥 − 𝑎)𝑘
]−𝑘
𝑘
𝑘=0
𝑥−1
]+𝛼 𝑥+1 ]
) 𝐹 (−],−𝛽 − ]; 𝛼 +1;
).
)(
𝑃](𝛼,𝛽) (𝑥) = (
]
2
𝑥+1
(26)
∞
]−𝑘
×∑(
) (𝑥 − 𝑎)𝑟 (𝑎 − 𝑏)]−𝑘−𝑟
𝑟
𝑟=0
Corollary 9. The extended Jacobi functions can be presented
as follows:
∞ 𝑟
]+𝛼 ]+𝛽
= 𝑐] ∑ ∑ (
)(
)
]−𝑘
𝑘
𝑟=0
𝑘=0
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = (−1)] 𝑐] (𝑎 − 𝑏)] (
]−𝑘
×(
) (𝑥 − 𝑎)𝑟 (𝑎 − 𝑏)]−𝑟
𝑟−𝑘
]−𝑟
= 𝑐 ∑(𝑥 − 𝑎) (𝑎 − 𝑏)
𝑥−𝑎
𝑥−𝑏 ]
) 𝐹 (−], −𝛽 − ]; 𝛼 + 1;
).
𝑎−𝑏
𝑥−𝑏
(25)
Remark 8. Taking 𝑎 = 1, 𝑏 = −1, and 𝑐 = 1/2 in
Corollary 7, we give the following hypergeometric function
representation for the 𝑔− Jacobi functions in [6]:
∞
𝑟=0
Remark 6. If we get 𝑎 = 1, 𝑏 = −1, and 𝑐 = 1/2 in
(20), then (20) is reduced to the hypergeometric function
representation satisfying the 𝑔− Jacobi functions in [6].
Proof. Applying (9) to (20), the proof follows.
𝑘=0
𝑟
𝑎−𝑥
]+𝛼
= 𝑐] (𝑎−𝑏)] (
),
) 𝐹 (−], 𝛼+ 𝛽+ ]+ 1; 𝛼 + 1;
]
𝑎−𝑏
(24)
×(
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
∞
∞
(−])𝑟 (𝛽+𝛼 +] +1)𝑟 𝑎 − 𝑥 𝑟
]+𝛼
(
)
)∑
= 𝑐] (𝑎−𝑏)] (
] 𝑟=0
𝑟!(𝛼 + 1)𝑟
𝑎−𝑏
which is the desired result.
]+𝛼
)
(𝑥; 𝑎, 𝑏, 𝑐) = 𝑐 (𝑎 − 𝑏) (
]
]
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
]+𝛼 𝛽+𝛼+]+𝑟
= 𝑐] ∑(𝑥 − 𝑎)𝑟 (𝑎 − 𝑏)]−𝑟 (
)(
)
]−𝑟
𝑟
𝑟=0
𝑘=0
]
(23)
Substituting (23) in (21), we get that
]
× ∑ ( ) {𝐷𝑥𝑘 [(𝑥 − 𝑎)𝛼+] ]} {𝐷𝑥]−𝑘 [(𝑏 − 𝑥)𝛽+] ]} .
𝑘
]
𝑟
𝑟+𝛼 ]+𝛽
𝛽+𝛼+]+𝑟
)(
).
(
)= ∑(
𝑟−𝑘
𝑘
𝑟
𝑘=0
(−𝑐)]
(𝑥 − 𝑎)−𝛼 (𝑏 − 𝑥)−𝛽
Γ (] + 1)
𝐹](𝛼,𝛽)
we have
× 𝐹 (−], 𝛼+𝛽+] +1; 𝛽 + 1;
𝑟
]+𝛼 ]+𝛽 ]−𝑘
)(
)(
)
∑(
]−𝑘
𝑘
𝑟−𝑘
𝑘=0
∞
𝑟
]+𝛼
𝑟+𝛼 ]+𝛽
)∑(
)(
).
