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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 info a b s t r a c t
Article history: The main object of this paper is to present generalizations of gamma, beta and Received 29 December 2009 hypergeometric functions. Some recurrence relations, transformation formulas, operation Received in revised form 13 April 2010 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 ∫ ∞ x−1 −1 Γp(x) := t exp −t − pt dt, (1) 0 Re(p) > 0. In 1997, Chaudhry et al. [2] presented the following extension of Euler’s beta function ∫ 1 [ ] x−1 y−1 p Bp (x, y) := t (1 − t) exp − dt, (2) 0 t(1 − t) (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:
∞ n − Bp (b + n, c − b) z Fp (a, b; c; z) = (a) B b c − b n n! n=0 ( , ) p ≥ 0; Re(c) > Re(b) > 0, ∞ n − Bp (b + n, c − b) z φp (b; c; z) = B b c − b n! n=0 ( , ) 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 4602 E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610 where (λ)ν denotes the Pochhammer symbol defined by Γ (λ + ν) (λ)0 ≡ 1 and (λ)ν := Γ (λ) and gave the Euler type integral representation ∫ 1 [ ] 1 b−1 c−b−1 −a p Fp (a, b; c; z) = t (1 − t) (1 − zt) exp − dt B (b, c − b) 0 t(1 − t) p > 0; p = 0 and |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) = 2F1 (a, b; c; z). In this paper, we consider the following generalizations of gamma and Euler’s beta functions
∫ ∞ p (α, β) := x−1 ; ; − − Γp (x) t 1F1 α β t dt (3) 0 t Re(α) > 0, Re(β) > 0, Re(p) > 0, Re(x) > 0, ∫ 1 −p (α, β) := x−1 − y−1 ; ; Bp (x, y) t (1 t) 1F1 α β dt, (4) 0 t(1 − t) (Re(p) > 0, Re(x) > 0, Re(y) > 0, Re(α) > 0, Re(β) > 0) (α,α) = (α,α) = (α,α) = respectively. It is obvious by (1), (3) and (2), (4) that, Γp (x) Γp (x), Γ0 (x) Γ (x), Bp (x, y) Bp (x, y) and (α, β) = B0 (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.
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 Γ (β) ∫ 1 (α, β) = α−s−1 − β−α−1 Γp (s) Γpµ2 (s)µ (1 µ) dµ. Γ (α)Γ (β − α) 0
Proof. Using the integral representation of confluent hypergeometric function, we have
∞ 1 ∫ ∫ pt (α, β) Γ (β) s−1 −ut− α−1 β−α−1 = u − Γp (s) u e t (1 t) dtdu. Γ (α)Γ (β − α) 0 0 Now using a one-to-one transformation (except possibly at the boundaries and maps the region onto itself) ν = ut, µ = t = 1 in the above equality and considering that the Jacobian of the transformation is J µ , we get
∫ ∞ ∫ 1 2 Γ (β) − − − pµ − − − − (α, β) = s 1 v v α s 1 − β α 1 Γp (s) v e dvµ (1 µ) dµ. Γ (α)Γ (β − α) 0 0 From the uniform convergence of the integrals, the order of integration can be interchanged to yield that
∫ 1 [∫ ∞ 2 ] Γ (β) − − − pµ − − − − (α, β) = s 1 v v α s 1 − β α 1 Γp (s) v e dv µ (1 µ) dµ Γ (α)Γ (β − α) 0 0 ∫ 1 Γ (β) α−s−1 β−α−1 = Γpµ2 (s)µ (1 − µ) dµ. Γ (α)Γ (β − α) 0 Whence the result. 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])
∫ 1 Γ (β) − − − − Γ (β)Γ (α − s)Γ (s) Γ (α, β)(s) = Γ (s)µα s 1(1 − µ)β α 1dµ = . (5) Γ (α)Γ (β − α) 0 Γ (α)Γ (β − s) (α, β) The next theorem gives the Mellin transform representation of the function Bp (x, y) in terms of the ordinary beta function and Γ (α, β) (s) .
