recursion formulas for g1 and g2 horn hypergeometric functions

recursion formulas for g1 and g2 horn hypergeometric functions

Miskolc Mathematical Notes
Vol. 16 (2015), No. 2, pp. 1153–1162
HU e-ISSN 1787-2413
DOI: 10.18514/MMN.2015.1306
RECURSION FORMULAS FOR G1 AND G2 HORN
HYPERGEOMETRIC FUNCTIONS
RECEP ŞAHIN AND SUSAN RIDHA SHAKOR AGHA
Received 04 September, 2014
Abstract. The aim of this paper is to present various recursion formulas for Horn hypergeometric
functions by the contiguous relations of hypergeometric series.These recursion formulas allow us
to state the functions G1 and G2 Horn hypergeometric functions as a combination of themselves.
2010 Mathematics Subject Classification: 33C20; 11J72
Keywords: recursion formulas, Horn hypergeometric functions, contiguous relation
1. I NTRODUCTION
The first G1 and G2 Horn hypergeometric functions were defined by the series
[3–5]
1
X
xm yp
.˛/mCp .ˇ/p m ˇ 0 m p
G1 ˛; ˇ; ˇ 0 I x; y D
mŠ pŠ
m;pD0
jxj < r ; jyj < s ; r C s D 1
and
1
X
.˛/m ˛ 0 p .ˇ/p
G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y D
m
ˇ0
m;pD0
m
xm yp
p mŠ pŠ
jxj < 1 ; jyj < 1;
respectively. Recently, Opps et al: [1] have obtained some recursion formulas for the
function F2 by the contiguous relation of the Gauss hypergeometric function 2 F1 : In
[6], Wang gave some recursion formulas for Appell hypergeometric functions. The
aim of our present investigation is to construct various recursion formulas for each of
Horn hypergeometric functions G1 and G2 :
Recall that gamma function is defined in [2, 3] by
Z1
.n/ D t n
1
e t dt ; Re .n/ > 0:
0
c 2015 Miskolc University Press
1154
RECEP ŞAHIN AND SUSAN RIDHA SHAKOR AGHA
The Pochhammer symbol ./n is denoted by
./n WD . C 1/ ::: . C n
1/ ; .n 2 N W D f1; 2; 3; :::g/ and ./0 WD 1
and its well known form is also given in [1] as
. 1/n
; .n 2 N W D f1; 2; 3; :::g/ :
.1 /n
It easily follows from (1.1) that
./
nD
./m
n
D ./m . C m/
(1.1)
n;
(1.2)
for m; n 2 N.
2. R ECURSION FORMULAS OF G1
In this section, we give some recursion formulas for the function G1 . We start the
following theorem.
Theorem 1. Recursion formulas for the function G1 are as follows W
G1 ˛ C n; ˇ; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y
C .ˇ/
1ˇ
C ˇ ˇ0
0
n
X
x
G1 ˛ C k; ˇ
kD1
n
X
1
y
(2.1)
1; ˇ 0 C 1I x; y
G1 ˛ C k; ˇ C 1; ˇ 0
1I x; y
kD1
and
G1 ˛
n; ˇ; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y
.ˇ/
n
X1
0
1ˇ x
(2.2)
G1 ˛
k; ˇ
1; ˇ 0 C 1I x; y
kD0
ˇ ˇ0
y
1
n
X1
G1 ˛
k; ˇ C 1; ˇ 0
1I x; y
kD0
Proof. From the definition of the function G1 and transformation