= 𝑐] ∑(𝑥 − 𝑎)𝑟 (𝑎 − 𝑏)]−𝑟 (
]
−
𝑟
𝑟−𝑘
𝑘
𝑟=0
𝑘=0
(21)
Using the following identity:
(1 − 𝑥)𝛽+] (1 − 𝑥)𝛼+𝑟 = (1 − 𝑥)𝛽+𝛼+]+𝑟 ,
]+𝛽
)
]
(22)
𝑏−𝑥
).
𝑏−𝑎
(27)
Remark 10. Taking 𝑎 = 1, 𝑏 = −1, and 𝑐 = 1/2 in
Corollary 9, we give the following hypergeometric function
representation for the 𝑔− Jacobi functions in [6]:
𝑃](𝛼,𝛽) (𝑥) = (−1)] (
]+𝛽
)
]
1+𝑥
).
× 𝐹 (−], 𝛼 + 𝛽 + ] + 1; 𝛽 + 1;
2
(28)
4
Abstract and Applied Analysis
Corollary 11. The EJFs hold the following hypergeometric
function:
]+𝛽 𝑥−𝑎 ]
)
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = (−1)] 𝑐] (𝑎 − 𝑏)] (
)(
]
𝑏−𝑎
× 𝐹 (−], −𝛼 − ]; 𝛽 + 1;
𝑥−𝑏
).
𝑥−𝑎
where 0 ≤ |𝑡| < 1. Writing (𝑎 − 𝑥)/(𝑎 − 𝑏) instead of 𝑡 in the
last, we get
(𝑥−𝑎) (𝑏−𝑥) 𝑌󸀠󸀠 (𝑥)+[𝑏 (𝛼+1)+𝑎 (𝛽+1)−𝑥 (𝛼+𝛽+2)]
(29)
× 𝑌󸀠 (𝑥) +] (𝛼 + 𝛽 + ] + 1) 𝑌 (𝑥) = 0.
(35)
Proof. Using (9) and (27), we complete the proof immediately.
Remark 12. Taking 𝑎 = 1, 𝑏 = −1, and 𝑐 = 1/2 in
Corollary 11, we give the following hypergeometric function
representation for the 𝑔− Jacobi functions in [6]:
𝑥+ 1
]+𝛽 𝑥−1 ]
) 𝐹 (−],−𝛼−]; 𝛽+ 1;
).
𝑃](𝛼,𝛽) (𝑥) = (
)(
]
2
𝑥−1
(30)
Theorem 13. For the extended Jacobi functions, one has
Thus, the extended Jacobi functions having the hypergeometric function (20) satisfy the above differential equation.
Multiplying (32) by (𝑥 − 𝑎)𝛼 (𝑏 − 𝑥)𝛽 and rearranging, we have
the second differential equation.
Theorem 15. The extended Jacobi functions satisfy the following properties:
(i) lim] → 𝑛 𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = 𝐹𝑛(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐),
(ii) 𝐹](𝛼,𝛽) (−𝑥; 𝑎, 𝑏, 𝑐) = (−1)] 𝐹](𝛼,𝛽) (𝑥; −𝑎, −𝑏, 𝑐),
(iii) 𝐹](𝛼,𝛽) (𝑎; 𝑎, 𝑏, 𝑐) = 𝑐] (𝑎 − 𝑏)] ( ]+𝛼
] ),
(iv) 𝐹](𝛼,𝛽) (𝑏; 𝑎, 𝑏, 𝑐) = (−1)] 𝑐] (𝑎 − 𝑏)] ( ]+𝛽
] ),
(v) (𝑑/𝑑𝑥) 𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
(𝛼+1,𝛽+1)
𝐹]−1
(𝑥; 𝑎, 𝑏, 𝑐),
𝑑 (𝛼,𝛽)
𝐹
(𝑥; 𝑎, 𝑏, 𝑐)
(𝑥 − 𝑏)
𝑑𝑥 ]
(𝛼+1,𝛽)
(𝑥; 𝑎, 𝑏, 𝑐) ,
𝑑 (𝛼,𝛽)
𝐹
(𝑥; 𝑎, 𝑏, 𝑐)
(𝑥 − 𝑎)
𝑑𝑥 ]
(31)
Proof. Consider the following.