Theorem 2.2. Mellin transform representation of the new generalized beta function is given by ∫ ∞ s−1 (α, β) = + + (α, β) p Bp (x, y) dp B(s x, y s)Γ (s), (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
∫ ∞ ∫ ∞ ∫ 1 −p s−1 (α, β) = s−1 x−1 − y−1 ; ; p Bp (x, y) dp p t (1 t) 1F1 α β dtdp. (7) 0 0 0 t(1 − t) From the uniform convergence of the integral, the order of integration in (7) can be interchanged. Therefore, we have
∫ ∞ ∫ 1 ∫ ∞ −p s−1 (α, β) = x−1 − y−1 s−1 ; ; p Bp (x, y) dp t (1 t) p 1F1 α β dpdt. (8) 0 0 0 t(1 − t) Now using the one-to-one transformation (except possibly at the boundaries and maps the region onto itself) ν = p = t in (8), we get, t(1−t) , µ ∫ ∞ ∫ 1 ∫ ∞ s−1 (α, β) = (s+x)−1 − (y+s)−1 s−1 ; ; − p Bp (x, y) dp µ (1 µ) dµ ν 1F1(α β ν)dν. 0 0 0 Therefore, using (5), we have ∫ ∞ s−1 (α, β) = + + (α, β) p Bp (x, y) dp B(s x, y s)Γ (s). 0
(α, β) Corollary 2.3. By the Mellin inversion formula, we have the following complex integral representation for Bp (x, y):
1 ∫ i∞ (α, β) = + + (α, β) −s Bp (x, y) B(s x, y s)Γ (s) p ds. 2πi −i∞
− Remark 2.1. Putting s = 1 and considering that (α, β) 1 = Γ (β)Γ (α 1) in (6), we get Γ ( ) Γ (α)Γ (β−1) ∫ ∞ Γ (β)Γ (α − 1) (α, β) = + + Bp (x, y) dp B(x 1, y 1) . 0 Γ (α)Γ (β − 1) (α,α) Letting Bp (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 (α, β) = 2x−1 2y−1 ; ; − 2 2 Bp (x, y) 2 cos θ sin θ 1F1 α β p sec θ csc θ dθ, 0 ∫ ∞ ux−1 1 B(α, β) (x, y) = F α; β; −2p − p u + du. p x+y 1 1 0 (1 + u) u 4604 E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
Proof. Letting t = cos2 θ in (4), we get ∫ 1 −p (α, β) = x−1 − y−1 ; ; Bp (x, y) t (1 t) 1F1 α β dt 0 t(1 − t) ∫ π/2 2x−1 2y−1 2 2 = 2 cos θ sin θ 1F1(α; β; −p sec θ csc θ)dθ. 0 = u On the other hand, letting t 1+u in (4), we get ∫ 1 −p (α, β) = x−1 − y−1 ; ; Bp (x, y) t (1 t) 1F1 α β dt 0 t(1 − t) ∫ ∞ ux−1 1 = F α; β; −2p − p u + du. x+y 1 1 0 (1 + u) u
Theorem 2.5. For the new generalized beta function, we have the following functional relation: (α, β) + + (α, β) + = (α, β) Bp (x, y 1) Bp (x 1, y) Bp (x, y) .
Proof. Direct calculations yield
∫ 1 −p (α, β) + + (α, β) + = x − y−1 ; ; Bp (x, y 1) Bp (x 1, y) t (1 t) 1F1 α β dt 0 t(1 − t) ∫ 1 − x−1 y p + t (1 − t) 1F1 α; β; dt 0 t(1 − t) ∫ 1 − x y−1 x−1 y p = t (1 − t) + t (1 − t) 1F1 α; β; dt 0 t(1 − t) ∫ 1 −p = x−1 − y−1 ; ; = (α, β) t (1 t) 1F1 α β dt Bp (x, y) . 0 t(1 − t)
Whence the result. Theorem 2.6. For the product of two new generalized gamma function, we have the following integral representation: π ∞ ∫ 2 ∫ (α, β) (α, β) = 2(x+y)−1 2x−1 2y−1 Γp (x)Γp (y) 4 r cos θ sin θ 0 0 2 2 p 2 2 p × 1F1 α; β; −r cos θ − 1F1 α; β; −r sin θ − drdθ. (9) r2 cos2 θ r2 sin2 θ
Proof. Substituting t = η2 in (3), we get ∫ ∞ − p Γ (α, β)(x) = 2 η2x 1 F α; β; −η2 − dη. p 1 1 2 0 η Therefore ∫ ∞ ∫ ∞ − − p p Γ (α, β)(x)Γ (α, β)(y) = 4 η2x 1ξ 2y 1 F α; β; −η2 − F α; β; −ξ 2 − dηdξ. p p 1 1 2 1 1 2 0 0 η ξ Letting η = r cos θ and ξ = r sin θ in the above equality, π ∞ ∫ 2 ∫ (α, β) (α, β) = 2(x+y)−1 2x−1 2y−1 Γp (x)Γp (y) 4 r cos θ sin θ 0 0 2 2 p 2 2 p × 1F1 α; β; −r cos θ − 1F1 α; β; −r sin θ − drdθ. r2 cos2 θ r2 sin2 θ
Remark 2.2. Putting p = 0 and α = β in (9), we get the classical relation between the gamma and beta functions: Γ (x)Γ (y) B (x, y) = . Γ (x + y) 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: ∞ − (y)n B(α, β)(x, 1 − y) = B(α, β) (x + n, 1) , p n! p n=0 Re(p) > 0.
