Generalized Beta Function Defined by Wright Function

Transcription

1 Generalized Beta Function Defined by Wright Function ENES ATA Ahi Evran University, Deartment of Mathematics, Kırşehir, Turkey. arxiv:83.3v [math.ca] 5 Mar 8 Abstract The main object of this aer is to resent generalizations of gamma, beta and hyergeometric functions. Some recurrence relations, transformation formulas, oeration formulas and integral reresentations are obtained for these new generalizations. Keywords: Gamma function, Beta function, Wright function, Gauss hyergeometric function, Confluent hyergeometric function, Mellin transform Introduction In recent years, some extensions of the well known secial functions have been considered by several authors [, 3, 4, 5, 6, 9, ]. In 994, Chaudhry and Zubair [] have indroduced the following extension of gamma function Γ x t x ex t, Re >. t In 997, Chaudhry et al. [5] resented the following extension of Euler s beta function [ B x,y t x t y ex ], t t Re >, Rex >, Rey > and they roved that this extension has connections with the Macdonald, error and Whittakers function. It clearly seems that Γ x Γx and B x,y Bx,y, where Γx and Bx,y are the classical gamma and beta functions []. Afterwards, Chaudhry et al. [7] used B x,y to extend the Gauss hyergeometric function and the confluent hyergeometric function as follows: F a,b;c;z n a n B b+n,c b Bb,c b z n 3 ; Rec > Reb >, Φ b;c;z n B b+n,c b z n Bb,c b 4

2 ; Rec > Reb >, where λ v denotes the Pochhammer symbol defined by and gave the Euler tye integral reresentations F a,b;c;z Bb,c b λ and λ v Γλ+v Γλ t b t c b zt a ex [ ] t t > ; and arg z < π < ; Rec > Reb >, Φ b;c;z Bb,c b [ ] t b t c b ex zt t t > ; and Rec > Reb >. They called these functions extended Gauss hyergeometric function EGHF and extended confluent hyergeometric function ECHF, resectively. They have discussed the differentiation roerties and Mellin transform of F a,b;c;z and Φ b;c;z and obtained transformation formulas, recurrence relations, summation and asymtotic formulas for these functions. In this aer, we use Wright function to define new generalizations of gamma and beta functions, which defined in [8] as Ψ α,β;z where α, β, C and Reα >, Reβ >. n z n Γαn+β, 5 We start our investigation introducing the following generalizations of gamma and beta functions as Ψ Γ α,β x t x Ψ α,β; t 6 t and Rex >, Reα >, Reβ >, Re >, Ψ B α,β x,y t x t y Ψ α,β; 7 t t Rex >, Rey >, Reα >, Reβ >, Re >. We call the new generalizations of gamma and beta functions as Ψ-gamma and Ψ-beta functions, resectively. It is obvious that, Ψ Γ, x Γ x, Ψ Γ, x Γx,

3 and Ψ B, x,y B x,y, Ψ B, x,y Bx,y, where Γ x,b x,y are generalized functions defined as and, resectively. In section, different integral reresentations and roerties of Ψ-beta function are obtained. Additionally, relations of Ψ-gamma and Ψ-beta functions are discussed. In Section 3, we define Ψ-Gauss and Ψ-confluent hyergeometric functions, using Ψ B α,β x,y obtain the integral reresentations of these Ψ-Gauss and Ψ-confluent hyergeometric functions. Furthermore we discussed the differentiation roerties, Mellin transforms, transformation formulas, recurrence relations, summation formulas for these new hyergeometric functions. Proerties of Ψ-gamma and Ψ-beta functions The next theorem gives the Mellin transform reresentation of the function Ψ B α,β x,y in terms of the ordinary beta function and Ψ Γ α,β s. Theorem.. Mellin transform reresentation of the Ψ-beta function is given by M [Ψ B α,β x,y ] Bx+s,y +s Ψ Γ α,β s, Res >, Rex+s >, Rey +s >, Re >, Reα >, Reβ >. Proof. Multilying 7 by s and integrating with resect to from to, we get M [Ψ B α,β x,y ] s t x t y Ψ α,β; d. 8 t t From the uniform convergence of the integral, the order of integration in 8 can be interchanged. Therefore, we have M [Ψ B α,β x,y ] t x t y s Ψ α,β; d. 9 t t Now using the one-to-one transformation excet ossibly at the boundaries and mas of the region onto itself v in 9, we get, t t Therefore, we have M [Ψ B α,β x,y ] which comletes the roof. t x+s t y+s M [Ψ B α,β x,y ] Bx+s,y +s Ψ Γ α,β s, v s Ψ α,β; vdv. 3

4 Corollary.. By the Mellin inversion formula, we have the following comlex integral reresentation for Ψ B α,β x,y : Ψ B α,β x,y πi +i i Bx+s,y +s Ψ Γ α,β s s ds. Theorem.3. For the Ψ-beta function, we have the following integral reresentations: π Ψ B α,β x,y cos x θsin y θ Ψ α,β; sec θcsc θ dθ, Ψ B α,β u x x,y Ψ +u x+y α,β; u+ du. u Proof. Letting t cos θ in 7, we get Ψ B α,β x,y π On the other hand, letting t u +u Ψ B α,β x,y This comletes the roof. t x t y Ψ α,β; t t cos x θsin y θ Ψ α,β; sec θcsc θ dθ. in 7, we get t x t y Ψ α,β; u x +u x+y Ψ α,β; t t u+ u du. Theorem.4. For the Ψ-beta function, we have the following functional relation: Ψ B α,β Proof. Direct calculations yield Ψ B α,β x,y ++ Ψ B α,β x,y ++ Ψ B α,β x+,y t x t y Ψ α,β; x+,y Ψ B α,β x,y. + t t [ t x t y +t x t y ] Ψ α,β; t t t x t y Ψ α,β; t t Ψ B α,β x, y hence the result. t x t y Ψ α,β; t t 4

