Properties and Applications of Extended Hypergeometric Functions

Transcription

1 Ingeniería y Ciencia ISSN: ISSN-e: ing. cienc., vol. 1, no. 19, pp , enero-junio This a open-access article distributed under the terms of the Creative Commons Attribution License. Properties Applications of Extended Hypergeometric Functions Daya K. Nagar 1, Raúl Alejro Morán-Vásquez 2 Arjun K. Gupta 3 Received: , Acepted: Available online: MSC:33C9 Abstract In this article, we study several properties of extended Gauss hypergeometric extended confluent hypergeometric functions. We derive several integrals, inequalities establish relationship between these other special functions. We also show that these functions occur naturally in statistical distribution theory. Key words: Beta distribution; extended beta function; extended confluent hypergeometric function; extended Gauss hypergeometric function; gamma distribution; Gauss hypergeometric function. 1 Ph.D. in Science, dayaknagar@yahoo.com, Universidad de Antioquia, Medellín, Colombia. 2 Magíster en Matemáticas, alejromoran77@gmail.com, Universidade de São Paulo, São Paulo, Brasil. 3 Ph.D. in Statistics, gupta@bgsu.edu, Bowling Green State University, Bowling Green, Ohio, USA. Universidad EAFIT 11

2 Properties Applications of Extended Hypergeometric Functions Propiedades y aplicaciones de Funciones Hipergeométricas Extendida Resumen En este artículo estudiamos varias propiedades de las funciones hipergeométrica de Gauss extendida e hipergeométrica confluente extendida. Derivamos varias integrales, desigualdades y establecemos relaciones entre estas y otras funciones especiales. También mostramos que estas funciones ocurren naturalmente en la teoría de distribuciones estadísticas. Palabras clave: Distribución beta; función beta extendida; función hipergeométrica confluente extendida; función hipergeométrica de Gauss extendida; distribución gamma; función hipergeométrica de Gauss. 1 Introduction The classical beta function, denoted by Ba, b), is defined see Luke 1]) by the Euler s integral Ba, b) = t a 1 1 t) b 1 dt, = Γa)Γb), Rea) >, Reb) >. 1) Γa + b) Based on the beta function, the Gauss hypergeometric function, denoted by F a, b; c; z), the confluent hypergeometric function, denoted by Φb; c; z), for Rec) > Reb) >, are defined as see Luke 1]), F a, b; c; z) = Φb; c; z) = 1 1 t b 1 1 t) c b 1 1 zt) a dt, arg1 z) < π, 2) t b 1 1 t) c b 1 expzt) dt. 3) Using the series expansions of 1 zt) a expzt) in 2) 3), respectively, the series representations of F a, b; c; z) Φb; c; z), 12 Ingeniería y Ciencia

3 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta for Rec) > Reb) >, are obtained as F a, b; c; z) = n= Φb; c; z) = a) n Bb + n, c b) n= z n, z < 1, 4) n! Bb + n, c b) z n n!, 5) respectively. In 1997, Chaudhry et al. 2] extended the classical beta function to the whole complex plane by introducing in the integr of 1) the exponential factor exp σ/t1 t)], with Reσ) >. Thus, the extended beta function is defined as Ba, b; σ) = t a 1 1 t) b 1 exp σ ] dt, Reσ) >. 6) t1 t) If we take σ = in 6), then for Rea) > Reb) > we have Ba, b; ) = Ba, b). Further, replacing t by 1 t in 6), one can see that Ba, b; σ) = Bb, a; σ). The rationale justification for introducing this function are given in Chaudhry et al. 2] where several properties a statistical application have also been studied. Miller 3] further studied this function has given several additional results. In 24, Chaudhry et al. 4] gave definitions of the extended Gauss hypergeometric function the extended confluent hypergeometric function, denoted by F σ a, b; c; z) Φ σ b; c; z), respectively. These definitions were developed by considering the extended beta function 6) instead of beta function 1) that appear in the general term of the series 4) 5). Thus, for Rec) > Reb) >, F σ a, b; c; z) Φ σ b; c; z) are defined by F σ a, b; c; z) = n= a) n Bb + n, c b; σ) z n, σ, z < 1, 7) n! ing.cienc., vol. 1, no. 19, pp , enero-junio

