Extended Jacobi Functions via Riemann-Liouville Fractional Derivative

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 𝑓 with order 𝜇 defined by 𝐷^𝜇𝑓(𝑡)∶=𝐷^𝑚𝐽^𝑚−𝜇𝑓(𝑡), (1) where 𝑚∈ℕ, 𝑚−1≤𝜇<𝑚, and 𝐽^𝑚−𝜇𝑓(𝑡)1∶=Γ𝑚−𝜇^𝑡₀(𝑡−𝜏)^{𝑚−𝜇−1}𝑓(𝜏)𝑑𝜏 (2) is the Riemann-Liouville fractional integral of 𝑓 with order 𝑚−𝜇. Here, Γ denotes the classical gamma function. It is easy to see that the fractional derivative of the power function 𝑓(𝑡)=𝑡^𝛼 is given by 𝐷^𝜇𝑡^𝛼=Γ(𝛼+1)Γ𝑡𝛼−𝜇+1^𝛼−𝜇, (3) where 𝛼≥−1,𝜇≥0,and𝑡>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: 𝐷^𝜇𝜑(𝑡)𝑓(𝑡)=^∞_𝑘=0𝜇𝑘𝜑^(𝑘)(𝑡)𝐷^𝜇−𝑘𝑓(𝑡). (4) 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: 𝑑^𝛼𝑑𝑡^𝛼𝑓(𝑥(𝑡=𝑑))^𝛼𝑓(𝑥)𝑑𝑥^𝛼𝑑𝑥𝑑𝑡^𝛼. (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) As usual, (6) has a solution of the hypergeometric function defined by 𝐹(𝑎,𝑏;𝑐;x)=^∞_𝑘=0(𝑎)_𝑘(𝑏)_𝑘(𝑐)_𝑘𝑥𝑘!^𝑘;|𝑥|<1, (7) where (𝜆)_𝑘 is the Pochhammer symbol (𝜆)_𝑘=𝜆(𝜆+1)⋯(𝜆+𝑘−1),(𝜆)₀=1. (8) Furthermore, we have the following transformation for the hypergeometric functions 𝐹(𝑎,𝑏;𝑐;𝑥)=(1−𝑥)^−𝑎𝐹𝑎,𝑐−𝑏;𝑐;−𝑥,1−𝑥 (9) where |−𝑥/(1−𝑥)|<1,|𝑥|<1.

Fujiwara [5] studied the polynomial 𝐹_𝑛^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐) which is called the extended Jacobi polynomial (EJP) and defined by the Rodrigues formula 𝐹_𝑛^(𝛼,𝛽)()=()𝑥;𝑎,𝑏,𝑐−𝑐^𝑛()𝑛!𝑥−𝑎^−𝛼(𝑏−𝑥)^−𝛽×𝑑^𝑛𝑑𝑥^𝑛(𝑥−𝑎)^𝑛+𝛼(𝑏−𝑥)^𝑛+𝛽, (10) where 𝛼,𝛽>−1 and 𝑐>0. The EJPs 𝐹_𝑛^(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_𝑎(𝑥−𝑎)^𝛼(𝑏−𝑥)^𝛽𝐹_𝑛^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)𝐹_𝑚^(𝛼,𝛽)()=𝑐𝑥;𝑎,𝑏,𝑐𝑑𝑥_𝑚+𝑛(−1)^𝛼+𝛽+1(𝑎−𝑏)^{𝑚+𝑛+𝛼+𝛽+1}()Γ×Γ×Γ𝛼+𝑛+1𝛽+𝑛+1𝑛!𝛼+𝛽+2𝑛+1𝛼+𝛽+𝑛+1⁻¹𝛿_𝑚,𝑛, (11) where 𝛿_𝑚,𝑛 is the Kronecker delta and min{ℝ(𝛼),ℝ(𝛽)}>−1;𝑚,𝑛∈ℕ₀∶=ℕ∪{0}.

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 𝛼,𝛽>−1 and 𝑐>0. The extended Jacobi functions are defined to be as the following Rodrigues formula: 𝐹_𝜈^(𝛼,𝛽)()=()𝑥;𝑎,𝑏,𝑐−𝑐^𝜈Γ()()𝜈+1𝑥−𝑎^−𝛼(𝑏−𝑥)^−𝛽×𝑑^𝜈𝑑𝑥^𝜈(𝑥−𝑎)^𝜈+𝛼(𝑏−𝑥)^𝜈+𝛽, (12) where 𝜈>0 and 𝑑^𝜈/𝑑𝑥^𝜈is the Riemann-Liouville fractional differentiation operator.