m p
.˛ C 1/mCp D .˛/mCp 1 C C
˛ ˛
we can get the following relation:
G1 ˛ C 1; ˇ; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y
C .ˇ/
Cˇ
0
1 ˇ xG1
ˇ 0 1 yG1
˛ C 1; ˇ
0
(2.3)
1; ˇ C 1I x; y
˛ C 1; ˇ C 1; ˇ 0
1I x; y :
RECURSION FORMULAS FOR G1 AND G2 HORN HYPERGEOMETRIC FUNCTIONS
1155
By applying this contiguous relation to function G1 with the parameter ˛ C 2 , we
have
G1 ˛ C 2; ˇ; ˇ 0 I x; y
0
D G1 ˛ C 1; ˇ; ˇ 0 I x; y C .ˇ/ 1 ˇ xG1 ˛ C 2; ˇ 1; ˇ 0 C 1I x; y
0
Cˇ ˇ
yG1 ˛ C 2; ˇ C 1; ˇ 0 1I x; y
1
D G1 ˛; ˇ; ˇ 0 I x; y
0 C .ˇ/ 1 ˇ x G1 ˛ C 1; ˇ 1; ˇ 0 C 1I x; y C G1 ˛ C 2; ˇ 1; ˇ 0 C 1I x; y
0
Cˇ ˇ
yŒG1 ˛ C 1; ˇ C 1; ˇ 0 1I x; y C G1 ˛ C 2; ˇ C 1; ˇ 0 1I x; y 
1
G1 ˛ C n; ˇ; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y
C .ˇ/
n
X
0
1ˇ x
C ˇ ˇ0
G1 ˛ C k; ˇ
kD1
n
X
1
y
1; ˇ 0 C 1I x; y
G1 ˛ C k; ˇ C 1; ˇ 0
1I x; y
kD1
If we compute the function G1 with the parameter ˛ C n by relation (2.3) for n
times, we find the formula given by (2.1). Replacing ˛ by ˛ 1 in the contiguous
relation (2.3), we get
G1 ˛ 1; ˇ; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y .ˇ/ 1 ˇ 0 xG1 ˛; ˇ 1; ˇ 0 C 1I x; y
ˇ ˇ 0 1 yG1 ˛; ˇ C 1; ˇ 0 1I x; y :
If we apply this relation to the function G1 with the parameter ˛ n for n times,
we obtain the recursion formula (2.2) similar to the proof of formula (2.1).
By the same contiguous relations (2.1) and (2.2) we can express the functions G1
in the above theorem in other forms.
Theorem 2. The function G1 satisfies the recursion formulas W
!
!
n X
n i
X
n n i
0
.ˇ/i k ˇ 0 k i
G1 ˛ C n; ˇ; ˇ I x; y D
i
k
(2.4)
i D0 kD0
G1 ˛
x k y i G1 ˛ C i C k; ˇ C i k; ˇ 0 C k
!
!
n i
n X
X
n n i
0
.ˇ/i k ˇ 0 k i
n; ˇ; ˇ I x; y D
i
k
i I x; y
(2.5)
i D0 kD0
. x/k . y/i G1 ˛ C i C k; ˇ C i
k; ˇ 0 C k
i I x; y :
1156
RECEP ŞAHIN AND SUSAN RIDHA SHAKOR AGHA
Proof. By the induction method, we only prove the recursion formula given by
(2.4). For n D 1, formula (2.4) is satisfied. Assume that the result (2.4) is true for
n D t. Then, we need to show that the relation (2.4) is satisfied for n D t C 1. Setting
n D t in (2.4), we have
t X
t i
X
t
G1 ˛ C t; ˇ; ˇ 0 I x; y D
i
t i
k .ˇ/i k
ˇ0
k i
xk yi
i D0 kD0
k; ˇ 0 C k
G1 ˛ C i C k; ˇ C i
i I x; y :
Replacing ˛ by ˛ C 1 in the above relation, we obtain
0
G1 ˛ C t C 1; ˇ; ˇ I x; y D
t X
t i
X
t
i
t i
k .ˇ/i k
ˇ0
k i
xk yi
i D0 kD0
G1 ˛ C i C k C 1; ˇ C i
k; ˇ 0 C k
i I x; y :
In the above equality, we apply the contiguous relation (2.3) with the transformations
˛ ! ˛ C i C k; ˇ ! ˇ C i k; ˇ 0 ! ˇ 0 C k i: Using the relations
!
!
