CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 235 (2011) 4601–4610 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 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 Z 1 x−1 −1 Γp.x/ VD 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 Z 1 [ ] x−1 y−1 p Bp .x; y/ VD 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/ D Γ .x/ and B0 .x; y/ D B .x; y/. Afterwards, Chaudhry et al. [8] used Bp .x; y/ to extend the hypergeometric functions (and confluent hypergeometric functions) as follows: 1 n X Bp .b C n; c − b/ z Fp .a; bI cI z/ D .a/ B b c − b n nW nD0 . ; / p ≥ 0I Re.c/ > Re.b/ > 0; 1 n X Bp .b C n; c − b/ z φp .bI cI z/ D B b c − b nW nD0 . ; / p ≥ 0I 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 C ν/ (λ)0 ≡ 1 and (λ)ν VD Γ (λ) and gave the Euler type integral representation Z 1 [ ] 1 b−1 c−b−1 −a p Fp .a; bI cI z/ D t .1 − t/ .1 − zt/ exp − dt B .b; c − b/ 0 t.1 − t/ p > 0I p D 0 and jarg .1 − z/j < π < pI 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; bI cI z/ and obtained transformation formulas, recurrence relations, summation and asymptotic formulas for this function. Note that F0 .a; bI cI z/ D 2F1 .a; bI cI z/. In this paper, we consider the following generalizations of gamma and Euler's beta functions Z 1 p (α; β/ VD x−1 I I − − Γp .x/ t 1F1 α β t dt (3) 0 t Re(α/ > 0; Re(β/ > 0; Re.p/ > 0; Re.x/ > 0; Z 1 −p (α; β/ VD x−1 − y−1 I I 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/ D .α,α/ D .α,α/ D 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/ D 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 D 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/ Z 1 (α; β/ D 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 1 Z Z pt (α; β/ Γ (β/ s−1 −ut− α−1 β−α−1 D 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 D ut; µ D t D 1 in the above equality and considering that the Jacobian of the transformation is J µ ; we get Z 1 Z 1 2 Γ (β/ − − − pµ − − − − (α; β/ D 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 Z 1 [Z 1 2 ] Γ (β/ − − − pµ − − − − (α; β/ D 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 Z 1 Γ (β/ α−s−1 β−α−1 D 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 D 0 in the above Theorem gives (see [9, p. 192]) Z 1 Γ (β/ − − − − Γ (β/Γ (α − s/Γ .s/ Γ (α; β/.s/ D Γ .s)µα s 1.1 − µ)β α 1dµ D : (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 Z 1 s−1 (α; β/ D C C (α; β/ p Bp .x; y/ dp B.s x; y s/Γ .s/; (6) 0 Re.s/ > 0; Re.x C s/ > 0; Re.y C 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 D 0 to p D 1, we get Z 1 Z 1 Z 1 −p s−1 (α; β/ D s−1 x−1 − y−1 I I 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 Z 1 Z 1 Z 1 −p s−1 (α; β/ D x−1 − y−1 s−1 I I 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 D p D t in (8), we get, t.1−t/ ; µ Z 1 Z 1 Z 1 s−1 (α; β/ D .sCx/−1 − .yCs/−1 s−1 I I − p Bp .x; y/ dp µ .1 µ/ dµ ν 1F1(α β ν/dν: 0 0 0 Therefore, using (5), we have Z 1 s−1 (α; β/ D C C (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 Z i1 (α; β/ D C C (α; β/ −s Bp .x; y/ B.s x; y s/Γ .s/ p ds: 2πi −i1 − Remark 2.1. Putting s D 1 and considering that .α; β/ 1 D Γ (β/Γ (α 1/ in (6), we get Γ . / Γ (α/Γ (β−1/ Z 1 Γ (β/Γ (α − 1/ (α; β/ D C C Bp .x; y/ dp B.x 1; y 1/ : 0 Γ (α/Γ (β − 1/ (α,α/ Letting Bp .x; y/ D Bp.x; y/, it reduces to Chaudhry's [2] interesting relation Z 1 Bp .x; y/ dp D B.x C 1; y C 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: Z π=2 (α; β/ D 2x−1 2y−1 I I − 2 2 Bp .x; y/ 2 cos θ sin θ 1F1 α β p sec θ csc θ dθ; 0 Z 1 ux−1 1 B(α; β/ .x; y/ D F αI βI −2p − p u C du: p xCy 1 1 0 .1 C u/ u 4604 E.