Theorem 2. The explicit form of the EJFs is given by 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈∞_𝑘=0𝑘()𝜈+𝛼𝜈−𝑘𝜈+𝛽𝑥−𝑎^𝑘(𝑥−𝑏)^𝜈−𝑘, (13) where 𝛼,𝛽>−1 and 𝑐>0.

Proof. If we use the Leibniz rule (4) in (12), then we have 𝐹_𝜈^(𝛼,𝛽)()=()𝑥;𝑎,𝑏,𝑐−𝑐^𝜈Γ()()𝜈+1𝑥−𝑎^−𝛼(𝑏−𝑥)^−𝛽×^∞_𝑘=0𝜈𝑘𝐷^𝑘_𝑥(𝑏−𝑥)^𝛽+𝜈𝐷_𝑥^𝜈−𝑘(𝑥−𝑎)^𝛼+𝜈. (14) It follows from the definition of fractional derivative that 𝐷_𝑥^𝜈−𝑘(𝑥−𝑎)^𝛼+𝜈=^∞_𝑟=0𝑟𝑎𝜈+𝛼_𝑟(−1)^𝑟Γ(𝜈+𝛼−𝑟+1)Γ()𝑥𝑘+𝛼−𝑟+1^{𝑘+𝛼−𝑟}. (15) From (15) and (14), we get that 𝐹_𝜈^(𝛼,𝛽)()=()𝑥;𝑎,𝑏,𝑐−𝑐^𝜈Γ()()𝜈+1𝑥−𝑎^−𝛼(𝑏−𝑥)^−𝛽×^∞_𝑘=0𝜈𝑘(−1)^𝑘Γ𝛽+𝜈+1Γ×()𝛽+𝜈−𝑘+1𝑏−𝑥^{𝛽+(𝜈−𝑘)}×^∞_𝑟=0𝑟𝑎𝜈+𝛼_𝑟(−1)^𝑟Γ(𝜈+𝛼−𝑟+1)Γ(𝑘+𝛼−𝑟+1)×𝑥^{𝑘+𝛼−𝑟}. (16) Using the fact that ^∞_𝑟=0Γ(𝛼+𝑘+1)Γ()𝑎𝑘+𝛼−𝑟+1𝑟!^𝑟(−1)^𝑟𝑥^{𝛼+𝑘−𝑟}=(𝑥−𝑎)^𝛼+𝑘, (17) the proof is completed.

Corollary 3. The another explicit form of the EJFs is given by 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈∞_𝑘=0(−1)^𝑘𝑘×()𝜈+𝛼𝜈+𝛽𝜈−𝑘𝑥−𝑎^𝜈−𝑘(𝑥−𝑏)^𝑘, (18)
where 𝛼,𝛽>−1 and 𝑐>0.

