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Extension of gamma, beta and hypergeometric functions
Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
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Extension of gamma, beta and hypergeometric functions Emine Özergin ∗ , Mehmet Ali Özarslan, Abdullah Altın Eastern Mediterranean University, Department of Mathematics, Gazimagusa, TRNC, Mersin 10, Turkey
article
abstract
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Article history: Received 29 December 2009 Received in revised form 13 April 2010
The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations. © 2010 Elsevier B.V. All rights reserved.
Keywords: Gamma function Beta function Hypergeometric function Confluent hypergeometric function Mellin transform
1. Introduction In recent years, some extensions of the well known special functions have been considered by several authors [1–7]. In 1994, Chaudhry and Zubair [1] have introduced the following extension of gamma function
Γp (x) :=
∞
∫
t x−1 exp −t − pt −1 dt ,
(1)
0
Re(p) > 0. In 1997, Chaudhry et al. [2] presented the following extension of Euler’s beta function Bp (x, y) :=
1
∫
[
t x−1 (1 − t )y−1 exp − 0
p t (1 − t )
]
dt ,
(2)
(Re(p) > 0, Re(x) > 0, Re(y) > 0) and they proved that this extension has connections with the Macdonald, error and Whittakers function. It is clearly seem that Γ0 (x) = Γ (x) and B0 (x, y) = B (x, y). Afterwards, Chaudhry et al. [8] used Bp (x, y) to extend the hypergeometric functions (and confluent hypergeometric functions) as follows: Fp (a, b; c ; z ) =
∞ − Bp (b + n, c − b) n =0
B (b, c − b)
(a)n
zn n!
p ≥ 0; Re(c ) > Re(b) > 0,
φp (b; c ; z ) =
∞ − Bp (b + n, c − b) z n n=0
B (b, c − b)
n!
p ≥ 0; Re(c ) > Re(b) > 0,
∗
Corresponding author. Tel.: +90 5428565066; fax: +90 3923651604. E-mail addresses: [email protected] (E. Özergin), [email protected] (M.A. Özarslan), [email protected] (A. Altın).
0377-0427/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2010.04.019
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where (λ)ν denotes the Pochhammer symbol defined by
(λ)0 ≡ 1 and (λ)ν :=
Γ (λ + ν) Γ (λ)
and gave the Euler type integral representation Fp (a, b; c ; z ) = p > 0; p = 0
B ( b, c − b)
and
1
∫
1
[
t b−1 (1 − t )c −b−1 (1 − zt )−a exp − 0
p
]
t (1 − t )
dt
|arg (1 − z )| < π < p; Re(c ) > Re(b) > 0.
They called these functions extended Gauss hypergeometric function (EGHF) and extended confluent hypergeometric function (ECHF), respectively. They have discussed the differentiation properties and Mellin transforms of Fp (a, b; c ; z ) and obtained transformation formulas, recurrence relations, summation and asymptotic formulas for this function. Note that F0 (a, b; c ; z ) = 2 F1 (a, b; c ; z ). In this paper, we consider the following generalizations of gamma and Euler’s beta functions (α, β)
Γp
(x) :=
∞
∫
t x−1 1 F1 α; β; −t −
p
dt t Re(α) > 0, Re(β) > 0, Re(p) > 0, Re(x) > 0,
(3)
0
1
−p dt , t (1 − t ) 0 (Re(p) > 0, Re(x) > 0, Re(y) > 0, Re(α) > 0, Re(β) > 0) β) B(α, (x, y) := p
∫
t x−1 (1 − t )y−1 1 F1
α; β;
(α,α)
respectively. It is obvious by (1), (3) and (2), (4) that, Γp (α, β)
(4)
(x) = Γp (x), Γ0(α,α) (x) = Γ (x), B(α,α) (x, y) = Bp (x, y) and p
(x, y) = B (x, y) . In Section 2, different integral representations and some properties of new generalized Euler’s beta function are obtained. Additionally, relations of new generalized gamma and beta functions are discussed. In Section 3, we generalize the (α, β) hypergeometric function and confluent hypergeometric function, using Bp (x, y) obtain the integral representations of this new generalized Gauss hypergeometric functions. Furthermore we discussed the differentiation properties, Mellin transforms, transformation formulas, recurrence relations, summation formulas for these new hypergeometric functions. B0
2. Some properties of gamma and beta functions It is important and useful to obtain different integral representations of the new generalized beta function, for later use. Also it is useful to discuss the relationships between classical gamma functions and new generalizations. For p = 0, we have (α, β) the following integral representation for Γp (x). Theorem 2.1. For the new generalized gamma function, we have (α, β)
Γp
Γ (β) (s) = Γ (α)Γ (β − α)
1
∫ 0
Γpµ2 (s)µα−s−1 (1 − µ)β−α−1 dµ.
Proof. Using the integral representation of confluent hypergeometric function, we have (α, β)
Γp
Γ (β) (s) = Γ (α)Γ (β − α)
∞
∫
1
∫
0
pt us−1 e−ut − u t α−1 (1 − t )β−α−1 dtdu.
0
Now using a one-to-one transformation (except possibly at the boundaries and maps the region onto itself) ν = ut , µ = t in the above equality and considering that the Jacobian of the transformation is J = µ1 , we get
Γp(α, β) (s) =
Γ (β) Γ (α)Γ (β − α)
∞
∫ 0
1
∫
v s−1 e−v−
pµ2
v
dvµα−s−1 (1 − µ)β−α−1 dµ.
0
From the uniform convergence of the integrals, the order of integration can be interchanged to yield that (α, β)
Γp
Γ (β) (s) = Γ (α)Γ (β − α)
∫
Γ (β) = Γ (α)Γ (β − α)
∫
Whence the result.
1
0
0
∞
[∫
v
pµ2 s−1 −v− v e
]
dv µα−s−1 (1 − µ)β−α−1 dµ
0 1
Γpµ2 (s)µα−s−1 (1 − µ)β−α−1 dµ.
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The case p = 0 in the above Theorem gives (see [9, p. 192])
Γ (α, β) (s) =
Γ (β) Γ (α)Γ (β − α)
1
∫
Γ (s)µα−s−1 (1 − µ)β−α−1 dµ =
0
Γ (β)Γ (α − s)Γ (s) . Γ (α)Γ (β − s) (α, β)
The next theorem gives the Mellin transform representation of the function Bp function and Γ (α, β) (s) .
(5)
(x, y) in terms of the ordinary beta
Theorem 2.2. Mellin transform representation of the new generalized beta function is given by ∞
∫
β) ps−1 B(α, (x, y) dp = B(s + x, y + s)Γ (α, β) (s), p
(6)
0
Re(s) > 0, Re(x + s) > 0, Re(y + s) > 0, Re(p) > 0, Re(α) > 0, Re(β) > 0. Proof. Multiplying (4) by ps−1 and integrating with respect to p from p = 0 to p = ∞, we get ∞
∫
β) ps−1 B(α, p
(x, y) dp =
∞
∫
p
0
s−1
1
∫
0
t
x −1
(1 − t )
y−1
1 F1
0
−p α; β; dtdp. t (1 − t )
(7)
From the uniform convergence of the integral, the order of integration in (7) can be interchanged. Therefore, we have ∞
∫
β) ps−1 B(α, (x, y) dp = p
1
∫
0
t x−1 (1 − t )y−1 0
∞
∫
ps−1 1 F1
α; β;
0
−p dpdt . t (1 − t )
(8)
Now using the one-to-one transformation (except possibly at the boundaries and maps the region onto itself) ν = p , µ = t in (8), we get, t (1−t ) ∞
∫
β) ps−1 B(α, (x, y) dp = p
1
∫
0
µ(s+x)−1 (1 − µ)(y+s)−1 dµ
∞
∫
0
ν s−1 1 F1 (α; β; −ν)dν. 0
Therefore, using (5), we have ∞
∫
β) ps−1 B(α, (x, y) dp = B(s + x, y + s)Γ (α, β) (s). p
0
(α, β)
Corollary 2.3. By the Mellin inversion formula, we have the following complex integral representation for Bp β) B(α, (x, y) = p
∫
1
i∞
2π i −i∞
B(s + x, y + s)Γ (α, β) (s) p−s ds.
