Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 5 Issue 4 (2013), Pages 65-77.
EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTIONS
AND THEIR APPLICATIONS
(COMMUNICATED BY HARI M. SRIVASTAVA)
MIN-JIE LUO, R. K. RAINA
(κ )
Abstract. By means of the extended beta function Bb,dl , we introduce new
extensions of the generalized hypergeometric functions and present some new
integral and series representations (including the one obtained by adopting the
well-known Ramanujan’s Master Theorem). Further generalizations are also
considered. We point out the usefulness of some of the the results by showing
their connections with other special functions and with a class of fractional
calculus operators.
1. Introduction and Prelimiaries
New extensions of some of the well-known special functions (e.g. gamma function, beta function, Gauss hypergeometric function, etc.) have been extensively
−1
studied in recent past. By inserting a regularization factor e−bt , Chaudhry and
Zubair [3] have introduced the following extension of the gamma function:
Z ∞
b
x−1
dt, ℜ (b) > 0,
(1.1)
t
exp −t −
Γb (x) =
t
0
and Chaudhry et al. [4] considered the extension of Euler’s beta function in the
following form:
Z 1
b
y−1
x−1
t
(1 − t)
exp −
Bb (x, y) =
dt, ℜ (b) > 0.
(1.2)
t (1 − t)
0
Later, Chaudhry et al. [5] used Bb (x, y) to extend the Gauss hypergeometric
function given by
Fb (α, β; γ; z) =
∞
X
(α)m
m=0
Bb (β + m, γ − β) z m
,
B (β, γ − β)
m!
(1.3)
(b ≥ 0; |z| < 1; ℜ (γ) > ℜ (β) > 0)
2010 Mathematics Subject Classification. 33A65, 33C90.
Key words and phrases. Extended beta function, Generalized hypergeometric functions, Ramanujan’s Master Theorem.
c
2013
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted July 2, 2013. Published November 4, 2013.
65
66
MIN-JIE LUO, R. K. RAINA
where (α)m denotes the Pochhammer symbol defined in terms of gamma functions
by
(
Γ (α + m)
1
m = 0; α ∈ C\ {0}
(α)m =
=
Γ (α)
α (α + 1) (α + 2) · · · (α + m − 1) m ∈ N; α ∈ C.
For b = 0, the function (1.3) evidently reduces to the usual Gauss hypergeometric
function.
Subsequently, Ozergin et al. [12] introduced generalizations of the gamma and
Euler’s beta functions given by
Z ∞
b
(α,β)
x−1
dt
(1.4)
Γb
(x) =
t
1 F1 α; β; −t −
t
0
(ℜ (α) > 0, ℜ (β) > 0, ℜ (b) > 0, ℜ (x) > 0)
and
1
tx−1 (1 − t)y−1 1 F1 α; β;
−b
dt
t (1 − t)
0
(ℜ (α) > 0, ℜ (β) > 0, ℜ (b) > 0, ℜ (x) > 0, ℜ (y) > 0) .
(α,β)
Bb
(x, y) =
Z
(1.5)
(α,β)
By applying Bb
(x, y), a slightly generalized version of (1.3) was given in [12].
In a recent interesting paper, Srivastava et al. [15] introduced a family of generalized incomplete hypergeometric functions by replacing one of the Pochhammer
symbols involving in the coefficient of the generalized hypergeometric function with
the incomplete Pochhammer symbols. For further works on the subject of the
incomplete Pochhammer symbols and the generalized incomplete hypergeometric
functions, one may also refer to the papers [14], [17], [18] and [19].
In the sequel, we shall be employing in our extended results the following definition due to Srivastava et al. [16] (see also [14] and [18]).
Definition 1.1. ([16, p.243]) Let a function Θ (κl ; z) be analytic within the disk
|z| < R (0 < R < ∞) and let its Taylor-Maclaurin coeficients be explicitly denoted
by the sequence {κl }l∈N0 . Suppose also that the function Θ (κl ; z) can be continued
analytically in the right half-plane ℜ (z) > 0 with the asymptotic property given as
follows:
∞
X zl