Proof. This formula can be proved similar to Theorem 2 by taking the following: 𝐹_𝜈^(𝛼,𝛽)()=()𝑥;𝑎,𝑏,𝑐−𝑐^𝜈Γ()()𝜈+1𝑥−𝑎^−𝛼(𝑏−𝑥)^−𝛽×^∞_𝑘=0𝜈𝑘𝐷^𝑘_𝑥(𝑥−𝑎)^𝛼+𝜈𝐷_𝑥^𝜈−𝑘(𝑏−𝑥)^𝛽+𝜈. (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_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝜈+𝛼×𝐹−𝜈,𝛼+𝛽+𝜈+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 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈∞_𝑘=0𝑘()𝜈+𝛼𝜈−𝑘𝜈+𝛽𝑥−𝑎^𝑘(𝑥−𝑎+𝑎−𝑏)^𝜈−𝑘=𝑐^𝜈∞_𝑘=0𝑘()𝜈+𝛼𝜈−𝑘𝜈+𝛽𝑥−𝑎^𝑘×^∞_𝑟=0𝑟)𝜈−𝑘(𝑥−𝑎^𝑟(𝑎−𝑏)^{𝜈−𝑘−𝑟}=𝑐^𝜈∞𝑟_𝑟=0𝑘=0𝑘×()𝜈+𝛼𝜈−𝑘𝜈+𝛽𝜈−𝑘𝑟−𝑘𝑥−𝑎^𝑟(𝑎−𝑏)^𝜈−𝑟=𝑐^𝜈∞_𝑟=0(𝑥−𝑎)^𝑟(𝑎−𝑏)^{𝑟𝜈−𝑟}_𝑘=0𝑘𝜈+𝛼𝜈−𝑘𝜈+𝛽𝜈−𝑘𝑟−𝑘=𝑐^𝜈∞_𝑟=0(𝑥−𝑎)^𝑟(𝑎−𝑏)^𝜈−𝑟𝜈+𝛼𝜈−𝑟^𝑟𝑘=0𝑘.𝑟+𝛼𝑟−𝑘𝜈+𝛽 (21) Using the following identity: (1−𝑥)^𝛽+𝜈(1−𝑥)^𝛼+𝑟=(1−𝑥)^{𝛽+𝛼+𝜈+𝑟}, (22) we have 𝑟=𝛽+𝛼+𝜈+𝑟^𝑟_𝑘=0𝑘𝑟+𝛼𝑟−𝑘𝜈+𝛽. (23) Substituting (23) in (21), we get that 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈∞_𝑟=0(𝑥−𝑎)^𝑟(𝑎−𝑏)^𝜈−𝑟𝑟𝜈+𝛼𝜈−𝑟𝛽+𝛼+𝜈+𝑟=𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝜈+𝛼^∞_𝑟=0(−𝜈)_𝑟𝛽+𝛼+𝜈+1_𝑟𝑟!(𝛼+1)_𝑟𝑎−𝑥𝑎−𝑏^𝑟=𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝐹𝜈+𝛼−𝜈,𝛼+𝛽+𝜈+1;𝛼+1;𝑎−𝑥,𝑎−𝑏 (24) which is the desired result.

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].

Corollary 7. The extended Jacobi functions (EJFs) hold the following hypergeometric function: 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈(𝑎−𝑏)^𝜈𝜈×𝜈+𝛼𝑥−𝑏𝑎−𝑏^𝜈𝐹−𝜈,−𝛽−𝜈;𝛼+1;𝑥−𝑎.𝑥−𝑏 (25)

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

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]: 𝑃_𝜈^(𝛼,𝛽)(𝑥)=𝜈𝜈+𝛼𝑥+12^𝜈𝐹−𝜈,−𝛽−𝜈;𝛼+1;𝑥−1𝑥+1. (26)

Corollary 9. The extended Jacobi functions can be presented as follows: 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=(−1)^𝜈𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝜈+𝛽×𝐹−𝜈,𝛼+𝛽+𝜈+1;𝛽+1;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_𝜈^(𝛼,𝛽)(𝑥)=(−1)^𝜈𝜈𝜈+𝛽×𝐹−𝜈,𝛼+𝛽+𝜈+1;𝛽+1;1+𝑥2. (28)

Corollary 11. The EJFs hold the following hypergeometric function: 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=(−1)^𝜈𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝜈+𝛽𝑥−𝑎𝑏−𝑎^𝜈×𝐹−𝜈,−𝛼−𝜈;𝛽+1;𝑥−𝑏.𝑥−𝑎 (29)

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]: 𝑃_𝜈^(𝛼,𝛽)(𝑥)=𝜈𝜈+𝛽𝑥−12^𝜈𝐹−𝜈,−𝛼−𝜈;𝛽+1;𝑥+1𝑥−1. (30)

Theorem 13. For the extended Jacobi functions, one has ()𝑑𝑥−𝑏𝐹𝑑𝑥^{𝜈(𝛼,𝛽)}(𝑥;𝑎,𝑏,𝑐)=𝜈𝐹_𝜈^(𝛼,𝛽)()(𝐹𝑥;𝑎,𝑏,𝑐+𝑐𝑎−𝑏)𝛽+𝜈^{(𝛼+1,𝛽)}_𝜈−1(),()𝑑𝑥;𝑎,𝑏,𝑐𝑥−𝑎𝐹𝑑𝑥^{𝜈(𝛼,𝛽)}(𝑥;𝑎,𝑏,𝑐)=𝜈𝐹_𝜈^(𝛼,𝛽)()()𝐹𝑥;𝑎,𝑏,𝑐−𝑐𝑎−𝑏)(𝛼+𝜈^{(𝛼,𝛽+1)}_𝜈−1(𝑥;𝑎,𝑏,𝑐). (31)