(i) By (20), we have
lim 𝐹(𝛼,𝛽)
]→𝑛 ]
= ]𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
(𝛼,𝛽+1)
− 𝑐 (𝑎 − 𝑏) (𝛼 + ]) 𝐹]−1
𝑐(] + 𝛼 + 𝛽 + 1)
(vi) (𝑑𝑘 /𝑑𝑥𝑘 ) 𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = 𝑐𝑘 (] + 𝛼 + 𝛽 + 1)𝑘
(𝛼+𝑘,𝛽+𝑘)
𝐹]−𝑘
(𝑥; 𝑎, 𝑏, 𝑐) for 𝑘 ∈ N.
= ]𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
+ 𝑐 (𝑎 − 𝑏) (𝛽 + ]) 𝐹]−1
=
(𝑥; 𝑎, 𝑏, 𝑐)
]+𝛼
= lim 𝑐] (𝑎 − 𝑏)] (
)
]→𝑛
]
(𝑥; 𝑎, 𝑏, 𝑐) .
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
× 𝐹 (−], 𝛼 + 𝛽 + ] + 1; 𝛼 + 1;
𝑎−𝑥
)
𝑎−𝑏
𝑎−𝑥
𝑛+𝛼
= 𝑐𝑛 (𝑎 − 𝑏)𝑛 (
)
) 𝐹 (−𝑛, 𝛼 + 𝛽 + 𝑛 + 1; 𝛼 + 1;
𝑛
𝑎−𝑏
= 𝐹𝑛(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) .
(𝑥 − 𝑎) (𝑏 − 𝑥) 𝑌󸀠󸀠 (𝑥)
+ [𝑏 (𝛼 + 1) + 𝑎 (𝛽 + 1) − 𝑥 (𝛼 + 𝛽 + 2)]
(32)
󸀠
× 𝑌 (𝑥) +] (𝛼 + 𝛽 + ] + 1) 𝑌 (𝑥) = 0
(36)
(ii) By (13), we get
𝐹](𝛼,𝛽) (−𝑥; 𝑎, 𝑏, 𝑐)
or
∞
]+𝛼 ]+𝛽
= 𝑐] ∑ (
)(
) (−𝑥 − 𝑎)𝑘 (−𝑥 − 𝑏)]−𝑘
]−𝑘
𝑘
𝑑
[(𝑥 − 𝑎)𝛼+1 (𝑏 − 𝑥)𝛽+1 𝑌󸀠 (𝑥)]
𝑑𝑥
𝛼
(33)
𝛽
+ ] (𝛼 + 𝛽 + ] + 1) (𝑥 − 𝑎) (𝑏 − 𝑥) 𝑌 (𝑥) = 0.
Proof. With the help of (6) and (7), the hypergeometric
function 𝐹(−], ] + 𝛼 + 𝛽 + 1; 𝛼 + 1; 𝑡) satisfies
𝑡 (1−𝑡) 𝐹󸀠󸀠 +[𝛼+1−𝑡 (𝛼+𝛽+2)] 𝐹󸀠 + ] (𝛼 + 𝛽 + ] + 1) 𝐹 = 0,
(34)
(37)
𝑘=0
= (−1)] 𝐹](𝛼,𝛽) (𝑥; −𝑎, −𝑏, 𝑐) .
(iii) It is enough to take 𝑥 = 𝑎 in (13).
(iv) It proof is enough to take 𝑥 = 𝑏 in (27).
(v) Using (20) and differentiating with respect to 𝑥, the
result follows.
(vi) Repeating 𝑘 times operation in (v), we obtain
(vi).