!
n 1
n 1
n
C
D
k
k 1
k
and
!
n
D 0 ; whe n k > n or k < 0
k
and with some simplifications, we deduce
G1 ˛ C t C 1; ˇ; ˇ 0 I x; y
D
t i
t X
X
t
t i
k .ˇ/i k
i
ˇ0
k i
x k y i G1 ˛ C i C k; ˇ C i
k; ˇ 0 C k
i D0 kD0
C
t X
t i
X
t
i
t i
k .ˇ/i k
ˇ0
k i
x kC1 y i .ˇ C i
k/
1
ˇ0 C k
i
1
i D0 kD0
G1 ˛ C 1 C i C k; ˇ
C
t X
t i
X
t
i
k; ˇ 0 C 1 C k
1Ci
t i
k .ˇ/i k
ˇ0
k i
i I x; y
x k y i C1 .ˇ C i
k/1 ˇ 0 C k
i D0 kD0
G1 ˛ C 1 C i C k; ˇ C 1 C i
D
t C1 t C1
X
Xi
i D0 kD0
k; ˇ 0
t C1 t i C1
.ˇ/i k
i
k
ˇ0
1Ck
k i
xk yi
i I x; y
i
1
i I x; y
RECURSION FORMULAS FOR G1 AND G2 HORN HYPERGEOMETRIC FUNCTIONS
G1 ˛ C i C k; ˇ C i
k; ˇ 0 C k
1157
i I x; y ;
where we replace k by k 1 in the second summation term and i by i 1 in the third
summation term in the first equality. So, we obtain the recursion formula (2.4). In a
similar manner, the relation (2.5) can be easily proved.
Theorem 3. For the function G1 , we have
G1 ˛; ˇ C n; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y
C ˛ ˇ0
1
y
n
X
(2.6)
G1 ˛ C 1; ˇ C k; ˇ 0
1I x; y
kD1
˛ˇ 0 x
n
X
kD1
.ˇ C k 1/ 1
G1 ˛ C 1; ˇ C k
.ˇ C k 1/
2; ˇ 0 C 1I x; y
and
G1 ˛; ˇ
n; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y
˛ ˇ0
y
1
n
X
(2.7)
G1 ˛ C 1; ˇ
k C 1; ˇ 0
1I x; y
kD1
0
˛ˇ x
n
X
.ˇ
kD1
k/ 1
G1 ˛ C 1; ˇ
.ˇ k/
k
1; ˇ 0 C 1I x; y :
Proof. Using the definition of the function G1 and the equality
p m
.ˇ C 1/p m D .ˇ/p m 1 C
ˇ ˇ
we can easily obtain the contiguous function
G1 ˛; ˇ C 1; ˇ 0 I x; y D G1 ˛; ˇ; ˇ 0 I x; y
C ˛ ˇ 0 1 yG1 ˛ C 1; ˇ C 1; ˇ 0
˛
.ˇ/
ˇ
1
0
ˇ xG1 ˛ C 1; ˇ
(2.8)
1I x; y
1; ˇ 0 C 1I x; y :
If we apply this contiguous relation for two times for the function G1 with the parameter ˇ C 2, we get
G1 ˛; ˇ C 2; ˇ 0 I x; y D G1 ˛; ˇ C 1; ˇ 0 I x; y