Remark 2.1. Putting s = 1 and considering that Γ (α, β) (1) = ∞
∫
β) B(α, (x, y) dp = B(x + 1, y + 1) p
0
(α,α)
Letting Bp
Γ (β)Γ (α−1) Γ (α)Γ (β−1)
in (6), we get
Γ (β)Γ (α − 1) . Γ (α)Γ (β − 1)
(x, y) = Bp (x, y), it reduces to Chaudhry’s [2] interesting relation
∞
∫
Bp (x, y) dp = B(x + 1, y + 1), 0
Re(x) > −1, Re(y) > −1, between the classical and the extended beta functions. Theorem 2.4. For the new generalized beta function, we have the following integral representations: π/2
∫
β) B(α, (x, y) = 2 p
cos2x−1 θ sin2y−1 θ 1 F1 α; β; −p sec2 θ csc2 θ dθ ,
0
(α, β)
Bp
(x, y) =
∫ 0
∞
ux−1
F x +y 1 1
(1 + u)
1 α; β; −2p − p u + du. u
(x, y):
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Proof. Letting t = cos2 θ in (4), we get β) B(α, (x, y) = p
1
∫
t x−1 (1 − t )y−1 1 F1
α; β;
0
π/2
∫ =2
−p dt t (1 − t )
cos2x−1 θ sin2y−1 θ 1 F1 (α; β; −p sec2 θ csc2 θ )dθ .
0 u On the other hand, letting t = 1+ in (4), we get u
β) B(α, (x, y) = p
1
∫ 0
∞
∫ = 0
−p dt t (1 − t ) ux − 1 1 du. 1 F1 α; β; −2p − p u + (1 + u)x+y u
t x−1 (1 − t )y−1 1 F1
α; β;
Theorem 2.5. For the new generalized beta function, we have the following functional relation: β) β) β) B(α, (x, y + 1) + B(α, (x + 1, y) = B(α, (x, y) . p p p
Proof. Direct calculations yield 1
−p dt t (1 − t ) 0 ∫ 1 −p dt + t x−1 (1 − t )y 1 F1 α; β; t (1 − t ) 0 ∫ 1 x = t (1 − t )y−1 + t x−1 (1 − t )y 1 F1 α; β;
β) β) B(α, (x, y + 1) + B(α, (x + 1, y) = p p
∫
t x (1 − t )y−1 1 F1
α; β;
0 1
∫
t x−1 (1 − t )y−1 1 F1
= 0
Whence the result.
α; β;
−p dt t (1 − t )
−p β) dt = B(α, (x, y) . p t (1 − t )
Theorem 2.6. For the product of two new generalized gamma function, we have the following integral representation: (α, β)
Γp
(α, β)
(x)Γp
(y) = 4
π
∫
∞
∫
2
0
r 2(x+y)−1 cos2x−1 θ sin2y−1 θ
0
× 1 F1 α; β; −r 2 cos2 θ −
p
r 2 cos2 θ
2 2 1 F1 α; β; −r sin θ −
p
r 2 sin2 θ
drdθ .
Proof. Substituting t = η2 in (3), we get (α, β)
Γp
(x) = 2
∞
∫
η
2x−1
1 F1
α; β; −η − 2
0
p
η2
dη.
Therefore
Γp(α, β) (x)Γp(α, β) (y) = 4
∞
∫ 0
∞
∫ 0
p p η2x−1 ξ 2y−1 1 F1 α; β; −η2 − 2 1 F1 α; β; −ξ 2 − 2 dηdξ . η ξ
Letting η = r cos θ and ξ = r sin θ in the above equality,
Γp(α, β) (x)Γp(α, β) (y) = 4
π
∫
2
0
∞
∫
r 2(x+y)−1 cos2x−1 θ sin2y−1 θ
0
× 1 F1 α; β; −r 2 cos2 θ −
p r 2 cos2 θ
2 2 1 F1 α; β; −r sin θ −
p r 2 sin2 θ
drdθ .
Remark 2.2. Putting p = 0 and α = β in (9), we get the classical relation between the gamma and beta functions: B (x, y) =
Γ (x)Γ (y) . Γ (x + y)
(9)
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Theorem 2.7. For the new generalized beta function, we have the following summation relation: β) B(α, (x, 1 − y) = p
∞ − (y)n
n!
n=0
β) B(α, (x + n, 1) , p
Re(p) > 0. Proof. From the definition of the new generalized beta function, we get β) B(α, (x, 1 − y) = p
1
∫
t x−1 (1 − t )−y 1 F1
α; β;
0
−p dt . t (1 − t )
Using the following binomial series expansion
( 1 − t ) −y =
∞ − tn (y)n , n! n=0
|t | < 1, we obtain β) B(α, (x, 1 − y) = p
− (y)n
1 ∞
∫ 0
n!
n =0
t x+n−1 1 F1
α; β;
−p dt . t (1 − t )
Therefore, interchanging the order of integration and summation and then using (4), we obtain β) B(α, (x, 1 − y) = p
∫ ∞ − (y)n n =0
=
n!
n!
t x+n−1 1 F1
α; β;
0
∞ − (y)n n =0
1
−p dt , t (1 − t )
B(α,β) (x + n, 1) . p
Now we obtain differential recurrence relations for generalized gamma and generalized beta functions defined by (3) and (4), respectively. Theorem 2.8. For the generalized gamma function, we have the following recurrence relation: (α, β)
d2 Γp
(x + 5)
−p
dp
(x + 3)
−β dp2 (α, β) d Γp (x + 1)
+p (α, β) d Γp (x + 3) dp2
−
(α, β)
d2 Γ p
dp
(α,β)
d Γp
(x + 2)
dp
+ α Γp(α, β) (x) = 0.
Proof. Taking derivatives under the integral symbol by using the Leibnitz rule, we get (α, β)
d2 Γp
(x + 5)
= 0
(x + 3)
−β dp2 (α, β) d Γp (x + 1)
+p (α, β) d Γp (x + 3) dp2
− ∫
(α, β)
d2 Γ p
(α, β)
d Γp
(x + 2)
dp
+ α Γp(α, β) (x) [ ] ∞ d2 z 2 dz t x−1 t 3 + pt + t + β t + p + α z dt , 2 dp
−p
dp
dp
dp
p p where z = 1 F1 α; β; −t − t . On the other hand, since z = 1 F1 α; β; −t − t is a solution of the equation
t 3 + pt
d2 z dp2
we get the result.
dz + t2 + βt + p + α z = 0, dp
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Theorem 2.9. For the generalized beta function, we have the following recurrence relation:
(α, β)
d2 Bp p
(x + 3, y + 3)
+β
dp2
(α, β)
d Bp
(x + 2, y + 2)
(α, β)
d Bp
(x + 1, y + 1)
+p
dp
dp
β) + α B(α, (x, y) = 0. p
Proof. Let S denote the left-hand side of the above assertion. Taking derivatives under the integral symbol in (4) by using the Leibnitz rule, we get 1
∫ S=
t
x −1
(1 − t )
y−1
pt (1 − t )
0
d2 z
[
dp2
+ (β t (1 − t ) + p)
where z = 1 F1 α; β; t (1−t ) . Since z = 1 F1 α; β; t (1−t ) pt (1 − t )
d2 z dp2
we get the result.