(|z| < R; 0 < R < ∞; κ0 = 1) ,
κl

l!
(1.6)
Θ (κl ; z) = l=0

1

ω

(ℜ (z) → ∞; M0 > 0; ω ∈ C) ,
M0 z exp (z) 1 + O
z
for some suitable constants M0 and ω depending essentially on the sequence {κl }l∈N0 .
(κ )
(κl )
The extended gamma function Γb l (z) and the extended beta function Bb
can then be expressed as (see [16, Equations (2.2) and (2.3)])
Z ∞
b
(κl )
z−1
t Θ κl ; −t −
Γb (z) =
dt,
t
0
(ℜ (z) > 0; ℜ (b) ≥ 0)
and
Z 1
b
(κl )
β−1
α−1
dt,
t
(1 − t)
Θ κl ; −
Bb (α, β) =
t (1 − t)
0
(α, β)
(1.7)
(1.8)
EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTIONS
67
(min {ℜ (α) , ℜ (β)} > 0; ℜ (b) ≥ 0) .
We shall also make use of the following definition of a two-parameter extension
of (1.8) due to Srivastava et al. [16, p.256, Eqn. (6.1)] (see also [16, Section 6] for
other related two-parameter definitions):
Z 1
b
d
(κ )
tα−1 (1 − t)β−1 Θ κl ; − −
Bb,dl (α, β) =
dt,
(1.9)
t 1−t
0
(min {ℜ (α) , ℜ (β)} > 0; min {ℜ (b) , ℜ (d)} ≥ 0) .
In this paper, we introduce some extended forms of the generalized hypergeometric functions by means of (1.9). Section 2 gives the extensions of Gauss hypergeometric functions and Section 3 treats extensions of the generalized hypergeometric
functions together with some of their fundamental properties. Mellin-Barnes type
integral representations are also derived by the application of the well-known Ramanujans Master Theorem. In Section 4, we consider further extensions and point
out relevant connections of some of the results with known (and new) results including a useful relationship with a class of fractional calculus operators.
2. Extended Gauss Hypergeometric Functions
(κ )
Using the extended beta function Bb,dl (α, β) defined by (1.9), we can easily form
another series representation of the Gauss hypergeometric function


∞
α1 , α2
X
(α1 )n (α2 )n z n

; z =
,
(2.1)
2 F1
(β1 )n
n!
n=0
β1
|z| < 1; α1 , α2 ∈ C; β1 ∈ C \ Z−
0 .
Let us replace
(κ )
Bb,dl (α2 + n, β1 − α2 )
(α2 )n
B (α2 + n, β1 − α2 )
=
−→
(β1 )n
B (α2 , β1 − α2 )
B (α2 , β1 − α2 )
in (2.1), then we obtain the extended form of the Gauss hypergeometric function
(2.1) in the following form:
(κ )
Definition 2.1. [16] The Extended Gauss hypergeometric function 2 F1 l is defined
by


(κ )
∞
α1 , α2
X
Bb,dl (α2 + n, β1 − α2 ) z n
(κl ) 

F
,
(2.2)
;
z;
b,
d
=
(α
)
2 1
1 n
B (α2 , β1 − α2 )
n!
n=0
β1
(|z| < 1; ℜ (β1 ) > ℜ (α2 ) > 0; min {ℜ (b) , ℜ (d)} ≥ 0) .
The integral representation of (2.2) is contained in the following:
Theorem 2.2.