Abstract and Applied Analysis
5
Corollary 16. As a consequence of Theorem 15(v) and (31), one
has the following recurrence relations:
𝑑 (𝛼,𝛽)
𝐹
(𝑥; 𝑎, 𝑏, 𝑐)
𝑑𝑥 ]
(𝛼+1,𝛽)
= 𝑐 {(𝛽 + ]) 𝐹]−1
(𝛼,𝛽+1)
(𝛼 + 𝛽 + (] +
2)) 𝐹](𝛼+1,𝛽+1)
(𝑥; 𝑎, 𝑏, 𝑐)} ,
(38)
(𝑥; 𝑎, 𝑏, 𝑐)
= {(𝛽 + (] + 1)) 𝐹](𝛼+1,𝛽) (𝑥; 𝑎, 𝑏, 𝑐)
+ (𝛼 + (] +
1)) 𝐹](𝛼,𝛽+1)
(𝑥; 𝑎, 𝑏, 𝑐)} .
(i) Rodrigues Formula. Consider the following.
Extended Jacobi functions:
(−𝑐)]
(𝑥; 𝑎, 𝑏, 𝑐) =
(𝑥 − 𝑎)−𝛼 (𝑏 − 𝑥)−𝛽
Γ (] + 1)
𝑑]
× ] {(𝑥 − 𝑎)]+𝛼 (𝑏 − 𝑥)]+𝛽 } .
𝑑𝑥
𝑑𝑛
× 𝑛 {(𝑥 − 𝑎)𝑛+𝛼 (𝑏 − 𝑥)𝑛+𝛽 } .
𝑑𝑥
(39)
(40)
]+𝛽
).
𝐹](𝛼,𝛽) (𝑏; 𝑎, 𝑏, 𝑐) = (−1)] 𝑐] (𝑎 − 𝑏)] (
]
𝑛+𝛽
).
𝐹𝑛(𝛼,𝛽) (𝑏; 𝑎, 𝑏, 𝑐) = (−𝑐)𝑛 (𝑎 − 𝑏)𝑛 (
𝑛
+ [𝑏 (𝛼 + 1) + 𝑎 (𝛽 + 1) − 𝑥 (𝛼 + 𝛽 + 2)]
∞
𝑘=0
(41)
]+𝛽
×(
) (𝑥 − 𝑎)𝑘 (𝑥 − 𝑏)]−𝑘 .
𝑘
(46)
(47)
(48)
(49)
×𝑌󸀠 (𝑥) + ] (𝛼 + 𝛽 + ] + 1) 𝑌 (𝑥) = 0.
Extended Jacobi polynomials:
(𝑥 − 𝑎) (𝑏 − 𝑥) 𝑌󸀠󸀠 (𝑥)
Extended Jacobi polynomials:
+ [𝑏 (𝛼 + 1) + 𝑎 (𝛽 + 1) − 𝑥 (𝛼 + 𝛽 + 2)]
𝑛+𝛼
)
𝑛−𝑘
(42)
𝑛+𝛽
×(
) (𝑥 − 𝑎)𝑘 (𝑥 − 𝑏)𝑛−𝑘 .
𝑘
(iii) Hypergeometric Representation. Consider the following.
Extended Jacobi functions:
]+𝛼
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = (𝑐)] (a − 𝑏)] (
)
]
× 𝐹 (−], 𝛼 + 𝛽 + ] + 1; 𝛼 + 1;
𝑛+𝛼
𝐹𝑛(𝛼,𝛽) (𝑎; 𝑎, 𝑏, 𝑐) = 𝑐𝑛 (𝑎 − 𝑏)𝑛 (
).
𝑛
(𝑥 − 𝑎) (𝑏 − 𝑥) 𝑌󸀠󸀠 (𝑥)
]+𝛼
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = 𝑐] ∑ (
)
]−𝑘
𝑘=0
Extended Jacobi polynomials:
(vi) Differential Equation. Consider the following.
Extended Jacobi functions:
(ii) Explicit Formula. Consider the following.
Extended Jacobi functions:
∞
(45)
Extended Jacobi polynomials:
(−𝑐)𝑛
(𝑥 − 𝑎)−𝛼 (𝑏 − 𝑥)−𝛽
𝑛!
𝐹](𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = 𝑐𝑛 ∑ (
(iv) Value at 𝑥 = 𝑎. Consider the following.
Extended Jacobi functions:
(v) Value at 𝑥 = 𝑏. Consider the following.