Proof. From the definition of the new generalized beta function, we get
∫ 1 −p (α, β) − = x−1 − −y ; ; Bp (x, 1 y) t (1 t) 1F1 α β dt. 0 t(1 − t) Using the following binomial series expansion
∞ n −y − t (1 − t) = (y)n , n! n=0 |t| < 1, we obtain ∫ 1 ∞ − (α, β) − (y)n x+n−1 p B (x, 1 − y) = t 1F1 α; β; dt. p n! t 1 − t 0 n=0 ( ) Therefore, interchanging the order of integration and summation and then using (4), we obtain ∞ ∫ 1 − (α, β) − (y)n x+n−1 p B (x, 1 − y) = t 1F1 α; β; dt, p n! t 1 − t n=0 0 ( ) ∞ − (y)n (α,β) = B (x + n, 1) . n! p n=0
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:
2 (α, β) 2 (α, β) (α,β) d Γp (x + 5) d Γp (x + 3) d Γp (x + 2) + p − β dp2 dp2 dp (α, β) (α, β) d Γp (x + 3) d Γp (x + 1) − − p + αΓ (α, β)(x) = 0. dp dp p
Proof. Taking derivatives under the integral symbol by using the Leibnitz rule, we get
2 (α, β) 2 (α, β) (α, β) d Γp (x + 5) d Γp (x + 3) d Γp (x + 2) + p − β dp2 dp2 dp (α, β) (α, β) d Γp (x + 3) d Γp (x + 1) − − p + αΓ (α, β)(x) dp dp p ∫ ∞ [ 2 ] − d z dz = tx 1 t3 + pt + t2 + βt + p + αz dt, 2 0 dp dp = ; ; − − p = ; ; − − p where z 1F1 α β t t . On the other hand, since z 1F1 α β t t is a solution of the equation
d2z dz t3 + pt + t2 + βt + p + αz = 0, dp2 dp we get the result. 4606 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:
2 (α, β) (α, β) (α, β) d Bp (x + 3, y + 3) d Bp (x + 2, y + 2) d Bp (x + 1, y + 1) p + β + p + αB(α, β) (x, y) = 0. dp2 dp dp 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 [ 2 ] − − d z dz S = tx 1 (1 − t)y 1 pt(1 − t) + (βt(1 − t) + p) + αz dt, 2 0 dp dp
− − where z = F ; ; p . Since z = F ; ; p is a solution of the equation 1 1 α β t(1−t) 1 1 α β t(1−t)
d2z dz pt(1 − t) + (βt(1 − t) + p) + αz = 0, dp2 dp we get the result.
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
∞ (α, β) n − Bp (b + n, c − b) z F (α, β) (a, b; c; z) := (a) p n B b c − b n! n=0 ( , ) and
∞ (α, β) n (α, β;p) − Bp (b + n, c − b) z 1F (b; c; z) := , 1 B b c − b n! n=0 ( , ) respectively. (α, β) ; ; (α, β;p) ; ; We call the Fp (a, b c z) by the generalized Gauss hypergeometric function (GGHF) and 1F1 (b c z) by the generalized confluent hypergeometric function (GCHF). Observe that [8], (α,α) ; ; = ; ; Fp (a, b c z) Fp (a, b c z) , (α, β) ; ; = ; ; F0 (a, b c z) 2F1 (a, b c z) , and (α,α;p) ; ; = (p) ; ; = ; ; 1F1 (b c z) 1F1 (b c z) φp (b c z) , (α, β;0) ; ; = ; ; 1F1 (b c z) 1F1 (b c z) .