Z 1
α1 , α2
1
(κl ) 
β −α −1

tα2 −1 (1 − t) 1 2
;
z;
b,
d
=
F
2 1
B (α2 , β1 − α2 ) 0
β1
b
d
−α1
· (1 − zt)
Θ κl ; − −
dt, (2.3)
t 1−t
(ℜ (β1 ) > ℜ (α2 ) > 0; ℜ (b) > 0, ℜ (d) > 0; b = d = 0, |arg (1 − z)| < π) .
68
MIN-JIE LUO, R. K. RAINA
(κ )
Proof. Replacing the extended beta function Bb,dl (α2 + n, β1 − α2 ) in (2.2) by its
integral representation given by (1.9), and then interchanging the order of summation and integration (which can be justified due to the absolute convergence
of the integral and the series involved), the integral representation (2.3) follows
immediately after some necessary simplification.
For d = b, the integral representation (2.3) corresponds to the known result [16,
p. 244, Eq. (2.6)].
3. Extended Generalized Hypergeometric Functions
In this section, we consider the generalized hypergeometric function and extend
(κ )
it by using the extended beta function Bb,dl (α, β).
The generalized hypergeometric function with p numerator and q denominator
parameters is defined by (see, e. g. [8, p. 27])


α1 , · · · , αp

; z
p Fq (α1 , · · · , αp ; β1 , · · · , βq ; z) = p Fq
β1 , · · · , βq
∞
X
(α1 )k · · · (αp )k z k
,
(3.1)
=
(β1 )k · · · (βq )k k!
k=0
(αl , βj ∈ C, βj 6= 0, −1, −2, · · · , l = 1, · · · , p; j = 1, · · · , q)
which is absolutely convergent for all values of z ∈ C, if p ≤ q. When P
p = q + 1,
q
the series is absolutely convergent for |z| < 1 and for |z| = 1, when ℜ( j=1 βj −
Pp
it is conditionally convergent for |z| = 1 (z 6= 1), if −1 <
l=1
Pqαl ) > 0,Pwhile
p
ℜ( j=1 βj − l=1 αl ) ≤ 0. More details can be found in [2], [10] and [13].
(κ )
In terms of the extended beta function Bb,dl (α, β) defined by (1.9), we can
construct a suitable extension of (3.1). The following cases need to be considered:
(1) For p = q+1, the coefficients of p Fq (α1 , · · · , αp ; β1 , · · · , βq ; z) can be rewritten as
q
q
Y
Y
B (αj+1 + n, βj − αj+1 )
(αj+1 )
= (α1 )n
, (n ∈ N0 ) .
(α1 )n
(β
)
B (αj+1 , βj − αj+1 )
j
j=1
j=1
By substituting the extended beta function (1.9) for each B(αj+1 + n, βj −
αj+1 ), we get the coefficients
(α1 )n
(κ )
q
Y
Bb,dl (αj+1 + n, βj − αj+1 )
j=1
B (αj+1 , βj − αj+1 )
, (n ∈ N0 ) .
(2) For p = q, the coefficients of our extension are simply
(κ )
q
Y
Bb,dl (αj + n, βj − αj )
j=1
B (αj , βj − αj )
, (n ∈ N0 ) .
(3) When p < q, the only reasonable construction of coefficients is
(κl )
p
1 Y Bb,d (αj + n, βr+j − αj )
, (n ∈ N0 ) .
(βi )n j=1
B (αj , βr+j − αj )
i=1
r
Y
EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTIONS
69
We can now give the formal definition of our extended generalized hypergeometric
function as follows:
Definition 3.1. For suitably constrained (real or complex) parameters αj , j =
1, · · · p; βi , i = 1, · · · , q, we define the extended generalized hypergeometric function
by


α1 , · · · , αp
(κl )
(α1 , · · · , αp ; β1 , · · · , βq ; z; b, d) = p Fq(κl ) 
; z; b, d
p Fq
β1 , · · · , βq
 ∞
(κl )
q
Y Bb,d (αj+1 + m, βj − αj+1 ) z m
X



,
(α
)

1
m

B (αj+1 , βj − αj+1 )
m!

m=0
j=1




(|z| < 1; p = q + 1; ℜ (βj ) > ℜ (αj+1 ) > 0)



(κl )

q
∞ Y

X
Bb,d (αj + m, βj − αj ) z m


,
B (αj , βj − αj )
m!
(3.2)
= m=0 j=1




(z ∈ C; p = q; ℜ (βj ) > ℜ (αj ) > 0)