Extended Jacobi Functions:
Extended Jacobi polynomials:
𝐹𝑛(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) =
𝑎−𝑥
).
𝑎−𝑏
(44)
]+𝛼
).
𝐹](𝛼,𝛽) (𝑎; 𝑎, 𝑏, 𝑐) = 𝑐] (𝑎 − 𝑏)] (
]
Now, we compare some properties of the extended Jacobi
functions with extended Jacobi polynomials.
𝐹](𝛼,𝛽)
𝑛+𝛼
)
𝐹𝑛(𝛼,𝛽) (𝑥; 𝑎, 𝑏, 𝑐) = (𝑐)𝑛 (𝑎 − 𝑏)𝑛 (
𝑛
× 𝐹 (−𝑛, 𝛼 + 𝛽 + 𝑛 + 1; 𝛼 + 1;
(𝑥; 𝑎, 𝑏, 𝑐)
+ (𝛼 + ]) 𝐹]−1
Extended Jacobi polynomials:
(50)
× 𝑌󸀠 (𝑥) + 𝑛 (𝛼 + 𝛽 + 𝑛 + 1) 𝑌 (𝑥) = 0.
(vii) Derivation. Consider the following.
Extended Jacobi functions:
𝑑 (𝛼,𝛽)
(𝛼+1,𝛽+1)
𝐹
(𝑥; 𝑎, 𝑏, 𝑐) = 𝑐 (] + 𝛼 + 𝛽 + 1) 𝐹]−1
(𝑥; 𝑎, 𝑏, 𝑐) .
𝑑𝑥 ]
(51)
Extended Jacobi polynomials:
𝑎−𝑥
).
𝑎−𝑏
(43)
𝑑 (𝛼,𝛽)
1
(𝛼+1,𝛽+1)
𝐹𝑛 (𝑥; 𝑎, 𝑏, 𝑐) = (𝑛 + 𝛼 + 𝛽 + 1) 𝐹𝑛−1
(𝑥; 𝑎, 𝑏, 𝑐) .
𝑑𝑥
2
(52)
6
Abstract and Applied Analysis
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:
𝑡𝜇 (1−𝑡𝜇 ) 𝐷2𝜇 𝑌 (𝑡)+[𝑐− (𝑎+ 𝑏+1) 𝑡𝜇 ] 𝐷𝜇 [𝑌 (𝑡)]−𝑎𝑏𝑌 (𝑡) = 0,
(53)
Definition 20. The generalized extended Jacobi functions
(GEJFs) are defined by
𝜇
2 𝐹1
(−], 𝑎 + 𝑏 + (] + 1) ; 𝑎 + 1;
= 𝑦0 (
𝑎 − 𝑡 𝑝 ∞ [ 𝑘−1
𝑎 − 𝑡 𝑝+𝑘𝜇
−1
(𝑝)] (
,
) + ∑ 𝑦0 ∏𝑔𝑗 (𝑝) 𝑓𝑗+1
)
𝑎−𝑏
𝑎−𝑏
𝑗=0
𝑘=1
]
[
(58)
where
𝑓𝑘 (𝑝) =
where 0 < 𝜇 ≤ 1.