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 (𝑥−𝑎)(𝑏−𝑥)𝑌^′′(𝑥)+𝑏(𝛼+1)+𝑎𝛽+1−𝑥𝛼+𝛽+2×𝑌^′(𝑥)+𝜈𝛼+𝛽+𝜈+1𝑌(𝑥)=0 (32) or 𝑑)𝑑𝑥(𝑥−𝑎^𝛼+1(𝑏−𝑥)^𝛽+1𝑌^′(𝑥())+𝜈𝛼+𝛽+𝜈+1𝑥−𝑎^𝛼(𝑏−𝑥)^𝛽𝑌(𝑥)=0. (33)

Proof. With the help of (6) and (7), the hypergeometric function 𝐹(−𝜈,𝜈+𝛼+𝛽+1;𝛼+1;𝑡) satisfies 𝑡(1−𝑡)𝐹^′′+𝛼+1−𝑡𝛼+𝛽+2𝐹^′+𝜈𝛼+𝛽+𝜈+1𝐹=0, (34) 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)+𝑎𝛽+1−𝑥𝛼+𝛽+2×𝑌^′(𝑥)+𝜈𝛼+𝛽+𝜈+1𝑌(𝑥)=0. (35) 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_𝜈→𝑛𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝐹_𝑛^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐), (ii)𝐹_𝜈^(𝛼,𝛽)(−𝑥;𝑎,𝑏,𝑐)=(−1)^𝜈𝐹_𝜈^(𝛼,𝛽)(𝑥;−𝑎,−𝑏,𝑐), (iii)𝐹_𝜈^(𝛼,𝛽)(𝑎;𝑎,𝑏,𝑐)=𝑐^𝜈(𝑎−𝑏)^𝜈(_𝜈^𝜈+𝛼), (iv)𝐹_𝜈^(𝛼,𝛽)(𝑏;𝑎,𝑏,𝑐)=(−1)^𝜈𝑐^𝜈(𝑎−𝑏)^𝜈_𝜈^𝜈+𝛽, (v)(𝑑/𝑑𝑥)𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐(𝜈+𝛼+𝛽+1)𝐹^{(𝛼+1,𝛽+1)}_𝜈−1(𝑥;𝑎,𝑏,𝑐), (vi)(𝑑^𝑘/𝑑𝑥^𝑘)𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝑘(𝜈+𝛼+𝛽+1)_𝑘𝐹^{(𝛼+𝑘,𝛽+𝑘)}_𝜈−𝑘(𝑥;𝑎,𝑏,𝑐)for𝑘∈ℕ.

Proof. Consider the following.
(i) By (20), we have lim_𝜈→𝑛𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=lim_𝜈→𝑛𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝜈+𝛼×𝐹−𝜈,𝛼+𝛽+𝜈+1;𝛼+1;𝑎−𝑥𝑎−𝑏=𝑐^𝑛(𝑎−𝑏)^𝑛𝑛𝐹𝑛+𝛼−𝑛,𝛼+𝛽+𝑛+1;𝛼+1;𝑎−𝑥𝑎−𝑏=𝐹_𝑛^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐). (36)
(ii) By (13), we get 𝐹_𝜈^(𝛼,𝛽)(−𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈∞_𝑘=0𝑘()𝜈+𝛼𝜈−𝑘𝜈+𝛽−𝑥−𝑎^𝑘(−𝑥−𝑏)^𝜈−𝑘=(−1)^𝜈𝐹_𝜈^(𝛼,𝛽)(𝑥;−𝑎,−𝑏,𝑐). (37)
(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).

Corollary 16. As a consequence of Theorem 15(v) and (31), one has the following recurrence relations: 𝑑𝐹𝑑𝑥^{𝜈(𝛼,𝛽)}()𝐹𝑥;𝑎,𝑏,𝑐=𝑐𝛽+𝜈^{(𝛼+1,𝛽)}_𝜈−1()+()𝐹𝑥;𝑎,𝑏,𝑐𝛼+𝜈^{(𝛼,𝛽+1)}_𝜈−1(,()𝐹𝑥;𝑎,𝑏,𝑐)𝛼+𝛽+𝜈+2_𝜈^{(𝛼+1,𝛽+1)}()=()𝐹𝑥;𝑎,𝑏,𝑐𝛽+𝜈+1_𝜈^{(𝛼+1,𝛽)}()+((𝐹𝑥;𝑎,𝑏,𝑐𝛼+𝜈+1))_𝜈^{(𝛼,𝛽+1)}(𝑥;𝑎,𝑏,𝑐). (38)