4 Properties Applications of Extended Hypergeometric Functions Φ σ b; c; z) = n= Bb + n, c b; σ) z n, σ, 8) n! respectively. Further, using the integral representation of the extended beta function 6) in 7) 8), Chaudhry et al. 4] obtained integral representations, for σ Rec) > Reb) >, of the extended Gauss hypergeometric function EGHF) the extended confluent hypergeometric function ECHF) as F σ a, b; c; z) = Φ σ b; c; z) = 1 1 t b 1 1 t) c b 1 exp σ ] dt, 1 zt) a t1 t) arg1 z) < π, 9) 1 t b 1 1 t) c b 1 exp zt ] σ dt, t1 t) 1) respectively. For σ = in 9), 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 1) 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. 4] Miller 3] found that extended forms of beta hypergeometric functions are related to the beta, Bessel Whittaker functions, also gave several alternative integral representations. In this article, we give several interesting results on extended beta, extended Gauss hypergeometric extended confluent hypergeometric functions show that they occur in a natural way in statistical distribution theory. This paper is divided into five sections. Section 2 deals with some well known definitions results on special functions. In Section 3, 14 Ingeniería y Ciencia

5 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta several properties of the extended beta, the extended Gauss hypergeometric the extended confluent hypergeometric functions have been studied. Section 4 deals with the integrals involving EGHF ECHF. Finally, applications of the extended Gauss hypergeometric the extended confluent hypergeometric functions are demonstrated in Section 5. 2 Some Known Definitions Results An integral representation of the type 2 modified Bessel function Gradshteyn Ryzhik 5, Eq ]) is given by K ν 2 ab) = 1 ) ν/2 a t ν 1 exp at + b )] dt, 11) 2 b t where Rea) > Reb) >. If we make the transformation t = 1 + u) 1 u in 2) 3) with the Jacobian Jt u) = 1 + u) 2, we obtain alternative integral representations for F a, b; c; z) Φb; c; z) as F a, b; c; z) = 1 u b u) a c du, 12) z)u] a 1 u b 1 expz1 + u) 1 u] Φb; c; z) = du, 13) 1 + u) c respectively. Putting z = 1 in 2) evaluating the resulting integral using 1), one obtains F a, b; c; 1) = Γc)Γc a b), Rec a b) >. 14) Γc a)γc b) In the remainder of this section we give several properties of extended beta, extended Gauss hypergeometric, extended confluent hypergeometric functions, most of them have been derived by Chaudhry et al. 2],4]. ing.cienc., vol. 1, no. 19, pp , enero-junio

6 Properties Applications of Extended Hypergeometric Functions Using the transformation t = 1 + u) 1 u in 6), with the Jacobian Jt u) = 1 + u) 2, we arrive at u a 1 exp σu + u 1 )] Ba, b; σ) = exp 2σ) du. 15) 1 + u) a+b For σ = with Rea) > Reb) >, the above expression gives the well-known integral representation of Ba, b) as Ba, b) = u a 1 du. 16) 1 + u) a+b If we take b = a in 15) compare the resulting expression with 11) we obtain an interesting relationship between the extended beta function the type 2 modified Bessel function as Ba, a; σ) = 2 exp 2σ)K a 2σ). 17) If we consider z = 1 in 9) compare the resulting expression with the representation 6), we find that the extended beta function EGHF are related by the expression F σ a, b; c; 1) = Bb, c b a; σ), Rec) > Reb) >. 18) Further, substituting c = a in 18) using 17), we obtain, for σ >, F σ a, b; a; 1) = Bb, b; σ) Bb, a b) = 2 exp 2σ) Bb, a b) K b2σ), 19) where Rea) > Reb) >. Note that 19) can also be obtained by taking z = 1 a = c in 2), then using the integral representation 11). In the integral representation of EGHF ECHF given in 9) 1), respectively, substituting t = 1 + u) 1 u, with the Jacobian 16 Ingeniería y Ciencia

