Kummer’s 24 Solutions of the Hypergeometric Differential Equation with the Aid of Fractional Calculus

We know that the hypergeometric function, which is a solution of the hypergeometric differential equation, is expressed in terms of the Riemann-Liouville fractional derivative (fD). The solution of the differential equation obtained by the Euler method takes the form of an integral, which is confirmed to be expressed in terms of the Riemann-Liouville fD of a function. We can rewrite this derivation such that we obtain the solution in the form of the Riemann-Liouville fD of a function. We present a derivation of Kummer’s 24 solutions of the hypergeometric differential equation by this method.


Introduction
The hypergeometric function is a solution of the hypergeometric differential equation, and is known to be expressed in terms of the Riemann-Liouville fractional derivative (fD) ( [1], p. 334). By the Euler method ( [2], Section 3.2), the solution of the hypergeometric differential equation is obtained in the form of an integral, which is confirmed to be expressed in terms of the Riemann-Liouville fD of a function. This shows that we can obtain the solution in the form of the Riemann-Liouville fD of a function. In fact, Nishimoto [3] obtained a solution of the hypergeometric differential equation in terms of the Liouville fD in the first step, and then expressed the obtained fD in terms of the hypergeometric function in the second step. His calculation in the second step is unacceptable. In [4], he gave a derivation of Kummer's 24 solutions of the hypergeometric differential equation ([5], Formula 15.5.4) ( [6], Section 2.2) by his method. In the present paper, we show that the desired solutions are obtained by using the Riemann-Liouville fD in place of the Liouville fD.
In a preceding paper [7], we discussed the Riemann-Liouville fD and the Liouville fD as analytic continuations of the respective fractional integrals (fIs), on the basis of the papers by Lavoie et al [1] [8], and those by Nishimoto [3] and Campos [9], respectively. In Section 2, we define these fIs of a function  (1) and (2), respectively, and give their properties which we use later. The notation +  is defined at the end of this section.
In Section 3, following [1] [3] [7]- [9], the Riemann-Liouville fD, , of order ν ∈  , are defined in the form of a contour integral, for a function ( ) f ζ which is analytic on a neighborhood of the path of integration. They are defined such that they are analytic continuations of the corresponding fI as a function of ν ∈  . In the present paper, the fI and fD are operated to a function of the form for γ ∈  and 2 γ ∈  . The analytic continuations of ( ) are then shown to be analytic as a function of ν as well as of γ and 2 γ . In the present paper, we use this fact in the calculation. In the following, we use fD to represent fI and fD as a whole.
In [1], the expression of the hypergeometric function: ( ) 2 1 , ; ; F a b c z in terms of the Riemann-Liouville fD is given. In Sections 4 and 4.1, its derivation is presented with the aid of the method using the Riemann-Liouville fD. In Sections 4.2-4.4 and 5, Kummer's 24 solutions of the hypergeometric differential equation are derived in two ways in the present method.
In a separate paper [10], a method of obtaining the asymptotic expansion of the Riemann-Liouville fD is presented by using a relation of its expression via a path integral or a contour integral with the corresponding Liouville fD. It is then applied to obtain the asymptotic expansion of the confluent hypergeometric function which is a solution of Kummer's differential equation. In that paper, Kummer's 8 solutions of Kummer's differential equation are obtained by using the method which is adopted in the present paper to obtain the solutions of the hypergeometric differential equation.
We use notations  ,  and  , which represent the sets of all complex numbers, of all real numbers and of all integers, respectively. We use also the notations given by

Riemann-Liouville fD and Liouville fD
Following preceding papers [7] [10], we adopt the following definitions of the Riemann-Liouville fI, φ-dept Liouville fI and the corresponding fDs.

Riemann-Liouville fI on the Complex Plane
Let ξ ∈  and z ∈  . We denote the path of integration from ξ to z by ( ) and ( ) f ζ be continuous on a neighborhood of z ζ = . Then the Riemann-Liouville fI of order λ + ∈  is defined by is the gamma function.