−p
dz
+ (β t (1 − t ) + p)
−p
dz dp
]
+ α z dt ,
is a solution of the equation
+ α z = 0,
dp
3. Generalized Gauss hypergeometric and confluent hypergeometric functions In this section we use the new generalization (4) of beta functions to generalize the hypergeometric and confluent hypergeometric functions defined by Fp(α, β) (a, b; c ; z ) :=
∞ −
(α, β)
(a)n
(b + n, c − b) z n B (b, c − b) n!
Bp
n =0
and (α, β;p)
1 F1
(b; c ; z ) :=
(α, β) ∞ − Bp (b + n, c − b) z n
B (b, c − b)
n =0
n!
,
respectively. (α, β) (α, β;p) We call the Fp (a, b; c ; z ) by the generalized Gauss hypergeometric function (GGHF) and 1 F1 (b; c ; z ) by the generalized confluent hypergeometric function (GCHF). Observe that [8], Fp(α,α) (a, b; c ; z ) = Fp (a, b; c ; z ) , (α, β)
F0
(a, b; c ; z ) = 2 F1 (a, b; c ; z ) ,
and (p)
(α,α;p)
(b; c ; z ) = 1 F1 (b; c ; z ) = φp (b; c ; z ) ,
(α, β;0)
(b; c ; z ) = 1 F1 (b; c ; z ) .
1 F1 1 F1
3.1. Integral representations of the GGHF and GCHF The GGHF can be provided with an integral representation by using the definition of the new generalized beta function (4). We get Theorem 3.1. For the GGHF, we have the following integral representations: Fp(α, β) (a, b; c ; z ) :=
1 B(b, c − b)
1
∫
t b−1 (1 − t )c −b−1 1 F1
α; β;
0
−p (1 − zt )−a dt , t (1 − t )
(10)
Re(p) > 0; p = 0 and |arg (1 − z )| < π; Re(c ) > Re(b) > 0; Fp(α, β) (a, b; c ; z ) := Fp(α, β) (a, b; c ; z ) :=
1 B (b, c − b) 2 B (b, c − b)
∞
∫
ub−1 (1 + u)a−c [1 + u(1 − z )]−a 1 F1
1 α; β; −2p − p u + du, u
0 π
∫
2
0
sin2b−1 ν cos2c −2b−1 ν 1 − z sin2 ν
−a
1 F1
α; β;
−p sin2 ν cos2 ν
dν.
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4607
Proof. Direct calculations yield Fp(α, β) (a, b; c ; z ) :=
∞ −
(α, β)
(a)n
Bp
n =0
=
1
( b + n, c − b ) z n B(b, c − b) n! ∫ ∞ 1 − b+n−1 c −b−1 t (1 − t ) (a)n 1 F1 α; β;
B(b, c − b) n=0
0
−p t (1 − t )
zn n!
dt
− ∞ −p (zt )n = t dt (1 − t ) (a)n 1 F1 α; β; B(b, c − b) 0 t ( 1 − t ) n =0 n! ∫ 1 1 −p = t b−1 (1 − t )c −b−1 1 F1 α; β; (1 − zt )−a dt . B(b, c − b) 0 t (1 − t ) ∫
1
1
b−1
c −b−1
t Setting u = 1− in (10), we get t
Fp(α, β) (a, b; c ; z ) =
∞
∫
1 B (b, c − b)
ub−1 (1 + u)a−c [1 + u(1 − z )]−a 1 F1
1 α; β; −2p − p u + du. u
0
On the other hand, substituting t = sin ν in (10), we have 2
Fp(α, β) (a, b; c ; z ) =
π
∫
2 B (b, c − b)
2
sin2b−1 ν cos2c −2b−1 ν 1 − z sin2 ν
−a
1 F1
α; β;
0
−p 2 sin ν cos2 ν
dν.
A similar procedure yields an integral representation of the GCHF by using the definition of the new generalized beta function. Theorem 3.2. For the GCHF, we have the following integral representations: (α, β;p)
1 F1
(b; c ; z ) :=
∫
1 B(b, c − b)
1
t
b −1
(1 − t )
c −b −1
e
zt
1 F1
0
−p α; β; dt , t (1 − t )
(11)
−p (1 − u)b−1 uc −b−1 z (1−u) e du, (b; c ; z ) := 1 F1 α; β; 1 F1 B ( b, c − b) u(1 − u) 0 p ≥ 0; and Re(c ) > Re(b) > 0. (α, β;p)
∫
1
Remark 3.1. Putting p = 0 in (10) and (11), we get the integral representations of the classical GHF and CHF. 3.2. Differentiation formulas for the new GGHF’s and new GCHF’s In this section, by using the formulas B(b, c − b) = bc B(b + 1, c − b) and (a)n+1 = a (a + 1)n , we obtain new formulas including derivatives of GGHF and GCHF with respect to the variable z. Theorem 3.3. For GGHF, we have the following differentiation formula:
(b)n (a)n (α, β) dn (α, β) Fp F (a, b; c ; z ) = (a + n, b + n; c + n; z ) . dz n (c )n p (α, β)
(a, b; c ; z ) with respect to z, we obtain (α, β) ∞ d (α, β) d − Bp (b + n, c − b) z n Fp (a, b; c ; z ) = (a)n dz dz n=0 B(b, c − b) n!
Proof. Taking the derivative of Fp
=
∞ − n =1
(α, β)
(a)n
Bp
(b + n, c − b) z n−1 . B(b, c − b) (n − 1)!
Replacing n → n + 1, we get (α, β) ∞ d (α, β) ba − Bp (b + n + 1, c − b) z n Fp (a, b; c ; z ) = (a + 1)n dz c n =0 B(b + 1, c − b) n!