7 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta Jt u) = 1+u) 2, alternative integral representations are obtained as F σ a, b; c; z) = exp 2σ) u b 1 exp σu + u 1 )] du 2) 1 + u) c a z)u] a Φ σ b; c; z) = exp 2σ) u b 1 zu 1 + u) exp c 1 + u σ u + 1 )] du. u 21) If we take σ = in 2) 21), we arrive at the representations 12) 13) of the classical Gauss hypergeometric function the classical confluent hypergeometric function, respectively. For arg1 z) < 1, the transformation formula is given by F σ a, b; c; z) = 1 z) a F σ a, c b; c; z ). 22) 1 z It is noteworthy that σ = in 22) gives the well-known transformation formula F a, b; c; z) = 1 z) a F a, c b; c; z ). 1 z Also, putting c = b in the above expression, one obtains F a, b; b; z) = 1 z) a. In the integral representation of the ECHF 1) consider the substitution 1 u = t, whose Jacobian is given by Jt u) = 1, to obtain Φ σ b; c; z) = expz) 1 u) b 1 u c b 1 exp zu ] σ du. u1 u) 23) By evaluating the integral in 23) using 1), Kummer s relation for extended confluent hypergeometric function is derived as Φ σ b; c; z) = expz)φ σ c b; c; z). 24) For σ =, the expression 24) reduces to the well known Kummer s first formula for the classical confluent hypergeometric function. ing.cienc., vol. 1, no. 19, pp , enero-junio

8 Properties Applications of Extended Hypergeometric Functions 3 Properties of the EGHF ECHF This section gives several properties of the the EGHF ECHF. Writing F σ a, b; c; z/a) in terms of integral representation using 9) taking a, we obtain lim a F σ a, b; c; z a ) = Φ σ b; c; z). Replacing exp σ/t) exp σ/1 t)] by their respective series expansions involving Laguerre polynomials L n σ) L ) n σ) n =, 1, 2,...) given in Miller 3, Eq. 3.4a, 3.4b], namely, exp σ ) = exp σ)t t L n σ)1 t) n, t < 1, n= exp σ ) = exp σ)1 t) 1 t L m σ)t m, t < 1, m= in 9) 1), integrating with respect to t using 2) 3), EGHF ECHF can also be expressed as F σ a, b; c; z) = exp 2σ) m,n= Bb + m + 1, c + n + 1 b) L m σ)l n σ)f a, b + m + 1; c + m + n + 2; z) Φ σ b; c; z) = exp 2σ) m,n= Bb + m + 1, c + n + 1 b) L m σ)l n σ)φb + m + 1; c + m + n + 2; z), respectively. 18 Ingeniería y Ciencia

9 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta Theorem 3.1. If z is such that z < 1, σ > c > b >, then F σ a, b; c; z) exp 4σ)F a, b; c; z) exp 1) F a, b; c; z). 25) 4σ Proof. It follows that for u > σ >, σu + u 1 2) implies that σu + u 1 ) 2σ exp σu + u 1 )] exp 2σ). Now, using this inequality in the representation given in 2), we get F σ a, b; c; z) exp 4σ) u b u) a c z)u] du a = exp 4σ)F a, b; c; z), where the last line has been obtained by using 12). inequality ln v v 1, v >, for v = 4σ, yields exp 4σ) exp 1), 4σ which gives the second part of the inequality. Further, the Using special cases of the Gauss hypergeometric function in 25), several inequalities for EGHF can be obtained. For example, application of F a, a + 12 ) ] 2a ; 2a + 1; z 2 = z F a, a + 12 ; 12 ) ; z = z) 2a + 1 z) 2a] 2 yield F σ a, a + 1 ) 2 ; 2a + 1; z exp 4σ) z ] 2a F σ a, a ; 1 ) 2 ; z exp 4σ) 1 + z) 2a + 1 z) 2a]. 2 ing.cienc., vol. 1, no. 19, pp , enero-junio

10 Properties Applications of Extended Hypergeometric Functions Further, using the Clausen s identity F a, b; a + b + 12 )] 2 ; z = 3 F 2 2a, 2b, a + b; 2a + 2b, a + b + 12 ) ; z in 25), one gets F σ a, b; a + b + 12 ; z )] 2 exp 8σ) 3 F 2 2a, 2b, a + b; 2a + 2b, a + b ; z ). If we put z = 1 in 25), then use 18) 14) in the resulting expression, we obtain Bb, d; σ) exp 4σ)Bb, d) exp 1) Bb, d), 4σ where d = c a b >. If we replace z = in 1) compare the resulting expression with 6), we see that the ECHF the extended beta function have the relationship Φ σ b; c; ) = Bb, c b; σ). Theorem 3.2. If α β are two scalars such that β α >, then β β ] α) c+1 zu α) Φ σ b; c; z) = u α) b 1 β u) c b 1 exp α β α ] σβ α)2 exp du. 26) u α)β u) Proof. Using the transformation t = u α)/β α) with the Jacobian β α) 1 in the representation 1), we obtain the result. If we consider β = 1 α = 1 in 26), we have another integral representation of extended confluent hypergeometric function as Φ σ b; c; z) = 2 c+1 expz/2) 2 Ingeniería y Ciencia