Now, we compare some properties of the extended Jacobi functions with extended Jacobi polynomials.(i) Rodrigues Formula. Consider the following. Extended Jacobi functions: 𝐹_𝜈^(𝛼,𝛽)()=()𝑥;𝑎,𝑏,𝑐−𝑐^𝜈Γ()()𝜈+1𝑥−𝑎^−𝛼(𝑏−𝑥)^−𝛽×𝑑^𝜈𝑑𝑥^𝜈(𝑥−𝑎)^𝜈+𝛼(𝑏−𝑥)^𝜈+𝛽. (39) Extended Jacobi polynomials: 𝐹_𝑛^(𝛼,𝛽)()=()𝑥;𝑎,𝑏,𝑐−𝑐^𝑛()𝑛!𝑥−𝑎^−𝛼(𝑏−𝑥)^−𝛽×𝑑^𝑛𝑑𝑥^𝑛(𝑥−𝑎)^𝑛+𝛼(𝑏−𝑥)^𝑛+𝛽. (40)(ii) Explicit Formula. Consider the following. Extended Jacobi functions: 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝜈∞_𝑘=0×𝑘()𝜈+𝛼𝜈−𝑘𝜈+𝛽𝑥−𝑎^𝑘(𝑥−𝑏)^𝜈−𝑘. (41) Extended Jacobi polynomials: 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=𝑐^𝑛∞_𝑘=0×𝑘()𝑛+𝛼𝑛−𝑘𝑛+𝛽𝑥−𝑎^𝑘(𝑥−𝑏)^𝑛−𝑘. (42)(iii) Hypergeometric Representation. Consider the following. Extended Jacobi functions: 𝐹_𝜈^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=(𝑐)^𝜈(a−𝑏)^𝜈𝜈𝜈+𝛼×𝐹−𝜈,𝛼+𝛽+𝜈+1;𝛼+1;𝑎−𝑥.𝑎−𝑏 (43) Extended Jacobi polynomials: 𝐹_𝑛^(𝛼,𝛽)(𝑥;𝑎,𝑏,𝑐)=(𝑐)^𝑛(𝑎−𝑏)^𝑛𝑛𝑛+𝛼×𝐹−𝑛,𝛼+𝛽+𝑛+1;𝛼+1;𝑎−𝑥.𝑎−𝑏 (44)(iv) Value at 𝑥=𝑎. Consider the following. Extended Jacobi functions: 𝐹_𝜈^(𝛼,𝛽)(𝑎;𝑎,𝑏,𝑐)=𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝜈+𝛼. (45) Extended Jacobi polynomials: 𝐹_𝑛^(𝛼,𝛽)(𝑎;𝑎,𝑏,𝑐)=𝑐^𝑛(𝑎−𝑏)^𝑛𝑛𝑛+𝛼. (46)(v) Value at 𝑥=𝑏. Consider the following. Extended Jacobi Functions: 𝐹_𝜈^(𝛼,𝛽)(𝑏;𝑎,𝑏,𝑐)=(−1)^𝜈𝑐^𝜈(𝑎−𝑏)^𝜈𝜈𝜈+𝛽. (47) Extended Jacobi polynomials: 𝐹_𝑛^(𝛼,𝛽)(𝑏;𝑎,𝑏,𝑐)=(−𝑐)^𝑛(𝑎−𝑏)^𝑛𝑛𝑛+𝛽. (48)(vi) Differential Equation. Consider the following. Extended Jacobi functions: (𝑥−𝑎)(𝑏−𝑥)𝑌^′′(𝑥)+𝑏(𝛼+1)+𝑎𝛽+1−𝑥𝛼+𝛽+2×𝑌^′(𝑥)+𝜈𝛼+𝛽+𝜈+1𝑌(𝑥)=0. (49) Extended Jacobi polynomials: (𝑥−𝑎)(𝑏−𝑥)𝑌^′′(𝑥)+[𝑏(𝛼+1)+𝑎𝛽+1−𝑥𝛼+𝛽+2×𝑌^′(𝑥)+𝑛𝛼+𝛽+𝑛+1𝑌(𝑥)=0. (50)(vii) Derivation. Consider the following. Extended Jacobi functions: 𝑑𝐹𝑑𝑥^{𝜈(𝛼,𝛽)}(𝑥;𝑎,𝑏,𝑐)=𝑐𝜈+𝛼+𝛽+1𝐹^{(𝛼+1,𝛽+1)}_𝜈−1(𝑥;𝑎,𝑏,𝑐). (51) Extended Jacobi polynomials: 𝑑𝐹𝑑𝑥^{𝑛(𝛼,𝛽)}()=1𝑥;𝑎,𝑏,𝑐2𝑛+𝛼+𝛽+1𝐹^{(𝛼+1,𝛽+1)}_𝑛−1(𝑥;𝑎,𝑏,𝑐). (52)