=
ba (α, β) Fp (a + 1, b + 1; c + 1; z ) . c
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Recursive application of this procedure gives us the general form:
(b)n (a)n (α, β) dn (α, β) F Fp ( a + n, b + n; c + n; z ) . (a, b; c ; z ) = dz n (c )n p Theorem 3.4. For GCHF, we have the following differentiation formula: dn
(α, β;p)
1F dz n 1
(b; c ; z ) =
(b)n (α, β;p) ( b + n; c + n; z ) . 1F (c )n 1
3.3. Mellin transform representation of the GGHF’s and GCHF’s In this section, we obtain the Mellin transform representations of the GGHF and GCHF. Theorem 3.5. For the GGHF, we have the following Mellin transform representation:
M Fp(α, β) (a, b; c ; z ) : s :=
Γ (α, β) (s)B(b + s, c + s − b) 2 F1 (a, b + s; c + 2s; z ) . B(b, c − b)
Proof. To obtain the Mellin transform, we multiply both sides of (10) by ps−1 and integrate with respect to p over the interval [0, ∞). Thus we get
M Fp(α, β) (a, b; c ; z ) : s :=
∞
∫
ps−1 Fp(α,β) (a, b; c ; z ) dp
0
=
1
∫
1 B(b, c − b)
t
b −1
(1 − t )
c −b −1
(1 − zt )
−a
∞
[∫
0
p
s−1
1 F1
0
] −p α; β; dp dt . t (1 − t )
(12)
p
Since substituting u = t (1−t ) in (12), ∞
∫
s −1
p
1 F1 α; β;
0
∫ ∞ −p dp = us−1 t s (1 − t )s 1 F1 (α; β; −u) du t (1 − t ) 0 ∫ ∞ = t s (1 − t )s 1 F1 (α; β; −u) du 0
= t s (1 − t )s Γ (α, β) (s). Thus we get
(α, β)
M Fp
=
( a, b ; c ; z ) : s =
1 B(b, c − b)
1
∫
t b+s−1 (1 − t )c +s−b−1 (1 − zt )−a Γ (α, β) (s)dt
0
1 Γ (α, β) (s)B(b + s, c + s − b) B(b, c − b) B(b + s, c + s − b)
1
∫
t b+s−1 (1 − t )c +2s−(b+s)−1 (1 − zt )−a dt
0
Γ (α, β) (s)B(b + s, c + s − b) = 2 F1 (a, b + s; c + 2s; z ) . B(b, c − b) (α, β)
Corollary 3.6. By the Mellin inversion formula, we have the following complex integral representation for Fp Fp(α, β) (a, b; c ; z ) =
1
∫
i∞
2π i −i∞
:
Γ (α, β) (s)B(b + s, c + s − b) −s 2 F1 (a, b + s; c + 2s; z ) p ds. B(b, c − b)
Theorem 3.7. For the new GCHF, we have the following Mellin transform representation:
M
(α, β;p)
1 F1
Γ (α, β) (s)B(b + s, c + s − b) (b; c ; z ) : s := 1 F1 (b + s; c + 2s; z ) . B(b, c − b) (α, β;p)
Corollary 3.8. By the Mellin inversion formula, we have the following complex integral representation for 1 F1 (α, β;p)
1 F1
( b; c ; z ) =
1
∫
i∞
2π i −i∞
Γ (α, β) (s)B(b + s, c + s − b) −s 1 F1 (b + s; c + 2s; z ) p ds. B(b, c − b)
:
E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
4609
3.4. Transformation formulas Theorem 3.9. For the new GGHF, we have the following transformation formula: Fp(α, β) (a, b; c ; z ) = (1 − z )−a Fp(α, β)
a, c − b ; b ;
z z−1
,
|arg(1 − z )| < π . Proof. By writing
[1 − z (1 − t )]−a = (1 − z )−a
−a
z
1+
1−z
t
and replacing t → 1 − t in (10), we obtain
−a ∫ 1 −p z (1 − z )−a t dt (1 − t )b−1 t c −b−1 1 − 1 F1 α; β; B(b, c − b) 0 z−1 t (1 − t ) Re(p) > 0; p = 0 and |z | < π; Re(c ) > Re(b) > 0. Fp(α, β) (a, b; c ; z ) =
Hence, (α, β)
Fp
(a, b; c ; z ) = (1 − z )
(α, β)
−a
a, c − b ; b ;
Fp
1 z
Remark 3.2. Note that, replacing z by 1 − Fp(α, β)
a, b ; c ; 1 −
1
z
z z−1
.
in Theorem 3.9, one easily obtains the following transformation formula
= z α Fp(α, β) (a, c − b; b; 1 − z )
|arg(z )| < π . z Furthermore, replacing z by 1+ in Theorem 3.9, we get the following transformation formula z
Fp(α, β)
a, b ; c ;
z 1+z
= (1 + z )a Fp(α, β) (a, c − b; b; z )
|arg(1 + z )| < π . Theorem 3.10. For the new GCHF, we have the following transformation formula: (α, β;p)
1 F1
(α, β;p)
(b; c ; z ) = exp(z )1 F1
(c − b; c ; −z ) .
Remark 3.3. Setting z = 1 in (10), we have the following relation between new defined hypergeometric and beta functions: Fp(α, β) (a, b; c ; 1) =
B(b, c − b) (α, β)
=
Bp
1
∫
1
t b−1 (1 − t )c −a−b−1 1 F1
α; β;
0
( b, c − a − b)
B(b, c − b)
−p dt t (1 − t )
.
3.5. Differential recurrence relations for GGHF’s and GCHF’s In this subsection we obtain some differential recurrence relations for GGHF’s and GCHF’s. We start with the following theorem: Theorem 3.11. For GGHF’s we have the following recurrence relation: (α, β)
pB(b + 3, c − b + 3)
d2 Fp
(a, b + 3; c + 6; z ) dp2 (α, β)
− pB(b + 1, c − b + 1)
dFp
(α, β)
− β B(b + 2, c − b + 2)
(a, b + 1; c + 2; z ) dp
dFp
(a, b + 2; c + 4; z )
+ α Fp(α, β) (a, b; c ; z ) = 0.
dp
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E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
Proof. Let S denote the left-hand side of the above assertion. Taking derivatives under the integral symbol in (10) by using the Leibnitz rule, we get 1
∫ S=
t
x −1
(1 − t )
y−1
[
pt (1 − t )
0
d2 z dp2
+ (β t (1 − t ) + p)
where z = 1 F1 α; β; t (1−t ) . Since z = 1 F1 α; β; t (1−t ) pt (1 − t )
d2 z dp2
we get the result.
−p
+ (β t (1 − t ) + p)
dz dp
−p
dz dp
]
+ α z dt ,
is a solution of the equation
+ α z = 0,
In a similar manner, we have the following for GCHF’s: Theorem 3.12. For GCHF’s we have the following recurrence relation: (α, β;p)
pB(b + 3, c − b + 3)
d21 F1
(b + 3; c + 6; z ) dp2 (α, β;p)
− pB(b + 1, c − b + 1)
d1 F 1
(α, β;p)
− β B(b + 2, c − b + 2)
(b + 1; c + 2; z ) dp
(α, β;p)
+ α 1 F1
d1 F1
(b + 2; c + 4; z ) dp
(b; c ; z ) = 0.
4. Concluding remarks By using the CHF, we have defined generalizations of gamma and beta functions. In their special cases, these generalizations include the extension of gamma and beta functions which were proposed in [1,2], respectively. Using the generalization of the beta function, we have generalized GHF and CHF, which include extended hypergeometric and confluent hypergeometric functions considered in [8]. We have investigated some properties of these generalized functions, most of which are analogous with the original functions. Most of the special functions of mathematical physics and engineering, such as Jacobi and Laguerre polynomials can be expressed in terms of GHF or CHF. Therefore, the corresponding extensions of several other familiar special functions are expected to be useful and need to be investigated. Acknowledgements The author is very grateful to the anonymous referees for many valuable comments and suggestions which helped to improve the draft. The present investigation was supported, in part, by the Ministry of National Education of TRNC under project MEKB-09-01. References [1] M.A. Chaudhry, S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994) 99–124. [2] M.A. Chaudhry, A. Qadir, M. Rafique, S.M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math. 78 (1997) 19–32. [3] M.A. Chaudhry, S.M. Zubair, On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms, J. Comput. Appl. Math. 59 (1995) 253–284. [4] M.A. Chaudhry, N.M. Temme, E.J.M. Veling, Asymptotic and closed form of a generalized incomplete gamma function, J. Comput. Appl. Math. 67 (1996) 371–379. [5] A.R. Miller, Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals, Rocky Mountain J. Math. 30 (2000) 703–714. [6] F. AL-Musallam, S.L. Kalla, Futher results on a generalized gamma function occurring in diffraction theory, Integral Transforms and Spec. Funct. 7 (3–4) (1998) 175–190. [7] M.A. Chaudhry, S.M. Zubair, Extended incomplete gamma functions with applications, J. Math. Anal. Appl. 274 (2002) 725–745. [8] M.A. Chaudhry, A. Qadir, H.M. Srivastava, R.B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2004) 589–602. [9] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Extension of gamma, beta and hypergeometric functions Emine Özergin ∗ , Mehmet Ali Özarslan, Abdullah Altın Eastern Mediterranean University, Department of Mathematics, Gazimagusa, TRNC, Mersin 10, Turkey
article
abstract
info
Article history: Received 29 December 2009 Received in revised form 13 April 2010
The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations. © 2010 Elsevier B.V. All rights reserved.