11 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta 1 zu 1+u) b 1 1 u) c b 1 exp 2 4σ ) 1 u 2 Theorem 3.3. If σ > c > b >, then Φ σ b; c; z) exp 4σ)Φb; c; z) exp 1) Φb; c; z). 4σ Proof. Similar to the proof of Theorem 3.1. du. 4 Integrals involving EGHF ECHF In this section we evaluate some integrals that are related to EGHF ECHF. Theorem 4.1. If σ, α > β >, Rec) > Reb) > Rea) >, then exp αx)x a 1 Φ σ b; c; βx) dx = Γa)α a F σ a, b; c; βα 1 ). 27) Proof. Using the integral representation 1) changing the order of integration, we have x a 1 exp αx)φ σ b; c; βx) dx 1 1 = t b 1 1 t) c b 1 exp σ ] t1 t) x a 1 exp α βt)x] dx dt. Now, integrating with respect to x using Euler s gamma integral then t using the representation 9), we get the desired result. Corollary 4.1. If σ >, α >, Rec) > Reb) > Rea) >, then x a 1 Γa)Bb a, c b; σ) Φ σ b; c; αx) dx = α a 28) ing.cienc., vol. 1, no. 19, pp , enero-junio

12 Properties Applications of Extended Hypergeometric Functions exp x)x a 1 Φ σ b; c; x) dx = Γa)Bb, c b a; σ). 29) Proof. Application of Kummer s relation 24) yields x a 1 Φ σ b; c; αx) dx = exp αx)x a 1 Φ σ c b; c; αx) dx. Evaluating the above integral by applying 27) then using the relation 18), we get 28). To prove 29) just take α = β = 1 in 27) use 18). Corollary 4.2. If σ > Rec) > Reb) >, then exp x)x c 1 Φ σ b; c; x) dx = 2Γc) exp 2σ)K b2σ). Proof. Just take a = c in 29), then use 17). Theorem 4.2. For σ, α < 1, Rea) > Red) > Rec) > Reb) >, we have x d 1 1 x) a d 1 F σ a, b; c; αx) dx = Bd, a d)f σ d, b; c; α). 3) Proof. Using 9) changing the order of integration x d 1 1 x) a d 1 F σ a, b; c; αx) dx 1 1 = t b 1 1 t) c b 1 exp σ ] t1 t) = x d 1 1 x) a d 1 dx dt 1 αtx) a Bd, a d) t b 1 1 t) c b 1 1 αt) d exp σ ] dt, t1 t) 22 Ingeniería y Ciencia

13 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta where the integral involving x has been evaluated using 2). Finally, using the representation 9), we arrive at the desired result. Corollary 4.3. For σ > Rea) > Rec) > Reb) >, we have x c 1 1 x) a c 1 F σ a, b; c; x) dx = 2Bc, a c) exp 2σ)K b2σ). Proof. Just take α = 1 c = d in 3), then use 19). Theorem 4.3. For σ >, Rec) > Reb) > Rea) > Red) >, we have = Proof. Using 3), one gets x d 1 1 x) a d 1 F σ a, b; c; 1 x) dx Bd, a d)bb, c + d a b; σ). 1 x) d 1 x a d 1 F σ a, b; c; 1 x) dx = Bd, a d)f σ d, b; c; 1). Now, replacing d a d by a d d, respectively, in the above expression, one gets 1 x) a d 1 x d 1 F σ a, b; c; 1 x) dx = Ba d, d)f σ a d, b; c; 1). Finally, substituting for F σ a d, b; c; 1) from 18), we get the desired result. Theorem 4.4. If σ >, α >, Rea) > Red) > Rec) > Reb) >, then x d 1 F σ a, b; c; αx) dx = α d Ba d, d)bb d, c b; σ). 31) ing.cienc., vol. 1, no. 19, pp , enero-junio