(κl )
p
r
∞ Y

X
Y

Bb,d (αj + m, βr+j − αj ) z m
1



,


(βi )m j=1
B (αj , βr+j − αj )
m!

m=0 i=1



(z ∈ C; r = q − p, p < q; ℜ (βr+j ) > ℜ (αj ) > 0)
The following theorem demonstrates that the form of the Euler type integral rep(κ )
resentation of p Fq l is very similar to that of the Euler type integral representation
of p Fq .
Theorem 3.2. For the extended generalized hypergeometric function defined by
(3.2), we have the following integral representation:


Z 1
α1 , · · · , αp
Γ (βq )
β −a −1
(κl ) 

; z; b, d =
tαp −1 (1 − t) q p
p Fq
Γ (αp ) Γ (βq − αp ) 0
β1 , · · · , βq


α1 , · · · , αp−1
b
d
(κl ) 

; zt; b, d Θ κl ; − −
· p−1 Fq−1
dt. (3.3)
t 1−t
β1 , · · · , βq−1
(ℜ (βq ) > ℜ (αp ) > 0, min{ℜ (b) , ℜ (d)} > 0; d = b = 0, |arg (1 − z)| < π)
Proof. We need to verify that formula (3.3) holds for three different expressions of
(κl )
(α1 , · · · , αp ; β1 , · · · , βq ; z; b, d) given in (3.2), respectively. Consider the case
p Fq
p = q + 1. In view of the representation that
(κ )
Bb,dl (αq+1 + m, βq − αq+1 )
Γ (βq )
=
B (αq+1 , βq − αq+1 )
Γ (αq+1 ) Γ (βq − αq+1 )
Z 1
d
b
βq −αq+1 −1
αq+1 +m−1
dt, (3.4)
t
(1 − t)
Θ κl ; − −
·
t 1−t
0
(m ∈ N0 , min{ℜ (b) , ℜ (d)} > 0, ℜ (βq ) > ℜ (αq+1 ) > 0)
70
MIN-JIE LUO, R. K. RAINA
we find that


α1 , · · · , αq+1
(κl ) 
; z; b, d
q+1 Fq
β1 , · · · ,
βq
Z 1
d
b
Γ (βq )
β −α
−1
tαq+1 −1 (1 − t) q q+1 Θ κl ; − −
=
Γ (αq+1 ) Γ (βq − αq+1 ) 0
t 1−t
·
∞
X
(α1 )m
m=0
q−1
Y
(κ )
Bb,dl (αj+1 + m, βj − αj+1 ) (zt)m
j=1
B (αj+1 , βj − αj+1 )
m!
dt
Z 1
Γ (βq )
tαq+1 −1 (1 − t)βq −αq+1 −1
=
Γ (αq+1 ) Γ (βq − αq+1 ) 0


α1 , · · · , αq
b
d
(κ )
; zt; b, d Θ κl ; − −
· q Fq−1l 
dt.
t 1−t
β1 , · · · , βq−1
(3.5)
It is clear that the relation (3.3) is also valid for p ≤ q, and this completes the
proof.
Remark. A multidimensional case of the Euler type integral representation of
(3.5) is given by


q
α1 , · · · , αq+1
Y
Γ (βj )
(κl ) 

; z; b, d =
q+1 Fq
Γ
(α
)
j+1 Γ (βj − αj+1 )
j=1
β1 , · · · ,
βq
Z 1
Z 1Y
q d
b
αj+1
βj −αj+1 −1
Θ κl ; − −
tj (1 − tj )
·
···
tj
1 − tj
0
0 j=1
−α1
· (1 − t1 t2 · · · tq z)
dt1 · · · dtq ,
which follows from the repeated application of the functional equation (3.5). If we
set d = b = 0, then this representation reduces to the one given in [7, p.132, Eq.
(4.2)].
Theorem 3.3. The following derivative formula holds for p ≤ q + 1.