Keywords: Gamma function Beta function Hypergeometric function Confluent hypergeometric function Mellin transform
1. Introduction In recent years, some extensions of the well known special functions have been considered by several authors [1–7]. In 1994, Chaudhry and Zubair [1] have introduced the following extension of gamma function
Γp (x) :=
∞
∫
t x−1 exp −t − pt −1 dt ,
(1)
0
Re(p) > 0. In 1997, Chaudhry et al. [2] presented the following extension of Euler’s beta function Bp (x, y) :=
1
∫
[
t x−1 (1 − t )y−1 exp − 0
p t (1 − t )
]
dt ,
(2)
(Re(p) > 0, Re(x) > 0, Re(y) > 0) and they proved that this extension has connections with the Macdonald, error and Whittakers function. It is clearly seem that Γ0 (x) = Γ (x) and B0 (x, y) = B (x, y). Afterwards, Chaudhry et al. [8] used Bp (x, y) to extend the hypergeometric functions (and confluent hypergeometric functions) as follows: Fp (a, b; c ; z ) =
∞ − Bp (b + n, c − b) n =0
B (b, c − b)
(a)n
zn n!
p ≥ 0; Re(c ) > Re(b) > 0,
φp (b; c ; z ) =
∞ − Bp (b + n, c − b) z n n=0
B (b, c − b)
n!
p ≥ 0; Re(c ) > Re(b) > 0,
∗
Corresponding author. Tel.: +90 5428565066; fax: +90 3923651604. E-mail addresses: [email protected] (E. Özergin), [email protected] (M.A. Özarslan), [email protected] (A. Altın).
0377-0427/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2010.04.019
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E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
where (λ)ν denotes the Pochhammer symbol defined by
(λ)0 ≡ 1 and (λ)ν :=
Γ (λ + ν) Γ (λ)
and gave the Euler type integral representation Fp (a, b; c ; z ) = p > 0; p = 0
B ( b, c − b)
and
1
∫
1
[
t b−1 (1 − t )c −b−1 (1 − zt )−a exp − 0
p
]
t (1 − t )
dt
|arg (1 − z )| < π < p; Re(c ) > Re(b) > 0.
They called these functions extended Gauss hypergeometric function (EGHF) and extended confluent hypergeometric function (ECHF), respectively. They have discussed the differentiation properties and Mellin transforms of Fp (a, b; c ; z ) and obtained transformation formulas, recurrence relations, summation and asymptotic formulas for this function. Note that F0 (a, b; c ; z ) = 2 F1 (a, b; c ; z ). In this paper, we consider the following generalizations of gamma and Euler’s beta functions (α, β)
Γp
(x) :=
∞
∫
t x−1 1 F1 α; β; −t −
p
dt t Re(α) > 0, Re(β) > 0, Re(p) > 0, Re(x) > 0,
(3)
0
1
−p dt , t (1 − t ) 0 (Re(p) > 0, Re(x) > 0, Re(y) > 0, Re(α) > 0, Re(β) > 0) β) B(α, (x, y) := p
∫
t x−1 (1 − t )y−1 1 F1
α; β;
(α,α)
respectively. It is obvious by (1), (3) and (2), (4) that, Γp (α, β)
(4)
(x) = Γp (x), Γ0(α,α) (x) = Γ (x), B(α,α) (x, y) = Bp (x, y) and p
(x, y) = B (x, y) . In Section 2, different integral representations and some properties of new generalized Euler’s beta function are obtained. Additionally, relations of new generalized gamma and beta functions are discussed. In Section 3, we generalize the (α, β) hypergeometric function and confluent hypergeometric function, using Bp (x, y) obtain the integral representations of this new generalized Gauss hypergeometric functions. Furthermore we discussed the differentiation properties, Mellin transforms, transformation formulas, recurrence relations, summation formulas for these new hypergeometric functions. B0
2. Some properties of gamma and beta functions It is important and useful to obtain different integral representations of the new generalized beta function, for later use. Also it is useful to discuss the relationships between classical gamma functions and new generalizations. For p = 0, we have (α, β) the following integral representation for Γp (x). Theorem 2.1. For the new generalized gamma function, we have (α, β)
Γp
Γ (β) (s) = Γ (α)Γ (β − α)
1
∫ 0
Γpµ2 (s)µα−s−1 (1 − µ)β−α−1 dµ.
Proof. Using the integral representation of confluent hypergeometric function, we have (α, β)
Γp
Γ (β) (s) = Γ (α)Γ (β − α)
∞
∫
1
∫
0
pt us−1 e−ut − u t α−1 (1 − t )β−α−1 dtdu.
0
Now using a one-to-one transformation (except possibly at the boundaries and maps the region onto itself) ν = ut , µ = t in the above equality and considering that the Jacobian of the transformation is J = µ1 , we get
Γp(α, β) (s) =
Γ (β) Γ (α)Γ (β − α)
∞
∫ 0
1
∫
v s−1 e−v−
pµ2
v
dvµα−s−1 (1 − µ)β−α−1 dµ.
0
From the uniform convergence of the integrals, the order of integration can be interchanged to yield that (α, β)
Γp
Γ (β) (s) = Γ (α)Γ (β − α)
∫
Γ (β) = Γ (α)Γ (β − α)
∫
Whence the result.
1
0
0
∞
[∫
v
pµ2 s−1 −v− v e
]
dv µα−s−1 (1 − µ)β−α−1 dµ
0 1
Γpµ2 (s)µα−s−1 (1 − µ)β−α−1 dµ.
E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
4603
The case p = 0 in the above Theorem gives (see [9, p. 192])
Γ (α, β) (s) =
Γ (β) Γ (α)Γ (β − α)
1
∫
Γ (s)µα−s−1 (1 − µ)β−α−1 dµ =
0
Γ (β)Γ (α − s)Γ (s) . Γ (α)Γ (β − s) (α, β)
The next theorem gives the Mellin transform representation of the function Bp function and Γ (α, β) (s) .
(5)
(x, y) in terms of the ordinary beta
Theorem 2.2. Mellin transform representation of the new generalized beta function is given by ∞
∫
β) ps−1 B(α, (x, y) dp = B(s + x, y + s)Γ (α, β) (s), p
(6)
0
Re(s) > 0, Re(x + s) > 0, Re(y + s) > 0, Re(p) > 0, Re(α) > 0, Re(β) > 0. Proof. Multiplying (4) by ps−1 and integrating with respect to p from p = 0 to p = ∞, we get ∞
∫
β) ps−1 B(α, p
(x, y) dp =
∞
∫
p
0
s−1
1
∫
0
t
x −1
(1 − t )
y−1
1 F1
0
−p α; β; dtdp. t (1 − t )
(7)
From the uniform convergence of the integral, the order of integration in (7) can be interchanged. Therefore, we have ∞
∫
β) ps−1 B(α, (x, y) dp = p
1
∫
0
t x−1 (1 − t )y−1 0
∞
∫
ps−1 1 F1
α; β;
0
−p dpdt . t (1 − t )
(8)
Now using the one-to-one transformation (except possibly at the boundaries and maps the region onto itself) ν = p , µ = t in (8), we get, t (1−t ) ∞
∫
β) ps−1 B(α, (x, y) dp = p
1
∫
0
µ(s+x)−1 (1 − µ)(y+s)−1 dµ
∞
∫
0
ν s−1 1 F1 (α; β; −ν)dν. 0
Therefore, using (5), we have ∞
∫
β) ps−1 B(α, (x, y) dp = B(s + x, y + s)Γ (α, β) (s). p
0
(α, β)
Corollary 2.3. By the Mellin inversion formula, we have the following complex integral representation for Bp β) B(α, (x, y) = p
∫
1
i∞
2π i −i∞
B(s + x, y + s)Γ (α, β) (s) p−s ds.
