Kummer’s 24 Solutions of the Hypergeometric Differential Equation with the Aid of Fractional Calculus ()
Received 30 November 2015; accepted 24 February 2016; published 29 February 2016

1. 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
,
and
, of order
, by (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,
and
, and the Liouville fD,
and
, of order
, are defined in the form of a contour integral, for a function
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
. In the present paper, the fI and fD are operated to a function of the form
for
and
. The analytic continuations of
and
are then shown to be analytic as a function of
as well as of
and
. 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:
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
,
,
,
for
and
, and
.
2. Riemann-Liouville fD and Liouville fD
Following preceding papers [7] [10] , we adopt the following definitions of the Riemann-Liouville fI, f-dept Liouville fI and the corresponding fDs.
2.1. Riemann-Liouville fI on the Complex Plane
Let
and
. We denote the path of integration from ξ to z by
, and use
to denote that the function
is integrable on
.
Definition 1. Let
,
,
and
be continuous on a neighborhood of
. Then the Riemann-Liouville fI of order
is defined by
(1)
where
is the gamma function.
2.2. Definition of f-Dept Liouville fI
Let
and
. We denote the half line
, by
, or by
. When
is locally integrable as a function of t in the interval
, we denote this by
.
Definition 2. Let
,
,
, and
. Let
be such that the integral
converges for
and diverges for
. We then call
the abscissa of conver-
gence, and denote it by
or
.
We then have
or
.
Lemma 1. Let
for
and
. Then
.
Definition 3. Let
and
. Let
and
be continuous on a neighborhood of
. Let
,
and
. Then we define
by
(2)
We call
the f-dept Liouville fI of
.
Definition 4. When the conditions in Definition 3 are satisfied, we define
for
by (1).
The following lemma was mentioned in [11] .
Lemma 2. Let
. Then ![]()
Proof. This is confirmed by comparing the second members of (1) and of (2). ,
2.3. Definitions of Riemann-Liouville fD and Liouville fD
Definition 5. The Riemann-Liouville fD:
for
and the Liouville fD:
for
, of order
satisfying
, are defined by
(3)
when the righthand side exists, where
, and
for
.
Here
for
denotes the greatest integer not exceeding x.
2.4. 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
are for the Liouville fI.
Lemma 3. Let
,
satisfy
, and
exist. Then
(4)
Proof. Proof for
and
is found in ( [12] , Section 2.2.6), where p and q appear in place of
and
, respectively. The proofs there apply for
and
if we replace p and q in the inequalities there by
and
, respectively. ,
Lemma 4. Let
,
and
satisfy
, and (i)
and
, or (ii)
and
. Then (4) holds valid for
.
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:
(5)
where
, δ = 0 if
, and
if
. When
, (5) shows the second equation in (4). ,
Lemma 5. Let
, and
exist. Then
.
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.
Corollary 1. Let
, and
exist. Then
(6)
(7)
3. Analytic Continuations of Riemann-Liouville fD and Liouville fD
3.1. Analytic Continuations of Riemann-Liouville fI
In [1] [7] [8] , analytic continuations of the Riemann-Liouville fI via contour integrals are discussed. In [7] ,
and
for
are defined as follows.
Definition 6. Let
be analytic on a neighborhood of the path
and on the point
, and
. Then
is defined by
(8)
for
, where the contour of integration is the Cauchy contour
shown in Figure 1(a), which starts from
, encircles the point z counterclockwise, and goes back to
, without crossing the path
. When
, we put
.
Definition 7. Let
,
,
, and
be analytic on a neighborhood of the path
and on the points
and z. Then
is defined by
(9)
for
, where
is the Pochhammer contour shown in Figure 1(b). When
, we put
. When
, we put
.
3.2. Analytic Continuations of Liouville fI
In [3] [7] [9] , the analytic continuation of Liouville fI:
is discussed. It is defined in [7] as follows.
Definition 8. Let
be analytic on a neighborhood of the path
, and
and
. Then
for
is defined by
(10)
where
. When
, we put
.
In [7] , another analytic continuation of Liouville fI:
was introduced. Here we define it for a function of the form
, where
,
,
, and
is an entire function.