C ˛ ˇ 0 1 yG1 ˛ C 1; ˇ C 2; ˇ 0 1I x; y
.ˇ C 1/ 1 0 ˇ xG1 ˛ C 1; ˇ; ˇ 0 C 1I x; y
˛
1
.ˇ C 1/
0
D G1 ˛; ˇ; ˇ I x; y
1158
RECEP ŞAHIN AND SUSAN RIDHA SHAKOR AGHA
C ˛ ˇ0
1
yG1 ˛ C 1; ˇ C 1; ˇ 0
1I x; y
.ˇ/ 1 0
ˇ xG1 ˛ C 1; ˇ 1; ˇ 0 C 1I x; y
ˇ
C ˛ ˇ 0 1 yG1 ˛ C 1; ˇ C 2; ˇ 0 1I x; y
.ˇ C 1/ 1 0
˛
ˇ xG1 ˛ C 1; ˇ; ˇ 0 C 1I x; y :
.ˇ C 1/
By iterating this method on G1 with ˇ C n for n times, we find
G1 ˛; ˇ C n; ˇ 0 I x; y D G1 ˛; ˇ C 1; ˇ 0 I x; y
˛
C ˛ ˇ0
n
X
y
1
G1 ˛ C 1; ˇ C k; ˇ 0
1I x; y
kD1
n
X
.ˇ C k
˛ˇ 0 x
1/ 1
G1 ˛ C 1; ˇ C k
.ˇ C k 1/
kD1
2; ˇ 0 C 1I x; y :
Replacing ˇ by ˇ
G1 ˛; ˇ
1 in contiguous relation (2.8), we obtain
1; ˇ I x; y D G1 ˛; ˇ; ˇ 0 I x; y ˛ ˇ 0 1 yG1 ˛ C 1; ˇ; ˇ 0
0
1I x; y
.ˇ 1/ 1 0
ˇ xG1 ˛ C 1; ˇ 2; ˇ 0 C 1I x; y :
.ˇ 1/
If we apply this relation to the function G1 with the parameter ˇ n for n times, we
obtain the recursion formulas (2.7).
C˛
Theorem 4. For the function G1 , the equalities
G1 ˛; ˇ; ˇ 0 C nI x; y D G1 ˛; ˇ; ˇ 0 I x; y
C ˛ .ˇ/
1x
n
X
G1 ˛ C 1; ˇ
(2.9)
1; ˇ 0 C kI x; y
kD1
˛ˇy
n
X
.ˇ 0 C k
kD1
.ˇ 0 C k
1/ 1
G1 ˛ C 1; ˇ C 1; ˇ 0 C k
1/
2I x; y
and
G1 ˛; ˇ; ˇ 0
nI x; y D G1 ˛; ˇ; ˇ 0 I x; y
˛ .ˇ/
1x
n
X
G1 ˛ C 1; ˇ
(2.10)
1; ˇ 0
k C 1I x; y
kD1
C ˛ˇy
n
X
.ˇ 0
kD1
hold.
.ˇ 0
k/ 1
G1 ˛ C 1; ˇ C 1; ˇ 0
k/
k
1I x; y
RECURSION FORMULAS FOR G1 AND G2 HORN HYPERGEOMETRIC FUNCTIONS
1159
Proof. If we use following equalities
G1 ˛; ˇ; ˇ 0 C 1I x; y D G1 ˛; ˇ; ˇ 0 I x; y
C ˛ .ˇ/
1 xG1
˛ C 1; ˇ
1; ˇ 0 C 1I x; y
.ˇ 0 / 1
yG1 ˛ C 1; ˇ C 1; ˇ 0
ˇ0
1I x; y D G1 ˛; ˇ; ˇ 0 I x; y
˛ˇ
G1 ˛; ˇ; ˇ 0
˛ .ˇ/
C ˛ˇ
1 xG1
˛ C 1; ˇ
1I x; y
1; ˇ 0 I x; y
.ˇ 0 1/ 1
yG1 ˛ C 1; ˇ C 1; ˇ 0
.ˇ 0 1/
2I x; y
we obtain recursion formulas given by (2.9) and (2.10).