Remark 2.1. Putting s = 1 and considering that Γ (α, β) (1) = ∞
∫
β) B(α, (x, y) dp = B(x + 1, y + 1) p
0
(α,α)
Letting Bp
Γ (β)Γ (α−1) Γ (α)Γ (β−1)
in (6), we get
Γ (β)Γ (α − 1) . Γ (α)Γ (β − 1)
(x, y) = Bp (x, y), it reduces to Chaudhry’s [2] interesting relation
∞
∫
Bp (x, y) dp = B(x + 1, y + 1), 0
Re(x) > −1, Re(y) > −1, between the classical and the extended beta functions. Theorem 2.4. For the new generalized beta function, we have the following integral representations: π/2
∫
β) B(α, (x, y) = 2 p
cos2x−1 θ sin2y−1 θ 1 F1 α; β; −p sec2 θ csc2 θ dθ ,
0
(α, β)
Bp
(x, y) =
∫ 0
∞
ux−1
F x +y 1 1
(1 + u)
1 α; β; −2p − p u + du. u
(x, y):
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Proof. Letting t = cos2 θ in (4), we get β) B(α, (x, y) = p
1
∫
t x−1 (1 − t )y−1 1 F1
α; β;
0
π/2
∫ =2
−p dt t (1 − t )
cos2x−1 θ sin2y−1 θ 1 F1 (α; β; −p sec2 θ csc2 θ )dθ .
0 u On the other hand, letting t = 1+ in (4), we get u
β) B(α, (x, y) = p
1
∫ 0
∞
∫ = 0
−p dt t (1 − t ) ux − 1 1 du. 1 F1 α; β; −2p − p u + (1 + u)x+y u
t x−1 (1 − t )y−1 1 F1
α; β;
Theorem 2.5. For the new generalized beta function, we have the following functional relation: β) β) β) B(α, (x, y + 1) + B(α, (x + 1, y) = B(α, (x, y) . p p p
Proof. Direct calculations yield 1
−p dt t (1 − t ) 0 ∫ 1 −p dt + t x−1 (1 − t )y 1 F1 α; β; t (1 − t ) 0 ∫ 1 x = t (1 − t )y−1 + t x−1 (1 − t )y 1 F1 α; β;
β) β) B(α, (x, y + 1) + B(α, (x + 1, y) = p p
∫
t x (1 − t )y−1 1 F1
α; β;
0 1
∫
t x−1 (1 − t )y−1 1 F1
= 0
Whence the result.
α; β;
−p dt t (1 − t )
−p β) dt = B(α, (x, y) . p t (1 − t )
Theorem 2.6. For the product of two new generalized gamma function, we have the following integral representation: (α, β)
Γp
(α, β)
(x)Γp
(y) = 4
π
∫
∞
∫
2
0
r 2(x+y)−1 cos2x−1 θ sin2y−1 θ
0
× 1 F1 α; β; −r 2 cos2 θ −
p
r 2 cos2 θ
2 2 1 F1 α; β; −r sin θ −
p
r 2 sin2 θ
drdθ .
Proof. Substituting t = η2 in (3), we get (α, β)
Γp
(x) = 2
∞
∫
η
2x−1
1 F1
α; β; −η − 2
0
p
η2
dη.
Therefore
Γp(α, β) (x)Γp(α, β) (y) = 4
∞
∫ 0
∞
∫ 0
p p η2x−1 ξ 2y−1 1 F1 α; β; −η2 − 2 1 F1 α; β; −ξ 2 − 2 dηdξ . η ξ
Letting η = r cos θ and ξ = r sin θ in the above equality,
Γp(α, β) (x)Γp(α, β) (y) = 4
π
∫
2
0
∞
∫
r 2(x+y)−1 cos2x−1 θ sin2y−1 θ
0
× 1 F1 α; β; −r 2 cos2 θ −
p r 2 cos2 θ
2 2 1 F1 α; β; −r sin θ −
p r 2 sin2 θ
drdθ .
Remark 2.2. Putting p = 0 and α = β in (9), we get the classical relation between the gamma and beta functions: B (x, y) =
Γ (x)Γ (y) . Γ (x + y)
(9)
E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
4605
Theorem 2.7. For the new generalized beta function, we have the following summation relation: β) B(α, (x, 1 − y) = p
∞ − (y)n
n!
n=0
β) B(α, (x + n, 1) , p
Re(p) > 0. Proof. From the definition of the new generalized beta function, we get β) B(α, (x, 1 − y) = p
1
∫
t x−1 (1 − t )−y 1 F1
α; β;
0
−p dt . t (1 − t )
Using the following binomial series expansion
( 1 − t ) −y =
∞ − tn (y)n , n! n=0
|t | < 1, we obtain β) B(α, (x, 1 − y) = p
− (y)n
1 ∞
∫ 0
n!
n =0
t x+n−1 1 F1
α; β;
−p dt . t (1 − t )
Therefore, interchanging the order of integration and summation and then using (4), we obtain β) B(α, (x, 1 − y) = p
∫ ∞ − (y)n n =0
=
n!
n!
t x+n−1 1 F1
α; β;
0
∞ − (y)n n =0
1
−p dt , t (1 − t )
B(α,β) (x + n, 1) . p
Now we obtain differential recurrence relations for generalized gamma and generalized beta functions defined by (3) and (4), respectively. Theorem 2.8. For the generalized gamma function, we have the following recurrence relation: (α, β)
d2 Γp
(x + 5)
−p
dp
(x + 3)
−β dp2 (α, β) d Γp (x + 1)
+p (α, β) d Γp (x + 3) dp2
−
(α, β)
d2 Γ p
dp
(α,β)
d Γp
(x + 2)
dp
+ α Γp(α, β) (x) = 0.
Proof. Taking derivatives under the integral symbol by using the Leibnitz rule, we get (α, β)
d2 Γp
(x + 5)
= 0
(x + 3)
−β dp2 (α, β) d Γp (x + 1)
+p (α, β) d Γp (x + 3) dp2
− ∫
(α, β)
d2 Γ p
(α, β)
d Γp
(x + 2)
dp
+ α Γp(α, β) (x) [ ] ∞ d2 z 2 dz t x−1 t 3 + pt + t + β t + p + α z dt , 2 dp
−p
dp
dp
dp
p p where z = 1 F1 α; β; −t − t . On the other hand, since z = 1 F1 α; β; −t − t is a solution of the equation
t 3 + pt
d2 z dp2
we get the result.
dz + t2 + βt + p + α z = 0, dp
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E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
Theorem 2.9. For the generalized beta function, we have the following recurrence relation:
(α, β)
d2 Bp p
(x + 3, y + 3)
+β
dp2
(α, β)
d Bp
(x + 2, y + 2)
(α, β)
d Bp
(x + 1, y + 1)
+p
dp
dp
β) + α B(α, (x, y) = 0. p
Proof. Let S denote the left-hand side of the above assertion. Taking derivatives under the integral symbol in (4) by using the Leibnitz rule, we get 1
∫ S=
t
x −1
(1 − t )
y−1
pt (1 − t )
0
d2 z
[
dp2
+ (β t (1 − t ) + p)
where z = 1 F1 α; β; t (1−t ) . Since z = 1 F1 α; β; t (1−t ) pt (1 − t )
d2 z dp2
we get the result.