α , · · · , αp

dn  (κl )  1

F
;
z;
b,
d
p
q

dz n 
β1 , · · · , βq


(α1 )n · · · (αp )n (κl ) α1 + n, · · · , αp + n

=
; z; b, d (n ∈ N0 ) . (3.6)
pF
(β1 )n · · · (βq )n q
β1 + n, · · · , βq + n
(κ )
Proof. Differentiating q+1 Fq l with respect to z, we obtain



α1 , · · · , αp

d 
(κl ) 
; z; b, d
q+1 Fq

dz 
β1 , · · · , βq
=
∞
X
m=1
(α1 )m
(κ )
q
Y
Bb,dl (αj+1 + m, βj − αj+1 ) z m−1
. (3.7)
B (αj+1 , βj − αj+1 )
(m − 1)!
j=1
EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTIONS
71
Replacing m → m + 1 in the right-hand side of (3.7), we are lead to



Qq
α1 , · · · , αq+1

d 
j=1 αj+1
(κl ) 

; z; b, d
= α1 Qq
q+1 Fq

dz 
j=1 βj
β1 , · · · ,
βq


α1 + 1, · · · , αq+1 + 1
; z; b, d . (3.8)
· q+1 Fq(κl ) 
β1 + 1, · · · ,
βq + 1
Recursive application of this procedure n-times gives us the general form (3.6).
Similarly, we can prove this result for the case p ≤ q.
For p = 2 and q = 1, we at once get



α , α2

dn  (κl )  1

;
z;
b,
d
F
2
1

dz n 
β1


α + n, α2 + n
(α1 )n (α2 )n (κl )  1
=
; z; b, d . (3.9)
2 F1
(β1 )n
β1 + n
If we put Θ (κl ; z) = 1 F1 (α; β; z) and b = d, then equation (3.9) reduces to





a, b
a + n, b + n

n 
(b)n (a)n (α,β) 
d
(α,β) 
F
Fb
; z =
; z ,

dz n  b
(c)n
c
c+n
which corresponds to the known result [12, Theorem 3.3].
Next, we derive the Mellin-Barnes type contour integral representation of the
function (3.2). We need the following well-known theorem which is widely used to
evaluate definite integrals and infinite series.
Theorem 3.4 (Ramanujan’s Master Theorem [1]). Assume f admits an expansion of the form:
f (x) =
∞
X
λ (k)
(−x)k (λ (0) 6= 0) .
k!
k=0
Then, the Mellin transform of f is given by
Z ∞
xs−1 f (x) dx = Γ (s) λ (−s) .
0
By means of the Ramanujan’s Master Theorem, we obtain the following MellinBarnes type integral representation.
Theorem 3.5. The Mellin-Barnes type integral representation of the function (3.2)
is given by


α1 , · · · , αp
(κl ) 
; z; b, d
p Fq
β1 , · · · , βq
72
MIN-JIE LUO, R. K. RAINA

Z Y
(κ )
q

Bb,dl (αj+1 − s, βj − αj+1 ) Γ (s) Γ (α1 − s)
1

−s


(−z) ds,


2πi
B
(α
,
β
−
α
)
Γ
(α
)
j+1
j
j+1
1



L1 j=1




(p = q + 1, ℜ (βj ) > ℜ (αj+1 ) > 0, j = 1, · · · , q, ℜ (α1 ) > 0)



Z Y
(κ )
q


Bb,dl (αj − s, βj − αj )

−s
 1
Γ (s) (−z) ds,
2πi
B
(α
,
β
−
α
)
j
j
j
=
L2 j=1




(p = q, ℜ (βj ) > ℜ (αj ) > 0, j = 1, · · · , q)