3. R ECURSION FORMULAS OF G2
In this section, we give some recursion formulas for the function G2 : We first
present the recursion formulas for the function G2 about the parameter ˛ and ˛ 0 :
Theorem 5. The function G2 satisfies the recursion formulas W
G2 ˛ C n; ˛ 0 ; ˇ; ˇ 0 I x; y D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
C .ˇ/
0
1ˇ x
n
X
G2 ˛ C k; ˛ 0 ; ˇ
(3.1)
1; ˇ 0 C 1I x; y
kD1
and
G2 ˛
n; ˛ 0 ; ˇ; ˇ 0 I x; y D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
.ˇ/
0
1ˇ x
n
X1
G2 ˛
(3.2)
k; ˛ 0 ; ˇ
1; ˇ 0 C 1I x; y :
kD0
Proof. By the definition of the function G2 , we get
G2 ˛ C 1; ˛ 0 ; ˇ; ˇ 0 I x; y D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
C .ˇ/
1ˇ
0
xG2 ˛ C 1; ˛ 0 ; ˇ
1; ˇ 0 C 1I x; y :
Applying this relation n times recursively, as the same as we have done in the proof
of Theorem 1, we immediately have complete the proof (3.1).
The function G2 in the above theorem can be expressed in other forms as follows:
Theorem 6. For the function G2 , the equalities
G2 ˛ C n; ˛ 0 ; ˇ; ˇ 0 I x; y
1160
RECEP ŞAHIN AND SUSAN RIDHA SHAKOR AGHA
!
n
X
n
.ˇ/
D
k
ˇ0
k
k
x k G2 ˛ C k; ˛ 0 ; ˇ
k; ˇ 0 C kI x; y
(3.3)
kD0
and
G2 ˛
n; ˛ 0 ; ˇ; ˇ 0 I x; y
!
n
X
n
.ˇ/
D
k
ˇ0
k
k
. x/k G2 ˛; ˛ 0 ; ˇ
k; ˇ 0 C kI x; y ; (3.4)
kD0
hold.
Proof. By the inductive method as we have done in the proof of Theorem 2, the
proof can be easily seen.
Theorem 7. The function G2 satisfies the following recursion formulas
G2 ˛; ˛ 0 C n; ˇ; ˇ 0 I x; y D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
C ˇ ˇ0
y
1
n
X
G2 ˛; ˛ 0 C k; ˇ C 1; ˇ 0
(3.5)
1I x; y
kD1
D
n
X
!
n
.ˇ/k ˇ 0
k
kD0
k
y k G2 ˛; ˛ 0 C k; ˇ C k; ˇ 0
kI x; y
and
G2 ˛; ˛ 0
n; ˇ; ˇ 0 I x; y D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
ˇ ˇ
0
y
1
n
X1
(3.6)
G2 ˛; ˛ 0
k; ˇ C 1; ˇ 0
1I x; y
kD0
D
n
X
kD0
!
n
.ˇ/k ˇ 0
k
k
. y/k G2 ˛; ˛ 0 ; ˇ C k; ˇ 0
Proof. By the definition of the function G2 , we get
G2 ˛; ˛ 0 C 1; ˇ; ˇ 0 I x; y D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
C ˇ ˇ 0 1 yG2 ˛; ˛ 0 C 1; ˇ C 1; ˇ 0
1I x; y
kI x; y
and
G2 ˛; ˛ 0
1; ˇ; ˇ 0 I x; y D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
ˇ ˇ 0 1 yG2 ˛; ˛ 0 ; ˇ C 1; ˇ 0
1I x; y :
By applying this relation for n times, as the same as we have done in the proof of
Theorem 1, we complete the proof.
RECURSION FORMULAS FOR G1 AND G2 HORN HYPERGEOMETRIC FUNCTIONS
1161
Now, we present the recursion formulas of the function G2 about the parameter ˇ
and ˇ 0 .