−p
dz
+ (β t (1 − t ) + p)
−p
dz dp
]
+ α z dt ,
is a solution of the equation
+ α z = 0,
dp
3. Generalized Gauss hypergeometric and confluent hypergeometric functions In this section we use the new generalization (4) of beta functions to generalize the hypergeometric and confluent hypergeometric functions defined by Fp(α, β) (a, b; c ; z ) :=
∞ −
(α, β)
(a)n
(b + n, c − b) z n B (b, c − b) n!
Bp
n =0
and (α, β;p)
1 F1
(b; c ; z ) :=
(α, β) ∞ − Bp (b + n, c − b) z n
B (b, c − b)
n =0
n!
,
respectively. (α, β) (α, β;p) We call the Fp (a, b; c ; z ) by the generalized Gauss hypergeometric function (GGHF) and 1 F1 (b; c ; z ) by the generalized confluent hypergeometric function (GCHF). Observe that [8], Fp(α,α) (a, b; c ; z ) = Fp (a, b; c ; z ) , (α, β)
F0
(a, b; c ; z ) = 2 F1 (a, b; c ; z ) ,
and (p)
(α,α;p)
(b; c ; z ) = 1 F1 (b; c ; z ) = φp (b; c ; z ) ,
(α, β;0)
(b; c ; z ) = 1 F1 (b; c ; z ) .
1 F1 1 F1
3.1. Integral representations of the GGHF and GCHF The GGHF can be provided with an integral representation by using the definition of the new generalized beta function (4). We get Theorem 3.1. For the GGHF, we have the following integral representations: Fp(α, β) (a, b; c ; z ) :=
1 B(b, c − b)
1
∫
t b−1 (1 − t )c −b−1 1 F1
α; β;
0
−p (1 − zt )−a dt , t (1 − t )
(10)
Re(p) > 0; p = 0 and |arg (1 − z )| < π; Re(c ) > Re(b) > 0; Fp(α, β) (a, b; c ; z ) := Fp(α, β) (a, b; c ; z ) :=
1 B (b, c − b) 2 B (b, c − b)
∞
∫
ub−1 (1 + u)a−c [1 + u(1 − z )]−a 1 F1
1 α; β; −2p − p u + du, u
0 π
∫
2
0
sin2b−1 ν cos2c −2b−1 ν 1 − z sin2 ν
−a
1 F1
α; β;
−p sin2 ν cos2 ν
dν.
E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
4607
Proof. Direct calculations yield Fp(α, β) (a, b; c ; z ) :=
∞ −
(α, β)
(a)n
Bp
n =0
=
1
( b + n, c − b ) z n B(b, c − b) n! ∫ ∞ 1 − b+n−1 c −b−1 t (1 − t ) (a)n 1 F1 α; β;
B(b, c − b) n=0
0
−p t (1 − t )
zn n!
dt
− ∞ −p (zt )n = t dt (1 − t ) (a)n 1 F1 α; β; B(b, c − b) 0 t ( 1 − t ) n =0 n! ∫ 1 1 −p = t b−1 (1 − t )c −b−1 1 F1 α; β; (1 − zt )−a dt . B(b, c − b) 0 t (1 − t ) ∫
1
1
b−1
c −b−1
t Setting u = 1− in (10), we get t
Fp(α, β) (a, b; c ; z ) =
∞
∫
1 B (b, c − b)
ub−1 (1 + u)a−c [1 + u(1 − z )]−a 1 F1
1 α; β; −2p − p u + du. u
0
On the other hand, substituting t = sin ν in (10), we have 2
Fp(α, β) (a, b; c ; z ) =
π
∫
2 B (b, c − b)
2
sin2b−1 ν cos2c −2b−1 ν 1 − z sin2 ν
−a
1 F1
α; β;
0
−p 2 sin ν cos2 ν
dν.
A similar procedure yields an integral representation of the GCHF by using the definition of the new generalized beta function. Theorem 3.2. For the GCHF, we have the following integral representations: (α, β;p)
1 F1
(b; c ; z ) :=
∫
1 B(b, c − b)
1
t
b −1
(1 − t )
c −b −1
e
zt
1 F1
0
−p α; β; dt , t (1 − t )
(11)
−p (1 − u)b−1 uc −b−1 z (1−u) e du, (b; c ; z ) := 1 F1 α; β; 1 F1 B ( b, c − b) u(1 − u) 0 p ≥ 0; and Re(c ) > Re(b) > 0. (α, β;p)
∫
1
Remark 3.1. Putting p = 0 in (10) and (11), we get the integral representations of the classical GHF and CHF. 3.2. Differentiation formulas for the new GGHF’s and new GCHF’s In this section, by using the formulas B(b, c − b) = bc B(b + 1, c − b) and (a)n+1 = a (a + 1)n , we obtain new formulas including derivatives of GGHF and GCHF with respect to the variable z. Theorem 3.3. For GGHF, we have the following differentiation formula:
(b)n (a)n (α, β) dn (α, β) Fp F (a, b; c ; z ) = (a + n, b + n; c + n; z ) . dz n (c )n p (α, β)
(a, b; c ; z ) with respect to z, we obtain (α, β) ∞ d (α, β) d − Bp (b + n, c − b) z n Fp (a, b; c ; z ) = (a)n dz dz n=0 B(b, c − b) n!
Proof. Taking the derivative of Fp
=
∞ − n =1
(α, β)
(a)n
Bp
(b + n, c − b) z n−1 . B(b, c − b) (n − 1)!
Replacing n → n + 1, we get (α, β) ∞ d (α, β) ba − Bp (b + n + 1, c − b) z n Fp (a, b; c ; z ) = (a + 1)n dz c n =0 B(b + 1, c − b) n!
