Extended Goursat s Hypergeometric Function

Transcription

1 International Journal of Mathematical Analysis Vol. 9, 25, no. 3, HIKARI Ltd, Extended Goursat s Hypergeometric Function Daya K. Nagar and Raúl Alejandro Morán-Vásquez Instituto de Matemáticas, Universidad de Antioquia Calle 67, No. 53 8, Medellín, Colombia S.A.) Arjun K. Gupta Department of Mathematics and Statistics Bowling Green State University, Bowling Green, Ohio , USA Copyright c 25 Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this article, we define an extended form of the Goursat s hypergeometric function and derive several results pertaining to it. We show that this function occurs naturally in statistical distribution theory. Mathematics Subject Classification: 33E99, 6E5 Keywords: Beta distribution; extended beta function; extended confluent hypergeometric function; extended Gauss hypergeometric function; gamma distribution; quotient; Gauss hypergeometric function Introduction The classical beta function, denoted by Ba, b), is defined see Luke 6]) by the Euler s integral Ba, b) = t a t) b dt, = Γa)Γb), Rea) >, Reb) >. ) Γa + b)

2 5 Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta Based on the beta function, the Gauss hypergeometric function, denoted by F a, b; c; z), and the confluent hypergeometric function, denoted by Φb; c; z), for Rec) > Reb) >, are defined as see Luke 6]), and F a, b; c; z) = Φb; c; z) = t b t) c b dt, arg z) < π, 2) zt) a t b t) c b expzt) dt. 3) It is also well know that, under certain conditions, the Goursat s function Goursat 3, p. 286]) or generalized hypergeometric function 2 F 2 a, b; c, d; z) Ga, b; c, d; z), is defined by Ga, b; c, d; z) = Ba, d a) v a v) d a Φb; c; zv) dv, 4) where Red) > Rea) >. Using the series expansions of zt) a and expzt) in 2) and 3), respectively, the following series representations of hypergeometric functions can be obtained: F a, b; c; z) = n= Φb; c; z) = a) n Bb + n, c b) n= Bb + n, c b) z n, z <, Rec) > Reb) >, 5) n! z n, Rec) > Reb) >. 6) n! In 997, Chaudhry et al. ] extended the classical beta function to the whole complex plane by introducing in the integrand of ) the exponential factor exp σ/t t)] with Reσ) >. Thus, the extended beta function is defined as Ba, b; σ) = t a t) b exp σ ] dt, Reσ) >. 7) t t) If we take σ = in 7), then for Rea) > and Reb) > we have Ba, b; ) = Ba, b). Further, replacing t by t in 7), one can see that Ba, b; σ) = Bb, a; σ). The rationale and justification for introducing this function are given in Chaudhry et al. ] where several properties and a statistical application have also been studied. Miller 7] further studied this function and has given several additional results. In 24, Chaudhry et al. 2] gave definitions of the extended Gauss hypergeometric function and the extended confluent hypergeometric function,

3 Extended Goursat s hypergeometric function 5 denoted by F σ a, b; c; z) and Φ σ b; c; z), respectively. These definitions were developed by considering the extended beta function 7) instead of beta function ) that appears in the general term of the series 5) and 6). They defined, for Rec) > Reb) >, the extended Gauss hypergeometric function and the extended confluent hypergeometric function as F σ a, b; c; z) = n= Φ σ b; c; z) = a) n Bb + n, c b; σ) n= Bb + n, c b; σ) z n, σ, z <, 8) n! z n, σ. 9) n! Further, using the integral representation of the extended beta function 7) in 8) and 9), Chaudhry et al. 2] obtained integral representations, for σ and Rec) > Reb) >, of extended Gauss hypergeometric function EGHF) and extended confluent hypergeometric function ECHF) as t b t) c b F σ a, b; c; z) = exp σ ] dt, zt) a t t) arg z) < π, ) and Φ σ b; c; z) = ] t b t) c b σ exp zt dt, ) t t) respectively. For σ = in ), we have F a, b; c; z) = F a, b; c; z). That is, the classical Gauss hypergeometric function is a special case of the extended Gauss hypergeometric function. Likewise, taking σ = in ) yields Φ b; c; z) = Φb; c; z), which means that the classical confluent hypergeometric function is a special case of the extended confluent hypergeometric function. Chaudhry et al. 2] and Miller 7] found that extended forms of beta and hypergeometric functions are related to the extended beta function, Bessel and Whittaker functions, and also gave several alternative integral representations. In this article, we define and study the extended Goursat s hypergeometric function and derive several results pertaining to it. We also show that this function occurs in a natural way in statistical distribution theory. An extended form of the Goursat s hypergeometric function has been defined in Section 2 and applications of the extended Goursat s hypergeometric function are discussed in Section 3.