14 Properties Applications of Extended Hypergeometric Functions Proof. Replacing F σ a, b; c; αx) by its integral representation 9) changing the order of integration, we get x d 1 F σ a, b; c; αx) dx 1 1 = t b 1 1 t) c b 1 exp σ ] x d 1 dx dt. t1 t) 1 + αtx) a Now, we integrate x using 16) then t using 6) to obtain the result. Corollary 4.4. If σ >, α > Rea) > Rec) > Reb) >, then x c 1 F σ a, b; c; αx) dx = 2Ba c, c)α c exp 2σ)K b c 2σ). Proof. Just take c = d in 31) use the relation 17). 5 Statistical Distributions In this section, we define the extended Gauss hypergeometric function the extended confluent hypergeometric function distributions. We study several properties of these new distributions their relationships with other known distributions. We also show that these distributions occur naturally as the distribution of the quotient U/V, where U V are independent, U has a gamma or beta type 2 distribution the rom variable V has an extended beta type 1 distribution. In the end, we derive results on products quotients of independent rom variables. First, we define the gamma, beta type 1 beta type 2 distributions. These definitions can be found in Johnson, Kotz Balakrishnan 6], Gupta Nagar 7]. A rom variable X is said to have a gamma distribution with parameters θ > ), κ > ), denoted by X Gaκ, θ), if its probabil- 24 Ingeniería y Ciencia

15 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta ity density function pdf) is given by {θ κ Γκ)} 1 x κ 1 exp x ), x >. 32) θ Note that for θ = 1, the above distribution reduces to a stard gamma distribution in this case we write X Gaκ). A rom variable X is said to have a beta type 1 distribution with parameters a, b), a >, b >, denoted as X B1a, b), if its pdf is given by {Ba, b)} 1 x a 1 1 x) b 1, < x < 1, where Ba, b) is the beta function. A rom variable X is said to have a beta type 2 distribution with parameters a, b), denoted as X B2a, b), a >, b >, if its pdf is given by {Ba, b)} 1 x a x) a+b), x >. 33) A rom variable X is said to have an extended beta type 1) distribution with parameters α, β λ, denoted by X EB1α, β; λ), if its pdf is given by Chaudhry et al. 2]), ] {Bα, β; λ)} 1 x α 1 1 x) β 1 λ exp, < x < 1, 34) x1 x) where Bα, β; λ) is the extended beta function defined by 6), λ >, < α, β <. For λ = with α > β >, the density 34) reduces to a beta type 1 density. Definition 5.1. A rom variable X is said to have an extended Gauss hypergeometric function distribution with parameters ν, α, β, γ σ, denoted by X EGHν, α, β, γ; σ), if its pdf is given by Bβ, γ β) Bν, α ν)bβ ν, γ β; σ) xν 1 F σ α, β; γ; x), x >, where α > ν >, γ > β > if σ > α > ν >, γ > β > ν > if σ =. ing.cienc., vol. 1, no. 19, pp , enero-junio

16 Properties Applications of Extended Hypergeometric Functions The following theorem derives the extended Gauss hypergeometric function distribution as the distribution of the ratio of two independent rom variables distributed as beta type 2 extended beta type 1. Theorem 5.1. Suppose that the rom variables U V are independent, U B2ν, γ) V EB1α, β; σ). Then U/V EGHν, ν + γ, ν + α, ν + α + β; σ). Proof. As U V are independent, by 33) 34), the joint density of U V is given by {Bν, γ)bα, β; σ)} 1 uν 1 v α 1 1 v) β 1 exp σ ], 1 + u) ν+γ v1 v) where u > < v < 1. Using the transformation X = U/V, with the Jacobian Ju x) = v, we obtain the joint density of V X as {Bν, γ)bα, β; σ)} 1 xν 1 v ν+α 1 1 v) β xv) ν+γ exp σ v1 v) where < v < 1 x >. Now, integration of the above expression with respect to v using 9) yields the desired result. If X EGHν, α, β, γ; σ), then EX h ) = Bβ, γ β) Bν, α ν)bβ ν, γ β; σ) ], x ν+h 1 F σ α, β; γ; x) dx. Now, evaluation of the above integral by using 31) yields EX h ) = Bν + h, α ν h)bβ ν h, γ β; σ). Bν, α ν)bβ ν, γ β; σ) where ν < Reh) < α ν if σ >, ν < Reh) < α ν Reh) < β ν if σ =. Next, we define study the extended confluent hypergeometric function distribution. 26 Ingeniería y Ciencia