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: 𝑡^𝜇1−𝑡^𝜇𝐷^2𝜇𝑌(𝑡)+𝑐−(𝑎+𝑏+1)𝑡^𝜇𝐷^𝜇[𝑌(𝑡)]−𝑎𝑏𝑌(𝑡)=0, (53) where 0<𝜇≤1.

Definition 18 (see [6]). The fractional hypergeometric matrix function is defined by ^𝜇₂𝐹₁(𝑎,𝑏;𝑐;𝑡)=𝑦₀𝑡^𝑝+^∞_𝑘=1𝑦₀^𝑘−1_𝑗=0𝑔_𝑗𝑝𝑓⁻¹_𝑗+1𝑝𝑡^{𝑘𝜇+𝑝}, (54) where 0<𝜇≤1 and 𝑓_𝑘𝑝=Γ𝑘𝜇+1+𝑝Γ()Γ𝑘−2𝜇+1+𝑝+𝑐𝑘𝜇+1+𝑝Γ(),𝑔𝑘−1𝜇+1+𝑝_𝑘𝑝=Γ𝑘𝜇+1+𝑝Γ()+()Γ𝑘−2𝜇+1+𝑝+𝑎𝑏𝑎+𝑏+1𝑘𝜇+1+𝑝Γ(),𝑘−1𝜇+1+𝑝 (55) and the following property 𝑓₀𝑝=Γ1+𝑝ΓΓ−2𝜇+1+𝑝+𝑐1+𝑝Γ−𝜇+1+𝑝=0 (56) holds for 𝜌>−1.

If we take −𝜈 instead of 𝑎,  𝑎+𝑏+𝜈+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)^𝜇𝐷^2𝜇𝑌(𝑡)+(−1)^𝜇(𝑎+1)(𝑎−𝑏)^𝜇−(𝑎+𝑏+2)(𝑎−𝑡)^𝜇×𝐷^𝜇[𝑌(𝑡)]+𝜈(𝑎+𝑏+𝜈+1)𝑌(𝑡)=0, (57) where 0<𝜇≤1/2.

Definition 20. The generalized extended Jacobi functions (GEJFs) are defined by ^𝜇₂𝐹₁−𝜈,𝑎+𝑏+(𝜈+1);𝑎+1;𝑎−𝑡𝑎−𝑏=𝑦₀𝑎−𝑡𝑎−𝑏^𝑝+^∞_𝑘=1𝑦₀^𝑘−1_𝑗=0𝑔_𝑗𝑝𝑓⁻¹_𝑗+1𝑝𝑎−𝑡𝑎−𝑏^{𝑝+𝑘𝜇}, (58) where 𝑓_𝑘𝑝=Γ𝑘𝜇+1+𝑝Γ()+()Γ𝑘−2𝜇+1+𝑝𝑎+1𝑘𝜇+1+𝑝Γ(),𝑔𝑘−1𝜇+1+𝑝_𝑘𝑝=Γ𝑘𝜇+1+𝑝Γ()((+()Γ𝑘−2𝜇+1+𝑝−𝜈𝑎+𝑏+𝜈+1))𝑎+𝑏+2𝑘𝜇+1+𝑝Γ(),𝑘−1𝜇+1+𝑝 (59) and the equation 𝑓₀𝑝=Γ1+𝑝Γ+()Γ−2𝜇+1+𝑝𝑎+11+𝑝Γ−𝜇+1+𝑝=0 (60) holds for 𝜌>−1.

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.