Z Y
(κ )
q
r

Bb,dl (αj − s, βj+r − αj ) Y

Γ (βi )
−s

 1
Γ (s) (−z) ds,


2πi
B
(α
,
β
−
α
)
Γ
(β
−
s)

j
j+r
j
i

i=1

L3 j=1


 (r = q − p, p < q, ℜ (β ) > (α ) > 0, ℜ (β ) > 0, i = 1, · · · , r)
r+j
j
i
(3.10)
where Li , i = 1, 2, 3 are Mellin-Barnes-type contours from −i∞ to i∞, with the
usual indentations in order to separate one set of poles from the other set of poles
of the integrand.
Proof. The result follows rather directly upon using the Ramanujan’s Master Theorem and the inverse Mellin transform.
There exists another kind of Mellin-Barnes type integral representation of the
function (3.2), if we can express the integrand of the contour integral in (3.10) into
a simpler form.
In what follows, we denote by
Cq (α, β; x) =
q
Y
Γ (xj + αj+1 ) Γ (βj − αj+1 + xj )
Γ (2xj + βj )
j=1
(3.11)
and
Hq (α, β; x) =
Cq (α, β; x)
.
Cq (α, β; 0)
(3.12)
Theorem 3.6. Let d = b with ℜ (b) > 0, then the Mellin-Barnes type integral
representation of the second kind for the function (3.2) with p = q + 1 is given by


+i∞
+i∞
Z
Z
q
α1 , · · · , αq+1
Y
1
(κ )
(κl ) 

; z; b =
Γ0 l (sj )
Hq (α, β; s)
···
q+1 Fq
q
(2πi)
j=1
β1 , · · · ,
βq
−i∞
−i∞


α1 , s1 + α2 , · · · , sq + αq+1
Pq
· q+1 Fq 
; z  b− j=1 sj ds1 · · · dsq , (3.13)
2s1 + β1 , · · · , 2sq + βq
where the function Hq (α, β; s) is defined by (3.12).
Proof. Following [12, Theorem 2.2], we can get the Mellin transform representation
of the extended beta function (1.8). Indeed, we have
(κl )
Bb
(αj+1 + m, βj − αj+1 )
1
=
2πi
Zi∞
−i∞
(κl )
B (sj + αj+1 + m, sj + βj − αj+1 ) Γ0
(sj ) b−sj dsj
EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTIONS
1
=
2πi
Zi∞
−i∞
B (sj + αj+1 , βj − αj+1 + sj )
73
(sj + αj+1 )m (κl )
(sj ) b−sj dsj , (3.14)
Γ
(2sj + βj )m 0
which in view of (3.2) and some elementary simplifications give the desired result
(3.13).
Remark. In view of (3.13), we get the expansion formula:


α1 , · · · , αq+1
(κl ) 
; zζ; b
q+1 Fq
β1 , · · · ,
βq


k
∞
−k, α2 , · · · , αq+1
X
(α1 )n
z
α
(κl ) 
; ζ; b , (3.15)
= (1 − z) 1
q+1 Fq
k!
z−1
β
,
·
·
·
,
β
k=0
1
q
which follows on applying Theorem 3.8 and [10, Eq.16.10.2].
4. Further results
Our definition of extended generalized hypergeometric function can further be
generalized to the following form:
Definition 4.1. For suitably constrained (real or complex) parameters αj , j =
1, · · · , p; βi , i = 1, · · · , q, we define the extended generalized hypergeometric functions by


(α1 , k1 ) , · · · , (αp , kp )
(κl ) 
; z; b, d
p Fq
β1 ,
···
βq ,
 ∞
(κ )
q
Y
Bb,dl (αj+1 + kj+1 m, βj − αj+1 ) z m

 X (α )

,

1 k1 m

B (αj+1 , βj − αj+1 )
m!
m=0

j=1




(|z| < 1; p = q + 1; ℜ (βj ) > ℜ (αj+1 ) > 0)


 ∞ q
(κl )


Y
X
Bb,d (αj + kj m, βj − αj ) z m


,
B (αj , βj − αj )
m!
= m=0 j=1
(4.1)




(z ∈ C; p = q; ℜ (βj ) > ℜ (αj ) > 0)



(κl )
p
∞ Y
r

Y
X

B
(α
+
k
j
j m, βr+j − αj ) z m
1

b,d


,


(βr )m j=1
B (αj , βr+j − αj )
m!