17 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta Definition 5.2. A rom 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 1 Φ σ α; β; x), x >, 35) Γν)Bα ν, β α; σ) where ν >, β > α > if σ > β > α > ν > if σ =. The extended confluent hypergeometric function distribution can be derived as the distribution of the quotient of independent gamma extended beta type 1 variables as given in the following theorem. Theorem 5.2. If U Gaa) V EB1b, c; σ) are independent, then X = U/V ECHa, a + b, a + b + c; σ). Proof. As U V are independent, from 32) 34), the joint density of U V is given by u a 1 v b 1 1 v) c 1 exp u σ/v1 v)], u >, < v < 1. Γa)Bb, c; σ) Making the transformation X = U/V, with the Jacobian Ju x) = v, we find the joint density of V X as x a 1 v a+b 1 1 v) c 1 exp vx σ/v1 v)], < v < 1, x >. Γa)Bb, c; σ) Now, the density of X is obtained by integrating the above expression with respect to v using the integral representation 1). By using 28), the expected value of X h, when X ECHν, α, β; σ), is derived as EX h ) = Γν + h)bα ν h, β α; σ), Γν)Bα ν, β α; σ) where Reν + h) > if σ > β > α > Reν + h) > if σ =. ing.cienc., vol. 1, no. 19, pp , enero-junio

18 Properties Applications of Extended Hypergeometric Functions In the remainder of this section we derive results on products quotients of independent rom variables. The derivation final result in each case involves extended forms of beta, confluent hypergeometric, Gauss hypergeometric or generalized hypergeometric functions showing ample applications of these functions further advancing statistical distribution theory. Theorem 5.3. Suppose that the rom variables X Y are independent, X Gaλ) Y ECHν, α, β; σ). Then, the pdf of R = Y/Y + X) is given by Bα, β α) Bν, λ)bα ν, β α; σ) rν 1 1 r) λ 1 F σ ν + λ, β α; β; r), where < r < 1. Proof. Since X Y are independent, from 32) 35), we write the joint density of X Y as Bα, β α) Γν)Γλ)Bα ν, β α; σ) xλ 1 y ν 1 exp x)φ σ α; β; y), where x > y >. Now, making the transformation S = Y + X R = Y/Y + X) with the Jacobian Jx, y r, s) = s using 24), we obtain the joint density of S R as Bα, β α) Γν)Γλ)Bα ν, β α; σ) rν 1 1 r) λ 1 s ν+λ 1 exp s)φ σ β α; β; rs), s >, < r < 1. Clearly, R S are not independent. Integrating the previous expression with respect to s by using 27) the density of R is obtained. Corollary 5.1. The density of W = X/Y is given by Bα, β α) w λ 1 Bν, λ)bα ν, β α; σ) 1 + w) F ν+λ σ where w >. ν + λ, β α; β; ) 1, 1 + w 28 Ingeniería y Ciencia

19 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta Theorem 5.4. Suppose that the rom variables U V are independent, U B2ν, γ) V EB1α, β; σ). Then Y = UV has the density Bα + γ, β) y ν 1 Bν, γ)bα, β; σ) 1 + y) F ν+γ σ ν + γ, β; α + β + γ; ) 1, y >. 1 + y Proof. As U V are independent, from 33) 34), the joint density of U V is given by {Bν, γ)bα, β; σ)} 1 uν 1 v α 1 1 v) β 1 exp σ ], 1 + u) ν+γ v1 v) where u > < v < 1. Using the transformation Y = UV, with the Jacobian Ju y) = 1/v, we obtain the joint density of V Y as {Bν, γ)bα, β; σ)} 1 y ν 1 v α+γ 1 1 v) β 1 exp σ/v1 v)], 1 + y) ν+γ 1 1 v)/1 + y)] ν+γ where < v < 1 y >. The marginal density of Y is obtained by integrating the above expression with respect to v using 9). Corollary 5.2. Suppose that the rom variables U V are independent, U B2ν, γ) V B1α, β). Then Y = UV has the density Bα + γ, β) y ν 1 Bν, γ)bα, β) 1 + y) F ν+γ ν + γ, β; α + β + γ; ) 1, y >. 1 + y The above corollary has also been derived in Nagar Zarrazola 8] Morán-Vásquez Nagar 9]. 6 Conclusion We have given several interesting properties of extended beta, extended Gauss hypergeometric extended confluent hypergeometric functions. We have also evaluated a number of integrals involving ing.cienc., vol. 1, no. 19, pp , enero-junio