=
ba (α, β) Fp (a + 1, b + 1; c + 1; z ) . c
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E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
Recursive application of this procedure gives us the general form:
(b)n (a)n (α, β) dn (α, β) F Fp ( a + n, b + n; c + n; z ) . (a, b; c ; z ) = dz n (c )n p Theorem 3.4. For GCHF, we have the following differentiation formula: dn
(α, β;p)
1F dz n 1
(b; c ; z ) =
(b)n (α, β;p) ( b + n; c + n; z ) . 1F (c )n 1
3.3. Mellin transform representation of the GGHF’s and GCHF’s In this section, we obtain the Mellin transform representations of the GGHF and GCHF. Theorem 3.5. For the GGHF, we have the following Mellin transform representation:
M Fp(α, β) (a, b; c ; z ) : s :=
Γ (α, β) (s)B(b + s, c + s − b) 2 F1 (a, b + s; c + 2s; z ) . B(b, c − b)
Proof. To obtain the Mellin transform, we multiply both sides of (10) by ps−1 and integrate with respect to p over the interval [0, ∞). Thus we get
M Fp(α, β) (a, b; c ; z ) : s :=
∞
∫
ps−1 Fp(α,β) (a, b; c ; z ) dp
0
=
1
∫
1 B(b, c − b)
t
b −1
(1 − t )
c −b −1
(1 − zt )
−a
∞
[∫
0
p
s−1
1 F1
0
] −p α; β; dp dt . t (1 − t )
(12)
p
Since substituting u = t (1−t ) in (12), ∞
∫
s −1
p
1 F1 α; β;
0
∫ ∞ −p dp = us−1 t s (1 − t )s 1 F1 (α; β; −u) du t (1 − t ) 0 ∫ ∞ = t s (1 − t )s 1 F1 (α; β; −u) du 0
= t s (1 − t )s Γ (α, β) (s). Thus we get
(α, β)
M Fp
=
( a, b ; c ; z ) : s =
1 B(b, c − b)
1
∫
t b+s−1 (1 − t )c +s−b−1 (1 − zt )−a Γ (α, β) (s)dt
0
1 Γ (α, β) (s)B(b + s, c + s − b) B(b, c − b) B(b + s, c + s − b)
1
∫
t b+s−1 (1 − t )c +2s−(b+s)−1 (1 − zt )−a dt
0
Γ (α, β) (s)B(b + s, c + s − b) = 2 F1 (a, b + s; c + 2s; z ) . B(b, c − b) (α, β)
Corollary 3.6. By the Mellin inversion formula, we have the following complex integral representation for Fp Fp(α, β) (a, b; c ; z ) =
1
∫
i∞
2π i −i∞
:
Γ (α, β) (s)B(b + s, c + s − b) −s 2 F1 (a, b + s; c + 2s; z ) p ds. B(b, c − b)
Theorem 3.7. For the new GCHF, we have the following Mellin transform representation:
M
(α, β;p)
1 F1
Γ (α, β) (s)B(b + s, c + s − b) (b; c ; z ) : s := 1 F1 (b + s; c + 2s; z ) . B(b, c − b) (α, β;p)
Corollary 3.8. By the Mellin inversion formula, we have the following complex integral representation for 1 F1 (α, β;p)
1 F1
( b; c ; z ) =
1
∫
i∞
2π i −i∞
Γ (α, β) (s)B(b + s, c + s − b) −s 1 F1 (b + s; c + 2s; z ) p ds. B(b, c − b)
:
E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
4609
3.4. Transformation formulas Theorem 3.9. For the new GGHF, we have the following transformation formula: Fp(α, β) (a, b; c ; z ) = (1 − z )−a Fp(α, β)
a, c − b ; b ;
z z−1
,
|arg(1 − z )| < π . Proof. By writing
[1 − z (1 − t )]−a = (1 − z )−a
−a
z
1+
1−z
t
and replacing t → 1 − t in (10), we obtain
−a ∫ 1 −p z (1 − z )−a t dt (1 − t )b−1 t c −b−1 1 − 1 F1 α; β; B(b, c − b) 0 z−1 t (1 − t ) Re(p) > 0; p = 0 and |z | < π; Re(c ) > Re(b) > 0. Fp(α, β) (a, b; c ; z ) =
Hence, (α, β)
Fp
(a, b; c ; z ) = (1 − z )
(α, β)
−a
a, c − b ; b ;
Fp
1 z
Remark 3.2. Note that, replacing z by 1 − Fp(α, β)
a, b ; c ; 1 −
1
z
z z−1
.
in Theorem 3.9, one easily obtains the following transformation formula
= z α Fp(α, β) (a, c − b; b; 1 − z )
|arg(z )| < π . z Furthermore, replacing z by 1+ in Theorem 3.9, we get the following transformation formula z
Fp(α, β)
a, b ; c ;
z 1+z
= (1 + z )a Fp(α, β) (a, c − b; b; z )
|arg(1 + z )| < π . Theorem 3.10. For the new GCHF, we have the following transformation formula: (α, β;p)
1 F1
(α, β;p)
(b; c ; z ) = exp(z )1 F1
(c − b; c ; −z ) .
Remark 3.3. Setting z = 1 in (10), we have the following relation between new defined hypergeometric and beta functions: Fp(α, β) (a, b; c ; 1) =
B(b, c − b) (α, β)
=
Bp
1
∫
1
t b−1 (1 − t )c −a−b−1 1 F1
α; β;
0
( b, c − a − b)
B(b, c − b)
−p dt t (1 − t )
.
3.5. Differential recurrence relations for GGHF’s and GCHF’s In this subsection we obtain some differential recurrence relations for GGHF’s and GCHF’s. We start with the following theorem: Theorem 3.11. For GGHF’s we have the following recurrence relation: (α, β)
pB(b + 3, c − b + 3)
d2 Fp
(a, b + 3; c + 6; z ) dp2 (α, β)
− pB(b + 1, c − b + 1)
dFp
(α, β)
− β B(b + 2, c − b + 2)
(a, b + 1; c + 2; z ) dp
dFp
(a, b + 2; c + 4; z )
+ α Fp(α, β) (a, b; c ; z ) = 0.
dp
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E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
Proof. Let S denote the left-hand side of the above assertion. Taking derivatives under the integral symbol in (10) by using the Leibnitz rule, we get 1
∫ S=
t
x −1
(1 − t )
y−1
[
pt (1 − t )
0
d2 z dp2
+ (β t (1 − t ) + p)
where z = 1 F1 α; β; t (1−t ) . Since z = 1 F1 α; β; t (1−t ) pt (1 − t )
d2 z dp2
we get the result.
−p
+ (β t (1 − t ) + p)
dz dp
−p
dz dp
]
+ α z dt ,
is a solution of the equation
+ α z = 0,
In a similar manner, we have the following for GCHF’s: Theorem 3.12. For GCHF’s we have the following recurrence relation: (α, β;p)
pB(b + 3, c − b + 3)
d21 F1
(b + 3; c + 6; z ) dp2 (α, β;p)
− pB(b + 1, c − b + 1)
d1 F 1
(α, β;p)
− β B(b + 2, c − b + 2)
(b + 1; c + 2; z ) dp
(α, β;p)
+ α 1 F1
d1 F1
(b + 2; c + 4; z ) dp
(b; c ; z ) = 0.
4. Concluding remarks By using the CHF, we have defined generalizations of gamma and beta functions. In their special cases, these generalizations include the extension of gamma and beta functions which were proposed in [1,2], respectively. Using the generalization of the beta function, we have generalized GHF and CHF, which include extended hypergeometric and confluent hypergeometric functions considered in [8]. We have investigated some properties of these generalized functions, most of which are analogous with the original functions. Most of the special functions of mathematical physics and engineering, such as Jacobi and Laguerre polynomials can be expressed in terms of GHF or CHF. Therefore, the corresponding extensions of several other familiar special functions are expected to be useful and need to be investigated. Acknowledgements The author is very grateful to the anonymous referees for many valuable comments and suggestions which helped to improve the draft. The present investigation was supported, in part, by the Ministry of National Education of TRNC under project MEKB-09-01. References [1] M.A. Chaudhry, S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994) 99–124. [2] M.A. Chaudhry, A. Qadir, M. Rafique, S.M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math. 78 (1997) 19–32. [3] M.A. Chaudhry, S.M. Zubair, On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms, J. Comput. Appl. Math. 59 (1995) 253–284. [4] M.A. Chaudhry, N.M. Temme, E.J.M. Veling, Asymptotic and closed form of a generalized incomplete gamma function, J. Comput. Appl. Math. 67 (1996) 371–379. [5] A.R. Miller, Reduction of a generalized incomplete gamma function, related Kampe de Feriet functions, and incomplete Weber integrals, Rocky Mountain J. Math. 30 (2000) 703–714. [6] F. AL-Musallam, S.L. Kalla, Futher results on a generalized gamma function occurring in diffraction theory, Integral Transforms and Spec. Funct. 7 (3–4) (1998) 175–190. [7] M.A. Chaudhry, S.M. Zubair, Extended incomplete gamma functions with applications, J. Math. Anal. Appl. 274 (2002) 725–745. [8] M.A. Chaudhry, A. Qadir, H.M. Srivastava, R.B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2004) 589–602. [9] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.