Definition 18 (see [6]). The fractional hypergeometric matrix
function is defined by
𝜇
2 𝐹1
𝑝
(𝑎, 𝑏; 𝑐; 𝑡) = 𝑦0 𝑡 +
∞
𝑘−1
−1
∑ 𝑦0 [∏𝑔𝑗 (𝑝) 𝑓𝑗+1
𝑗=0
𝑘=1
[
(𝑝)] 𝑡
]
𝑘𝜇+𝑝
,
𝑎−𝑡
)
𝑎−𝑏
Γ ((𝑘𝜇 + 1) + 𝑝)
Γ ((𝑘 − 2) 𝜇 + 1 + 𝑝)
+ (𝑎 + 1)
𝑔𝑘 (𝑝) =
Γ ((𝑘𝜇 + 1) + 𝑝)
− ] (𝑎 + 𝑏 + (] + 1))
Γ ((𝑘 − 2) 𝜇 + 1 + 𝑝)
+ (𝑎 + 𝑏 + 2)
(54)
Γ ((𝑘𝜇 + 1) + 𝑝)
,
Γ ((𝑘 − 1) 𝜇 + 1 + 𝑝)
(59)
Γ ((𝑘𝜇 + 1) + 𝑝)
,
Γ ((𝑘 − 1) 𝜇 + 1 + 𝑝)
where 0 < 𝜇 ≤ 1 and
Γ ((𝑘𝜇 + 1) + 𝑝)
Γ ((𝑘𝜇 + 1) + 𝑝)
𝑓𝑘 (𝑝) =
+𝑐
,
Γ ((𝑘 − 2) 𝜇 + 1 + 𝑝)
Γ ((𝑘 − 1) 𝜇 + 1 + 𝑝)
𝑔𝑘 (𝑝) =
Γ ((𝑘𝜇 + 1) + 𝑝)
+ 𝑎𝑏
Γ ((𝑘 − 2) 𝜇 + 1 + 𝑝)
+ (𝑎 + 𝑏 + 1)
Γ ((𝑘𝜇 + 1) + 𝑝)
,
Γ ((𝑘 − 1) 𝜇 + 1 + 𝑝)
(56)
holds for 𝜌 > −1.
If we take −] instead of 𝑎, 𝑎 + 𝑏 + ] + 1 instead of 𝑏, 𝑎 + 1
instead of 𝑐, and 𝑡 → (𝑎 − 𝑡)/(𝑎 − 𝑏) 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
(−1)2𝜇 (𝑎 − 𝑡)𝜇 [(𝑎 − 𝑏)𝜇 − (𝑎 − 𝑡)𝜇 ] 𝐷2𝜇 𝑌 (𝑡)
+ (−1)𝜇 [(𝑎 + 1) (𝑎 − 𝑏)𝜇 − (𝑎 + 𝑏 + 2) (𝑎 − 𝑡)𝜇 ]
× 𝐷𝜇 [𝑌 (𝑡)] + ] (𝑎 + 𝑏 + ] + 1) 𝑌 (𝑡) = 0,
where 0 < 𝜇 ≤ 1/2.
Γ (1 + 𝑝)
Γ (1 + 𝑝)
+ (𝑎 + 1)
=0
Γ (−2𝜇 + 1 + 𝑝)
Γ (−𝜇 + 1 + 𝑝)
(60)
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.
and the following property
Γ (1 + 𝑝)
Γ (1 + 𝑝)
+𝑐
=0
Γ (−2𝜇 + 1 + 𝑝)
Γ (−𝜇 + 1 + 𝑝)
𝑓0 (𝑝) =
holds for 𝜌 > −1.
(55)
𝑓0 (𝑝) =
and the equation
(57)
References
[1] K. S. Miller and B. Ross, An Introduction to the Fractional
Calculus and Fractional Differential Equations, John Wiley &
Sons, New York, NY, USA, 1993.
[2] K. B. Oldham and J. Spanier, The Fractional Calculus; Theory
and Applications of Differentiation and Integration to Arbitrary
Order, Mathematics in Science and Engineering, Academic
Press, New York, NY, USA, 1974.
[3] I. Podlubny, Fractional Differential Equations, Mathematics in
Science and Engineering, Academic Press, San Diego, Calif,
USA, 1999.
[4] G. Jumarie, “Fractional Euler’sintegral of first and second
kinds. Application to fractional Hermite’s polynomials and to
probability density of fractional orders,” Journal of Applied
Mathematics & Informatics, vol. 28, no. 1-2, pp. 257–273, 2010.
[5] I. Fujiwara, “A unified presentation of classical orthogonal
polynomials,” Mathematica Japonica, vol. 11, pp. 133–148, 1966.
[6] S. P. Mirevski, L. Boyadjiev, and R. Scherer, “On the RiemannLiouville fractional calculus, 𝑔-Jacobi functions and 𝐹-Gauss
functions,” Applied Mathematics and Computation, vol. 187, no.
1, pp. 315–325, 2007.
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