20 Properties Applications of Extended Hypergeometric Functions these function. Finally, we have shown that these functions occur in a natural way in statistical distribution theory. In a series of papers Castillo-Pérez his co-authors 1],11],12],13] have studied a generalization of the Gauss hypergeometric function defined by Ra, b; c; τ; z) = n= n= Bb + τ n, c b) z n, Rec) > Reb) >. n! Replacing Bb + τ n, c b) by Bb + τ n, c b; σ), an extended form of the above function can be defined as Bb + τ n, c b; σ) z n R σ a, b; c; τ; z) =, Rec) > Reb) >. n! The function defined above is a generalization of the extended Gauss hypergeometric function will be considered for further research. Acknowledgements The research work of DKN was supported by the Sistema Universitario de Investigación, Universidad de Antioquia under the project no. IN1182CE. References 1] Y. L. Luke, The special functions their approximations, Vol. I, ser. Mathematics in Science Engineering, Vol. 53. New York: Academic Press, ] M. A. Chaudhry, A. Qadir, M. Rafique, S. M. Zubair, Extension of Euler s beta function, J. Comput. Appl. Math., vol. 78, no. 1, pp , Online]. Available: 13, 15, 25 3] A. R. Miller, Remarks on a generalized beta function, J. Comput. Appl. Math., vol. 1, no. 1, pp , Online]. Available: 13, 14, 18 4] M. A. Chaudhry, A. Qadir, H. M. Srivastava, R. B. Paris, Extended hypergeometric confluent hypergeometric functions, Appl. Math. Comput., vol. 159, no. 2, pp , 24. Online]. Available: 13, 14, 15 3 Ingeniería y Ciencia

21 Daya K. Nagar, Raúl A. Morán-Vásquez Arjun K. Gupta 5] I. S. Gradshteyn I. M. Ryzhik, Table of integrals, series, products, 7th ed. Elsevier/Academic Press, Amsterdam, 27, translated from the Russian, Translation edited with a preface by Alan Jeffrey Daniel Zwillinger, With one CD-ROM Windows, Macintosh UNIX). 15 6] N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions. Vol. 2, 2nd ed., ser. Wiley Series in Probability Mathematical Statistics: Applied Probability Statistics. New York: John Wiley & Sons Inc., 1995, a Wiley-Interscience Publication. 24 7] A. K. Gupta D. K. Nagar, Matrix variate distributions, ser. Chapman & Hall/CRC Monographs Surveys in Pure Applied Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2, vol ] D. K. Nagar E. Zarrazola, Distributions of the product the quotient of independent Kummer-beta variables, Sci. Math. Jpn., vol. 61, no. 1, pp , 25. Online]. Available: contents/e-24-1/24-4.pdf 29 9] R. A. Morán-Vásquez D. K. Nagar, Product quotients of independent Kummer-gamma variables, Far East J. Theor. Stat., vol. 27, no. 1, pp , ] J. A. Castillo Pérez C. Jiménez Ruiz, Algunas representaciones simples de la función hipergeométrica generalizada 2 r 1 a, b; c; τ; x), Ingeniería y Ciencia, vol. 2, no. 4, pp , 26. Online]. Available: ] J. A. Castillo Pérez, Algunas integrales impropias con límites de integración infinitos que involucran a la generalizaciï 1 2n τ de la función hipergeométrica de gauss, Ingeniería y Ciencia, vol. 3, no. 5, pp , 27. Online]. Available: article/view/ ], Algunas integrales que involucran a la función hipergeométrica generalizada, Ingeniería y Ciencia, vol. 4, no. 7, pp. 7 22, 28. Online]. Available: article/view/ ] J. A. Castillo Pérez L. Galué, Algunas integrales indefinidas que contienen a la función hipergeométrica generalizada, Ingeniería y Ciencia, vol. 5, no. 9, pp , 29. Online]. Available: 3 ing.cienc., vol. 1, no. 19, pp , enero-junio