4 52 Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta 2 Extended Goursat s Hypergeometric Function In this section, we define an extended form of the Goursat s hypergeometric function by replacing the confluent hypergeometric by its extended form in 4). Following Chaudhry et al. 2], the extended Goursat s hypergeometric function G σ a, b; c, d; z) is defined by G σ a, b; c, d; z) = Ba, d a) v a v) d a Φ σ b; c; zv) dv, 2) where Rec) > Reb) >, and Red) > Rea) >. By taking σ = in 2) one gets G a, b; c, d; z) = 2 F 2 a, b; c, d; z), which means that the classical generalized hypergeometric function is a special case of the extended Goursat s function. Replacing Φ σ b; c; z) by its equivalent integral representation given in ) and changing the order of integration, the integral in 2) is re-written as G σ a, b; c, d; z) = t b t) c b exp σ ] t t) = Ba, d a) v a v) d a exp vzt) dv dt, t b t) c b exp σ t t) ] Φa; d; zt) dt, 3) where the last line has been obtained by using 3). For a = d, we have Φa;d;zt) = expzt), and the above expression gives G σ a,b;c,a;z) = Φ σ b; c; z). Taking z = in 3) and applying the definition of the extended beta function, we arrive at G σ a, b; c, d; ) = Bb, c b; σ). In the integral representation given in 3), by substituting t = + u) u, with the Jacobian Jt u) = + u) 2, an alternative integral representation is obtained as G σ a, b; c, d; z) = exp 2σ) u b + u) c exp σ u + u) ] Φ a; d; If we take σ = in the above expression, we arrive at Ga, b; c, d; z) = a; d; u b + u) c Φ zu + u ) du. ) zu du. + u

5 Extended Goursat s hypergeometric function 53 Replacing exp σ/t) and exp σ/ t)] by their respective series expansions involving Laguerre polynomials L n σ) L ) n σ) n =,, 2,...) given in Miller 7, Eq. 3.4a, 3.4b], namely, and exp σ ) = exp σ)t t L n σ) t) n, t <, n= exp σ ) = exp σ) t) t L m σ)t m, t <, m= in 3), an alternative representation for G σ a, b; c, d; z) is given by G σ a, b; c, d; z) = exp 2σ) m,n= Bb + m +, c + n + b)l m σ)l n σ) Ga, b + m + ; c + m + n + 2, d; z). The Mellin transform of G σ a, b; c, d; z), using 3), is given by = = σ s G σ a, b; c, d; z) dσ Γs) B b, c b) t b t) c b Φa; d; zt) t b+s t) c b+s Φa; d; zt) dt, exp σ t t) for Res) >. Now, evaluation of the above integral using 4) yields σ s G σ a, b; c, d; z) dσ = where Res + b) > and Res + c b) >. ] σ s dσ dt Γs)B b + s, c b + s) Ga, b + s; c + 2s, d; z), B b, c b) Theorem 2.. If σ >, c > b > and d > a >, then G σ a, b; c, d; z) exp 4σ)Ga, b; c, d; z) exp ) Ga, b; c, d; z). 4σ Proof. Recently, Nagar, Morán-Vásquez and Gupta 8] have shown that Φ σ b; c; z) exp 4σ)Φb; c; z) exp ) Φb; c; z), 4σ where σ > and c > b >. Now, using the above inequality in 2) and writing the resulting inequality using 4), we get the desired result.

6 54 Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta Theorem 2.2. If σ >, Rea) > α > and β >, then x α G σ a, b; c, d; βx) dx = β α Γα)Ba α, d a)bb α, c b; σ). Ba, d a) Proof. Using 3) and changing the order of integration, we have x α G σ a, b; c, d; βx) dx = t b t) c b exp σ ] t t) x α Φa; d; βxt) dx dt. Now, first integrating x using the result Nagar, Morán-Vásquez and Gupta 8]) x a Φb; c; αx) dx = Γa)Bb a, c b) α a and then t using Euler s beta integral, we get the desired result. 3 Statistical Distributions In this section, we give applications of the Goursat s hypergeometric function to statistical distribution theory. First, we define the gamma and the beta type distributions. These definitions can be found in Johnson, Kotz and Balakrishnan 5]. Definition 3.. A random variable X is said to have a gamma distribution with parameters θ > ), κ > ), denoted by X Gaκ, θ), if its probability density function pdf) is given by {θ κ Γκ)} x κ exp x ), x >. 4) θ Note that for θ =, the above distribution reduces to a standard gamma distribution and in this case we write X Gaκ). Definition 3.2. A random variable X is said to have a beta type distribution with parameters a, b), a >, b >, denoted as X Ba, b), if its pdf is given by where Ba, b) is the beta function. {Ba, b)} x a x) b, < x <, 5)