Definition of φ-Dept Liouville fI
Let z ∈  and φ ∈  . We denote the half line { } is locally integrable as a function of t in the interval ( ) 0, ∞ , we denote this by The following lemma was mentioned in [11].
Proof. This is confirmed by comparing the second members of (1) and of (2). 

Definitions of Riemann-Liouville fD and Liouville fD
Definition 5. The Riemann-Liouville fD: ( ) for ξ ∈  and the Liouville fD: when the righthand side exists, where Here x     for x ∈  denotes the greatest integer not exceeding x.

Index Law and Leibniz's Rule of Riemann-Liouville fI and Liouville fI
We use the following index law and Leibniz's rule, in Section 4.2. By Lemma 2, the formulas for e iφ ξ = ∞ ⋅ are for the Liouville fI.
Proof. Proof for ν ∈  and λ ∈  is found in ( [12], Section 2.2.6), where p and q appear in place of λ − and ν , respectively. The proofs there apply for p ∈  and q ∈  if we replace p and q in the inequalities there by Re p and Re q , respectively.  Re Re Re Proof. Proof of (4) for the case (i) is found in ( [7], Appendix A). In the case (ii), with the aid of this knowledge and formula (3), we prove the first equation in (4) in the following way: where Proof. By using the righthand side of (1), we see that both sides of the equation in this lemma are equal to This Leibniz's rule is given in ( [13], Section 5.5). The following corollary follows from this lemma.

Analytic Continuations of Riemann-Liouville fI
In [1] [7] [8], analytic continuations of the Riemann-Liouville fI via contour integrals are discussed. In [7], where the contour of integration is the Cauchy contour ( ) , C z ξ + shown in Figure 1(a), which starts from ξ , encircles the point z counterclockwise, and goes back to ξ , without crossing the path Definition 7. Let ( ) is the Pochhammer contour shown in Figure 1(b). When

Analyticity of Riemann-Liouville fD and Liouville fD
In this section, we consider functions ( ) The following Lemmas 6~10 are obtained by modifying the corresponding arguments given in Section 2 for the Riemann-Liouville fD and in Sections 3.1~3.3 for the Liouville fD in [7], with the aid of ( [14], Sections 3.1 and 3.2). Lemma 6.
In the following sections, we use

The Hypergeometric Function in Terms of Riemann-Liouville fD
Let a ∈ , b ∈ , c ∈  and z ∈  satisfy (i): The integral representation of ( ) 2 1 , ; ; F a b c z is given by when Re Re 0 c a > > , in ([5], Formula 15.5.4) ( [6], Section 2.5). In fact, we obtain (13) from (14) by expanding the righthand side of the latter in powers of z and then performing the integration term-by-term, when 1 z < .
This function is a solution of the hypergeometric differential equation: which has also another solution given by ( )

Solution of the Hypergeometric Differential Equation (15) with the Aid of Riemann-Liouville fD
The function ( ) 2 1 , ; ; F a b c z is known to be expressed in the form of (18) for 3 l = given below, in [1]. We now obtain the solutions of (15) expressed in terms of the Riemann-Liouville fD.
Proofs of the following two lemmas are presented in the following two sections. where the values a l , b l and c l are given in Table 1 Proof. We assume that a solution of (15) is expressed as ( ) ( ) If (i) or (ii) applies, we substitute this ( ) w z in (15), and use Lemma 3 and Corollary 1. We then obtain Putting 1 a λ = − and hence assuming Re 1 a < − , and applying 2 R D λ ξ − to (23), we obtain with the aid of Lemma 3. This equation requires that and ( ) ( ) l ∈  also are solutions of (15).
Proof. We first consider the case of 3 l = . We replace ( ) When we choose 1 c λ = − , this equation is reduced to (15) with a, b, c and w replaced by 3 a , 3 b , 3 c and u, respectively. In the case of 2 l = , we use ( ) ( ) 1 z u z λ − in place of ( ) z u z λ ⋅ . By using this lemma for 3 l = and 2 l = , we see that ( ) ( )  y ζ , respectively.  By Corollary 2 and Lemma 16, we obtain the following corollary.