Theorem 8. Recursion formulas for the function G2 are as follows
G2 ˛; ˛ 0 ; ˇ C n; ˇ 0 I x; y
D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y C ˛ 0 ˇ 0
y
1
n
X
(3.7)
G2 ˛; ˛ 0 C 1; ˇ C k; ˇ 0
1I x; y
kD1
˛ˇ 0 x
n
X
.ˇ C k
kD1
1/ 1
G2 ˛ C 1; ˛ 0 ; ˇ C k
.ˇ C k 1/
2; ˇ 0 C 1I x; y
and
G2 ˛; ˛ 0 ; ˇ
n; ˇ 0 I x; y
D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
(3.8)
˛0 ˇ0
y
1
n
X
G2 ˛; ˛ 0 C 1; ˇ
k C 1; ˇ 0
1I x; y
kD1
C ˛ˇ 0 x
n
X
kD1
.ˇ k/ 1
G2 ˛ C 1; ˛ 0 ; ˇ
.ˇ k/
k
1; ˇ 0 C 1I x; y :
Proof. From the definition of the function G2 , we have the following relation:
G2 ˛; ˛ 0 ; ˇ C 1; ˇ 0 I x; y
D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y C ˛ 0 ˇ 0 1 yG2 ˛; ˛ 0 C 1; ˇ C 1; ˇ 0 1I x; y
.ˇ/ 1
˛ˇ 0 x
G2 ˛ C 1; ˛ 0 ; ˇ 1; ˇ 0 C 1I x; y :
ˇ
If we apply this relation for n times, we can easily prove the theorem by the same
method as we have done in the proof of Theorem 3.
Theorem 9. Recursion formulas for the function G2 are as follows
G2 ˛; ˛ 0 ; ˇ; ˇ 0 C nI x; y
D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y C ˛ .ˇ/
1x
n
X
G2 ˛ C 1; ˛ 0 ; ˇ
(3.9)
1; ˇ 0 C kI x; y
kD1
˛ 0 ˇy
n
X
kD1
.ˇ 0 C k
.ˇ 0 C k
1/ 1
G2 ˛; ˛ 0 C 1; ˇ C 1; ˇ 0 C k
1/
2I x; y
and
G2 ˛; ˛ 0 ; ˇ; ˇ 0
nI x; y
(3.10)
1162
RECEP ŞAHIN AND SUSAN RIDHA SHAKOR AGHA
D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y
˛ .ˇ/
1x
n
X
G2 ˛ C 1; ˛ 0 ; ˇ
1; ˇ 0
k C 1I x; y
kD1
C ˛ 0 ˇy
n
X
.ˇ 0
kD1
.ˇ 0
k/ 1
G2 ˛; ˛ 0 C 1; ˇ C 1; ˇ 0
k/
k
1I x; y :
Proof. From the definition of the function G2 , we have the following relation:
G2 ˛; ˛ 0 ; ˇ; ˇ 0 C nI x; y
D G2 ˛; ˛ 0 ; ˇ; ˇ 0 I x; y C ˛ .ˇ/ 1 xG2 ˛ C 1; ˛ 0 ; ˇ 1; ˇ 0 C 1I x; y
.ˇ 0 / 1
˛ 0 ˇy
G2 ˛; ˛ 0 C 1; ˇ C 1; ˇ 0 1I x; y :
0
ˇ
If we apply this relation for n times, we can easily prove the theorem by the same
method as we have done in the proof of Theorem 3.
R EFERENCES
[1]
[2]
[3]
[4]
J. Horn, “Hypergeometrische funktionen zweier veräMath. Ann., vol. 105, pp. 381–407, 1931.
E. Rainville, Special functions. New York: Macmillan Company, 1971.
L. Slater, Generalized Hypergeometric Functions. Cambridge: Cambridge University Press, 1966.
H. Srivastava and P. Karlsson, Multiple Gaussian Hpergeometric Series. New York: Halsted Press
(John Wiley and Sons), 1985.
[5] H. Srivastava and H. Monacha, A Treatise on Generating Functions. New York: Halsted Press,
1984.
[6] X. Wang, “Recursion formulas for appell functions,” Integral Transforms Spec. Funct., vol. 23,
no. 6, pp. 421–433, 2012, doi: 10.1080/10652469.2011.596483.
Authors’ addresses
Recep Şahin
Kırıkkale University, Department of Mathematics, Yahşihan, 71450 Kırıkkale, Turkey
E-mail address: [email protected]
Susan Rıdha Shakor Agha
Kırıkkale University, Department of Mathematics, Yahşihan, 71450 Kırıkkale, Turkey
E-mail address: [email protected]