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:
1 ∫ 1 −p (α, β) ; ; := b−1 − c−b−1 ; ; − −a Fp (a, b c z) t (1 t) 1F1 α β (1 zt) dt, (10) B(b, c − b) 0 t (1 − t) Re(p) > 0; p = 0 and |arg (1 − z)| < π; Re(c) > Re(b) > 0; 1 ∫ ∞ 1 (α, β) ; ; := b−1 + a−c + − −a ; ; − − + Fp (a, b c z) u (1 u) [1 u(1 z)] 1F1 α β 2p p u du, B (b, c − b) 0 u π ∫ 2 − (α, β) 2 2b−1 2c−2b−1 2 −a p F (a, b; c; z) := sin ν cos ν 1 − z sin ν 1F1 α; β; dν. p 2 2 B (b, c − b) 0 sin ν cos ν E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610 4607
Proof. Direct calculations yield
∞ (α, β) n − Bp (b + n, c − b) z F (α, β) (a, b; c; z) := (a) p n B b c − b n! n=0 ( , ) ∞ ∫ 1 − n 1 − b+n−1 c−b−1 p z = (a) t (1 − t) 1F1 α; β; dt B b c − b n t 1 − t n! ( , ) n=0 0 ( ) ∫ 1 − ∞ n 1 b−1 c−b−1 p − (zt) = t (1 − t) 1F1 α; β; (a) dt B b c − b t 1 − t n n! ( , ) 0 ( ) n=0 ∫ 1 − 1 b−1 c−b−1 p −a = t (1 − t) 1F1 α; β; (1 − zt) dt. B(b, c − b) 0 t (1 − t) = t Setting u 1−t in (10), we get 1 ∫ ∞ 1 (α, β) ; ; = b−1 + a−c + − −a ; ; − − + Fp (a, b c z) u (1 u) [1 u(1 z)] 1F1 α β 2p p u du. B (b, c − b) 0 u On the other hand, substituting t = sin2 ν in (10), we have π ∫ 2 − (α, β) 2 2b−1 2c−2b−1 2 −a p F (a, b; c; z) = sin ν cos ν 1 − z sin ν 1F1 α; β; dν. p 2 2 B (b, c − b) 0 sin ν cos ν
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:
1 ∫ 1 −p (α, β;p) ; ; := b−1 − c−b−1 zt ; ; 1F1 (b c z) t (1 t) e 1F1 α β dt, (11) B(b, c − b) 0 t (1 − t)
− ∫ 1 (1 − u)b 1 uc−b−1 −p (α, β;p) ; ; := z(1−u) ; ; 1F1 (b c z) e 1F1 α β du, 0 B (b, c − b) u(1 − u) p ≥ 0; and Re(c) > Re(b) > 0.
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
− = c + − = + In this section, by using the formulas B(b, c b) b 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: n d (b)n (a) F (α, β) (a, b; c; z) = n F (α, β) (a + n, b + n; c + n; z) . n p p dz (c)n
(α, β) Proof. Taking the derivative of Fp (a, b; c; z) with respect to z, we obtain
∞ (α, β) n d d − Bp (b + n, c − b) z F (α, β) (a, b; c; z) = (a) dz p dz n B b c − b n! n=0 ( , ) ∞ (α, β) n−1 − Bp (b + n, c − b) z = (a) . n B b c − b n − 1 ! n=1 ( , ) ( ) Replacing n → n + 1, we get
∞ (α, β) n d ba − Bp (b + n + 1, c − b) z F (α, β) (a, b; c; z) = (a + 1) dz p c n B b + 1 c − b n! n=0 ( , ) ba = F (α, β) (a + 1, b + 1; c + 1; z) . c p 4608 E. Özergin et al. / Journal of Computational and Applied Mathematics 235 (2011) 4601–4610
Recursive application of this procedure gives us the general form: n d (b)n (a) F (α, β) (a, b; c; z) = n F (α, β) (a + n, b + n; c + n; z) . n p p dz (c)n
Theorem 3.4. For GCHF, we have the following differentiation formula: n d (α, β;p) (b)n (α, β;p) F (b; c; z) = F (b + n; c + n; z) . n 1 1 1 1 dz (c)n
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: (α, β) + + − (α, β) Γ (s)B(b s, c s b) M F (a, b; c; z) : s := 2F1 (a, b + s; c + 2s; z) . p 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 ∫ ∞ (α, β) ; ; : := s−1 (α,β) ; ; M Fp (a, b c z) s p Fp (a, b c z) dp 0 ∫ 1 [∫ ∞ − ] 1 b−1 c−b−1 −a s−1 p = t (1 − t) (1 − zt) p 1F1 α; β; dp dt. (12) B(b, c − b) 0 0 t(1 − t) Since substituting u = p in (12), t(1−t) ∫ ∞ − ∫ ∞ s−1 p s−1 s s p 1F1 α; β; dp = u t (1 − t) 1F1 (α; β; −u) du 0 t(1 − t) 0 ∫ ∞ s