Definition 9. Let (i):
be a function of the form stated above, where
, (ii):
be the modified Pochhammer contour shown in Figure 2, where
,
,
,
,
and
satisfy
, and (iii):
,
and
satisfy
,
,
, and
, Then
for
is defined by
(11)
where
. When
, we put
. When
,
is defined by analyticity.
3.3. Analyticity of Riemann-Liouville fD and Liouville fD
In this section, we consider functions
and
expressed by
(12)
where
,
,
and
.
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.
and
are analytic as a function of
as well as of
, and of
in the domains
and
, respectively.
Lemma 7.
and
are analytic as a function of
as well as of
and
.
Lemma 8. Let
exist. Then
exists and
.
Lemma 9. Let
exist. If
, then
exists and
.
Lemma 10. Lemmas 8 and 9 with
,
,
and
, replaced by
,
,
and
, respectively, are valid.
Remark 1. The statements related with
and
in Lemma 10 are proved by modifying the proofs of Theorems 3.1 and 3.3, respectively, in [7] .
In the following sections, we use
and
for the Riemann-Liouville fD.
4. The Hypergeometric Function in Terms of Riemann-Liouville fD
Let
,
,
and
satisfy (i):
or (ii):
and either
or
. In the case (i), the hypergeometric series
is defined by
(13)
where
for
and
, for
. In the case (ii), it is defined by
.
The integral representation of
is given by
(14)
when
, 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
.
This function is a solution of the hypergeometric differential equation:
(15)
which has also another solution given by
(16)
see ( [5] , Section 15.5.1) ( [6] , Section 2.2).
4.1. Solution of the Hypergeometric Differential Equation (15) with the Aid of Riemann-Liouville fD
The function
is known to be expressed in the form of (18) for
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.
Lemma 11. Let
and
for
be as follows:
(17)
(18)
where the values al, bl and cl are given in Table 1, and
are constants. Then
, for
and
, are solutions of (15).
Lemma 12. When
, we choose
, and then
given by (18) are expressed as
(19)
Corollary 2. When we put
for
, we have
![]()
(20)
![]()
(21)
Remark 2. The solutions
given in Corollary 2 satisfy
and
; see ( [5] , Formulas 15.5.3~15.5.4) ( [6] , Section 2.2). This is confirmed by noting that the solution of (15) in the form
with a fixed
and
is unique.
4.2. Proof of Lemma 11
Lemma 13 Let
, and (i):
and
, or (ii):
and
, or (iii):
,
and
. Then a solution of (15) is given by
(22)
Proof. We assume that a solution of (15) is expressed as
for
satisfying
. If (i) or (ii) applies, we substitute this
in (15), and use Lemma 3 and Corollary 1. We then obtain
(23)
Putting
and hence assuming
, and applying
to (23), we obtain
(24)
with the aid of Lemma 3. This equation requires that
(25)
and
when
. Now we obtain
if any of the three conditions in Lemma 13 is satisfied. Thus we obtain (22). When (iii) applies, we use Lemma 4 in place of Lemma 3. Then we have to use
. ,
Remark 3. The proof of Lemma 13 corresponds to the derivation, given in ( [2] , pp. 43-44), of an integral form of the solution of (15), where the method is called the Euler method.
Lemma 14. If
is a solution of (15), then
for
also are solutions of (15).
Proof. We first consider the case of
. We replace
by
in (15), then we obtain
(26)
When we choose
, this equation is reduced to (15) with a, b, c and w replaced by
,
,
and u, respectively. In the case of
, we use
in place of
. By using this lemma for
and
, we see that
and
are solutions of (15). This proves the case of
. ,
Proof of Lemma 11. The formula (18) for
follows from Lemmas 13 and 14 with the aid of Lemmas 7-10. ,
4.3. Expressions of
in Terms of the Hypergeometric Functions
We now present the expressions of
given in (18) in terms of the hypergeometric functions. We then obtain Kummer’s 24 solutions. In the following section, we give another derivation of them.
Proof of Lemma 12 is given at the first part of the proof of Lemma 15 below.
By using Lemmas 8, 9 and 10 and the middle member of (1), (18) is expressed as
(27)
Lemma 15. We choose
,
, and
.