m=0
i=1


(z ∈ C; r = q − p; p < q; ℜ (βr+j ) > ℜ (αj ) > 0)
where the new parameters k1 ∈ {0, 1}, kj , j = 2, · · · , p are non-negative integers.
Obviously, (4.1) reduces to (3.2), whenever, kj = 1, j = 1, · · · , p. To illustrate
its advantages, we first consider the following function:


(κ )
∞
(α1 , 1) , (α2 , k2 )
X
Bb,dl (α2 + k2 n, β1 − α2 ) z n
(κl ) 
. (4.2)
; z; b, d =
(α1 )n
2 F1
B (α2 , β1 − α2 )
n!
n=0
β1
Its integral representation can be written as
74
MIN-JIE LUO, R. K. RAINA
(κl )
2 F1


(α1 , 1) , (α2 , k2 )
1

; z; b, d =
B (α2 , β1 − α2 )
β1
Z 1
−α1
b
d
β −α −1
tα2 −1 (1 − t) 1 2
1 − ztk2
Θ κl ; − −
dt. (4.3)
t 1−t
0
(ℜ (β1 ) > ℜ (α2 ) > 0; min {ℜ (b) , ℜ (d)} > 0; b = d = 0, |arg (1 − z)| < π)
As an example, let us compute the case for k2 = 2 and z = 1. We have
Z 1
1
d
b
β1 −α2 −1
α2 −1
2 −α1
dt
Θ κl ; − −
t
(1 − t)
1−t
B (α2 , β1 − α2 ) 0
t 1−t
Z 1 α2 −1
β −α −1
t
(1 − t) 1 2
d
b
=
dt
(1 − t)−α1 (1 + t)−α1 Θ κl ; − −
B (α2 , β1 − α2 )
t 1−t
0
Z 1 α2 +k−1
∞ β −α −α −1
X
b
d
−α1
t
(1 − t) 1 2 1
dt
Θ κl ; − −
=
B (α2 , β1 − α2 )
t 1−t
k
0
k=0
k
∞
=
X (−1) (α1 ) (κ )
1
k
Bb,dl (α2 + k, β1 − α2 − α1 )
B (α2 , β1 − α2 )
k!
k=0
Γ (β1 − α2 − α1 ) Γ (α2 )
Γ (β1 )
=
Γ (α2 ) Γ (β1 − α2 )
Γ (β1 − α1 )
·
∞
X
(κ )
(α1 )k
Bb,dl (α2 + k, β1 − α2 − α1 ) (−1)k
k=0

α ,
Γ (β1 ) Γ (β1 − α2 − α1 ) (κl )  1
=
F
2
Γ (β1 − α2 ) Γ (β1 − α1 ) 1
We thus obtain the following:
B (α2 , β1 − α2 − α1 )

α2
; −1; b, d .
β1 − α1
k!
Theorem 4.2. If ℜ (β1 ) > ℜ (α2 ) > 0 and ℜ (β1 − α2 − α1 ) > 0, then


(α1 , 1) , (α2 , 2)
Γ (β1 ) Γ (β1 − α2 − α1 )
(κl ) 
; 1; b, d =
2 F1
Γ (β1 − α2 ) Γ (β1 − α1 )
β1


α1 ,
α2
(κl ) 
; −1; b, d . (4.4)
· 2 F1
β1 − α1
Remark. Theorem 4.2 is in fact a generalization of the following well-known result
[6, p.117, Theorem 2.3]:




a,
b
a, 2b , b+1
2
Γ
(c)
Γ
(c
−
a
−
b)


; −1 .
(4.5)
; 1 =
2 F1
3 F2
Γ (c − b) Γ (c − a)
c+1
c
c
−
a
,
2
2
It also shows some intrinsic relationship between the extended hypergeometric function and the usual generalized hypergeometric function.
EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTIONS
75
Theorem 4.3. For the extended generalized hypergeometric function (4.1), we have
the following formula holds true:


Z z αp −1
(α1 , k1 ) , · · · , (αp , kp )
β −a −1
t
(z − t) q p
(κl ) 
kp
1−βq

;
b,
d
F
;
cz
=
z
p q
B (αp , βq − αp )
0
β1 ,
··· ,
βq


(α1 , k1 ) , · · · , (αp−1 , kp−1 )
bz
dz
(κ )
· p−1 Fq−1l 
; ctkp ; b, d Θ κl ; − −
dt.
t
z−t
β1 ,
··· ,
βq−1
(4.6)
Proof. Denoting the right-hand side of (4.6) by A (z), then (as indicated before),
we just need to prove the case for p = q + 1. Let t = zv in A (z), we have then
Z 1
Γ (βq ) z βq −1
A (z) =
v αq+1 −1 (1 − v)βq −aq+1 −1
Γ (αp ) Γ (βq − αp ) 0


(α1 , k1 ) , · · · , (αq , kq )
d
b
(κl ) 
kq+1

dv
· q Fq−1
; c (zv)
; b, d Θ κl ; − −
v 1−v
β1 ,
··· ,
βq−1
(κl )
q−1
∞
Y Bb,d
X
(αj+1 + kj+1 m, βj − αj+1 )
Γ (βq ) z βq −1
(α1 )k1 m
Γ (αq+1 ) Γ (βq − αq+1 ) m=0
B (αj+1 , βj − αj+1 )
j=1
Z
m
1
cz kq+1
d
b
βq −aq+1 −1
αq+1 +kq+1 m−1
dv
v
(1 − v)
Θ κl ; − −
·
m!
v 1−v
0
m
(κ )
q
∞
Y
X
Bb,dl (αj+1 + kj+1 m, βj − αj+1 ) cz kq+1
βq −1
=z
(α1 )k1 m
B (αj+1 , βj − αj+1 )
m!
m=0
j=1


(α1 , k1 ) , · · · , (αq , kq ) , (αq+1 , kq+1 )
= z βq −1 q+1 Fq(κl ) 
; cz kq+1 ; b, d .
β1 ,
··· ,
βq−1 ,
βq
=
Theorem 4.4 has the significance and advantages of implementing the modified
definition (4.1) of the extended generalized hypergeometric function. It not only
enables us to formulate the classical results of hypergeometric functions in a new
way, but also provide some new important interpretations. We consider here one
of such new interpretations.
Srivastava et al. in [16] defined the following extended fractional derivative
operator:
Z z

dz
bz
1
−µ−1


Θ κl ; − −
f (t) dt,
(z − t)
 Γ (−µ)

t
z−t
0
µ,b,d
Dz,(κl ) {f (z)} =
(ℜ (µ) < 0) ,

o

m n

µ−m,b,d
d
D
{f
(z)}
, (m − 1 ≤ ℜ (µ) ≤ m (m ∈ N)) .
z,(κl )
dz m
(4.7)
The path of integration in (4.7) is a line in the complex t−plane from t = 0 to
t = z.
76
MIN-JIE LUO, R. K. RAINA
By using the operator (4.7), we can rewrite (4.6) as


(α1 , k1 ) , · · · , (αp , kp )
Γ (βq ) 1−βq
(κl ) 
; cz kp ; b, d =
z
p Fq
Γ (αp )
β1 ,
··· ,
βq



(α1 , k1 ) , · · · , (αp−1 , kp−1 )


−(β −α ),b,d
(κ )
tαp −1 p−1 Fq−1l 
; ctkp ; b, d .
· Dz,(κql ) p


β1 ,
··· ,
βq−1
(4.8)
Acknowledgments. The authors wish to thank the referee for useful suggestions.
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EXTENDED GENERALIZED HYPERGEOMETRIC FUNCTIONS
77
Min-Jie Luo
Department of Applied Mathematics, Donghua University, Shanghai 201620, P. R. China
E-mail address: mathwinnie@myopera.com
R. K. Raina
M.P. University of Agriculture and Technology, India, Present address: 10/11, Ganpati
Vihar, Opposite sector 5, Udaipur 313001, Rajasthan, India
E-mail address: rkraina 7@hotmail.com