s = t (1 − t) 1F1 (α; β; −u) du 0 = ts(1 − t)sΓ (α, β)(s). Thus we get 1 ∫ 1 (α, β) ; ; : = b+s−1 − c+s−b−1 − −a (α, β) M Fp (a, b c z) s t (1 t) (1 zt) Γ (s)dt B(b, c − b) 0 (α, β) ∫ 1 Γ (s)B(b + s, c + s − b) 1 + − + − + − − = tb s 1 (1 − t)c 2s (b s) 1 (1 − zt) adt B(b, c − b) B(b + s, c + s − b) 0 Γ (α, β)(s)B(b + s, c + s − b) = 2F1 (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 : 1 ∫ i∞ Γ (α, β)(s)B(b + s, c + s − b) (α, β) ; ; = + ; + ; −s Fp (a, b c z) 2F1 (a, b s c 2s z) p ds. 2πi −i∞ B(b, c − b)
Theorem 3.7. For the new GCHF, we have the following Mellin transform representation: (α, β) (α, β;p) Γ (s)B(b + s, c + s − b) M 1F (b; c; z) : s := 1F1 (b + s; c + 2s; z) . 1 B(b, c − b)
(α, β;p) Corollary 3.8. By the Mellin inversion formula, we have the following complex integral representation for 1F1 : 1 ∫ i∞ Γ (α, β)(s)B(b + s, c + s − b) (α, β;p) ; ; = + ; + ; −s 1F1 (b c z) 1F1 (b s c 2s z) p ds. 2πi −i∞ 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: − z F (α, β) (a, b; c; z) = (1 − z) a F (α, β) a, c − b; b; , p p z − 1 |arg(1 − z)| < π.
Proof. By writing −a − − z [1 − z(1 − t)] a = (1 − z) a 1 + t 1 − z and replacing t → 1 − t in (10), we obtain
− −a (1 − z) a ∫ 1 z −p (α, β) ; ; = − b−1 c−b−1 − ; ; Fp (a, b c z) (1 t) t 1 t 1F1 α β dt B(b, c − b) 0 z − 1 t(1 − t) Re(p) > 0; p = 0 and |z| < π; Re(c) > Re(b) > 0. Hence, (α, β) −a (α, β) z F (a, b; c; z) = (1 − z) F a, c − b; b; . p p z − 1
− 1 Remark 3.2. Note that, replacing z by 1 z in Theorem 3.9, one easily obtains the following transformation formula 1 F (α, β) a, b; c; 1 − = zαF (α, β) (a, c − b; b; 1 − z) p z p |arg(z)| < π.
z Furthermore, replacing z by 1+z in Theorem 3.9, we get the following transformation formula z F (α, β) a, b; c; = (1 + z)a F (α, β) (a, c − b; b; z) p 1 + z p |arg(1 + z)| < π.
Theorem 3.10. For the new GCHF, we have the following transformation formula: (α, β;p) ; ; = (α, β;p) − ; ; − 1F1 (b c z) exp(z)1F1 (c b c z) .
Remark 3.3. Setting z = 1 in (10), we have the following relation between new defined hypergeometric and beta functions:
1 ∫ 1 −p (α, β) ; ; = b−1 − c−a−b−1 ; ; Fp (a, b c 1) t (1 t) 1F1 α β dt B(b, c − b) 0 t(1 − t) (α, β) B (b, c − a − b) = p . B(b, c − b)
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:
2 (α, β) (α, β) d Fp (a, b + 3; c + 6; z) dFp (a, b + 2; c + 4; z) pB(b + 3, c − b + 3) − βB(b + 2, c − b + 2) dp2 dp (α, β) dFp (a, b + 1; c + 2; z) − pB(b + 1, c − b + 1) + αF (α, β) (a, b; c; z) = 0. dp p 4610 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 [ 2 ] − − d z dz S = tx 1 (1 − t)y 1 pt(1 − t) + (βt(1 − t) + p) + αz dt, 2 0 dp dp
− − where z = F ; ; p . Since z = F ; ; p is a solution of the equation 1 1 α β t(1−t) 1 1 α β t(1−t)
d2z dz pt(1 − t) + (βt(1 − t) + p) + αz = 0, dp2 dp we get the result. In a similar manner, we have the following for GCHF’s:
Theorem 3.12. For GCHF’s we have the following recurrence relation:
2 (α, β;p) (α, β;p) d F (b + 3; c + 6; z) d1F (b + 2; c + 4; z) pB(b + 3, c − b + 3) 1 1 − βB(b + 2, c − b + 2) 1 dp2 dp (α, β;p) + ; + ; d1F1 (b 1 c 2 z) (α, β;p) − pB(b + 1, c − b + 1) + α1F (b; c; z) = 0. dp 1
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.