Then
given by (18) is expressed as
, where
(28)
(29)
if k,
,
,
,
,
,
,
,
are those given in a row in Table 2.
Proof. We put
and
in (27). Then we obtain
(30)
when
, and also
. The data in the row
in Table 2, are so chosen that
given by (28) with the data is equal to (30). Lemma 12 follows from (30) with the aid of formula (19).
We put
and
in (27). We then obtain
(31)
when
,
and
. The data in the row
are taken from this equation.
We put
and
in (27). We then obtain
(32)
when
,
and
. The data in the row
are taken from this equation.
We put
in (28). Then we obtain
(33)
Applying this to the formula (28) for
and 3, we obtain the results in Table 2 for
and 5, respectively. ,
Remark 4. Let
given by (29), for
and
, be denoted by
when
. We show that they give Kummer’s 24 solutions of (15), which are
for
given in Theorem 1 below. They are related by
for
, and by
for
, where
,
,
,
,
,
,
,
,
,
, and
for
. Here
and
appear twice, and
and
do not appear. We note that when the formers are solutions of (15), the latters which are obtained from the formers by exchanging a and b, are obviously solutions of (15). By adding these in the set of solutions
, we have the 24 solutions of (15).
Remark 5. In Lemma 15, we have two expressions of
for different k. For instance for
and
, we have
, which is given in ( [5] , Formulas 15.3.3~15.3.5) ( [6] , Section 2.4.1).
Remark 6. When
, we have
, so that the equation (32) and the data for
in Table 2 are obtained by using the Liouville fD, and is given by Nishimoto in [4] . In that case, Nishimoto’s derivation is justified.
4.4. Solutions of (15) as a Function of
,
,
,
, and ![]()
In the following, there appear
,
,
and
for
. They are listed in Table 3.
Lemma 16. If
is a solution of (15), then
for
also are solutions of (15).
Proof. When
, we replace z and
by
and
, respectively, in (15). We then obtain the same equation with c, z and
replaced by
,
and
, respectively. We call the obtained Equation (15-2).
When
, we put
, and replace z and
by
and
or
, respectively, in (15). We then obtain
![]()
When we choose
, this equation is reduced to (15) with b, c, z and
replaced by
,
,
and
, respectively. We call the obtained Equation (15-3).
When
, we replace
and
by
and
, respectively, in (15-3). We then obtain
the same equation with
,
,
and
replaced by
,
,
and
, respectively.
When
, we replace
and
by
and
or
, respectively, in (15-2).
We then obtain the same equation with b,
,
and
replaced by
,
,
and
, respectively. We call the obtained Equation (15-5).
When
, we replace
and
by
and
, respectively, in (15-5). We then obtain
the same equation with
,
,
and
replaced by
,
,
and
,
respectively. ,
By Corollary 2 and Lemma 16, we obtain the following corollary.
Corollary 3. Let
for
represent the righthand side of the equation for
given in (20)~ (21). Then for
,
and
,
(34)
is a solution of (15).
We note here the following remark, which is used in obtaining Table 4 below.
Remark 7
,
for
and 5, and
for
and 6.
5. Kummer’s 24 Solutions of the Hypergeometric Differential Equation
By Corollary 3 and Lemma 7, we obtain the following theorem by the present method.
Theorem 1 We have 24 solutions of (15), which are expressed as
(35)
where the functions
of z and the values of
,
,
,
and
are listed in Table 4.
The values for
in Table 4 are obtained by comparing (35) with (20)~(21) in Corollary 2. By Corollary 3 and Remark 7, the functions
and the values for
are obtained with the aid of the following lemma.
Lemma 17. Let
,
,
,
and
for
represent
,
,
,
and
, respectively, as a function of a, b and c. Then the values of
,
,
,
and
and functions
of z for
are given by
(36)
(37)
(38)
(39)
where
and
.
The following lemma is well known, see ( [5] , Formulas 15.5.3~15.5.14) ([6] , Section 2.2).
Theorem 2. The solutions
given in Theorem 1 for
are related by
(40)
Proof. This is confirmed by using Lemma 16 or Corollary 3 with the aid of Remark 2. ,