7 Extended Goursat s hypergeometric function 55 Next, we define the extended confluent hypergeometric function distribution. Definition 3.3. A random variable X is said to have an extended confluent hypergeometric function distribution with parameters ν, α, β, σ), denoted by X ECHν, α, β; σ), if its pdf is given by Bα, β α)x ν Φ σ α; β; x), x >, 6) Γν)Bα ν, β α; σ) where ν >, β > α > if σ > and β > α > ν > if σ =. Definition 3.4. A random variable Y is said to have a generalized extended confluent hypergeometric function distribution with parameters ν, α, β, λ, σ), denoted by Y ECHν, α, β; λ; σ), if its pdf is given by Bα, β α)λ ν y ν Φ σ α; β; λy), y >, λ >, 7) Γν)Bα ν, β α; σ) where ν >, β > α > if σ > and β > α > ν > if σ =. For σ =, 6) and 7) reduce to the confluent hypergeometric function pdf and generalized confluent hypergeometric function pdf see Gupta and Nagar 4]). Note that the pdf 6) can be obtained from 7) by taking λ =. Conversely, 7) can be obtained from 6) by the transformation X = λy, λ >, with the Jacobian Jx y) = λ. The extended confluent hypergeometric function distribution can be derived as the distribution of the quotient of independent gamma and extended beta type variables Nagar, Morán-Vásquez and Gupta 8]). The next theorem derives the cumulative distribution function cdf) of X where X ECHν, α, β; σ). Theorem 3.. If X ECHν, α, β; σ), then P X x) = Bα, β α)xν G σ ν, α; β, ν + ; x), ν + )Bα ν, β α; σ) where G σ a, b; c, d; z) is the extended Goursat s function defined by 2). Proof. From 6), we have P X x) = Bα, β α) Γν)Bα ν, β α; σ) x t ν Φ σ α, β; t) dt. Now, taking y = t/x in the above integral and using the integral representation 2) in the resulting expression, we obtain the desired result.

8 56 Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta Theorem 3.2. Let the random variables X and Y be independent, X Gaλ) and Y ECHν, α, β; σ). Then, the pdf of S = Y + X is given by Bα, β α) Γν + λ)bα ν, β α; σ) sν+λ exp s)g σ ν, β α; β, ν + λ; s), s >, where G σ a, b; c, d; z) is the extended Goursat s hypergeometric function defined by 2). Proof. Since X and Y are independent, from 4) and 6), we get the joint pdf of X and Y as Bα, β α) Γν)Γλ)Bα ν, β α; σ) xλ y ν exp x)φ σ α; β; y), x >, y >. Making the transformation S = Y + X and R = Y/Y + X) with the Jacobian Jx, y s, r) = s, we obtain the joint pdf of S and R as Bα, β α) Γν)Γλ)Bα ν, β α; σ) rν r) λ s ν+λ exp s)φ σ β α; β; rs), where s > and < r <. Clearly, R and S are not independent. Integration of r in the joint pdf of S and R by using 2) yields the marginal pdf of S. Theorem 3.3. Let the random variables U and V be independent, U Bλ, γ) and V ECHν, α, β; σ). Then X = V/U has the pdf Bα, β α)bλ + ν, γ) Γν)Bλ, γ)bα ν, β α; σ) xν G σ λ + ν, α; β, λ + γ + ν; x), x >, where G σ a, b; c, d; z) is the extended Goursat s hypergeometric function defined by 2). Proof. Using 5), 6) and the independence of U and V, the joint pdf of U and V is given by Bα, β α)u λ u) γ v ν Φ σ α; β; v), < u <, v >. Γν)Bλ, γ)bα ν, β α; σ) Using the transformation X = V/U with the Jacobian Jv x) = u, we obtain the joint pdf of U and X as Bα, β α)x ν u λ+ν u) γ Φ σ α; β; xu), < u <, x >. Γν)Bλ, γ)bα ν, β α; σ) Integrating the above expression with respect to u, we find the marginal pdf of X as Bα, β α)x ν Γν)Bλ, γ)bα ν, β α; σ) u λ+ν u) γ Φ σ α; β; xu) du. Finally, evaluation of the above integral by using 2) yield the desired result.

9 Extended Goursat s hypergeometric function 57 Acknowledgments. The research work of DKN was supported by the Sistema Universitario de Investigación, Universidad de Antioquia under the project no. IN82CE. References ] M. Aslam Chaudhry, Asghar Qadir, M. Rafique and S. M. Zubair, Extension of Euler s beta function, J. Comput. Appl. Math., ), no., M. Aslam Chaudhry, Asghar Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 59 24), no. 2, ] E. Goursat, Mémosire sur les fonctions hypergéométriques d ordre supérieur, Ann. Sci. École Norm. Sup. Ser 2), 2883), ] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman & Hall/CRC, Boca Raton, 2. 5] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions-2, Second Edition, John Wiley & Sons, New York, ] Y. L. Luke, The Special Functions and Their Approximations, Vol., Academic Press, New York, ] Allen R. Miller, Remarks on a generalized beta function, J. Comput. Appl. Math., 998), no., ] Daya K. Nagar, Raúl Alejandro Morán-Vásquez and Arjun K. Gupta, Properties and applications of extended hypergeometric functions, Ingeniería y Ciencia, 24), no. 9, 3. Received: April, 25; Published: May 4, 25