5 Theorem.5. For the roduct of two Ψ-gamma function, we have the following integral reresentation: π Ψ Γ α,β x Ψ Γ α,β y 4 r x+y cos x θsin y θ Ψ α,β; r cos θ r cos θ Ψ α,β; r sin θ drdθ. r sin θ Proof. Substituting t η in 6, we get Ψ Γ α,β x η x Ψ α,β; η dη. η Therefore, Ψ Γ α,β x Ψ Γ α,β y 4 η x ξ y Ψ α,β; η η Ψ α,β; ξ ξ dηdξ. Letting η rcosθ and ξ rsinθ in the above equality, comletes the roof. π Ψ Γ α,β x Ψ Γ α,β y 4 r x+y cos x θsin y θ Ψ α,β; r cos θ r cos θ Ψ α,β; r sin θ drdθ r sin θ Theorem.6. For the Ψ-beta function, we have the following summation relation: Ψ B α,β x, y n Proof. From the definition of the Ψ-beta function, we get Ψ B α,β x, y Using the following binomial series exansion t y y n Ψ B α,β x+n,, Re >. t x t y Ψ α,β;. t t n t n y n, t <, 5

6 we obtain Ψ B α,β x, y n y n t x+n Ψ α,β;. t t Therefore, interchanging the order of integration and summation and then using 7, we obtain Ψ B α,β x, y which comletes the roof. n n y n y n t x+n Ψ α,β; t t Ψ B α,β x+n,, 3 Ψ-generalization of Gauss and confluent hyergeometric functions In this section, we use the Ψ-beta function 7 to define Ψ-generalization of Gauss and confluent hyergeometric functions as Ψ F α,β a,b;c;z Ψ B α,β b+n,c b z n a n Bb,c b n and Ψ Φ α,β b;c;z n Ψ B α,β b+n,c b z n Bb,c b, resectively. We call Ψ F α,β hyergeometric function. Observe that and a,b;c;z asψ-gauss hyergeometric function and Ψ Φ α,β b;c;z asψ-confluent Ψ F, a,b;c;z F a,b;c;z, Ψ F, a,b;c;z F a,b;c;z, Ψ Φ, b;c;z Φ b;c;z, Ψ Φ, b;c;z Φb;c;z, where F a,b;c;z,φ b;c;z are generalized functions defined as 3 and 4, resectively. 6

7 3. Integral reresentations The Ψ-Gauss hyergeometric function can be rovided with an integral reresentation by using the definition of the Ψ-beta function 7. Theorem 3.. Let Re > ; and arg z < π; Rec > Reb >, then Ψ F α,β a,b;c;z t b t c b Ψ α,β; zt a, Bb,c b t t a,b;c;z Bb,c b Ψ F α,β Ψ α,β; u b +u a c [+u z] a u+ du, u Ψ F α,β π a,b;c;z sin b θcos c b θ zsin θ a Bb,c b Ψ α,β; dθ. sin θcos θ Proof. Direct calculations yield Ψ F α,β a,b;c;z Setting u t t Ψ F α,β a,b;c;z Ψ B α,β b+n,c b a n Bb,c b n a n Bb,c b Bb,c b Bb,c b in, we get n z n t b+n t c b Ψ α,β; t b t c b Ψ α,β; t t t b t c b Ψ α,β; t t z n t t zt n a n n zt a. u b +u a c [+u z] a Ψ α,β; u+ du. Bb,c b u On the other hand, substituting t sin θ in, we have Ψ F α,β π a,b;c;z sin b θcos c b θ zsin θ a Bb,c b Ψ α,β; dθ. sin θcos θ A smilar rocedure yields an integral reresentation of the Ψ-confluent hyergeometric function by using the definition of the Ψ-beta function. 7

8 Theorem 3.. For the Ψ-confluent hyergeometric function, we have the following integral reresentations: Ψ Φ α,β b;c;z t b t c b e zt Ψ α,β;, Bb,c b t t b;c;z Bb,c b Ψ Φ α,β u c b u b e z u Ψ α,β; du, u u ; and Rec > Reb >. 3. Differentiation formulas In this section, by using the formulas, and Bb,c b c b Bb+,c b a n+ aa+ n we obtain new formulas including derivatives of Ψ-Gauss hyergeometric function and Ψ-confluent hyergeometric function with resect to the variable z. Theorem 3.3. For Ψ-Gauss hyergeometric function, we have the following differentiation formula: d n F α,β dz n a,b;c;z } a nb n c n Proof. Taking the derivative of Ψ F α,β d dz Relacing n n+, we get d dz F α,β a,b;c;z } d dz [ Ψ F α,β a+n,b+n;c+n;z ]. a,b;c;z with resect to z, we obtain { } Ψ B α,β b+n,c b z n a n Bb,c b n Ψ B α,β b+n,c b z n a n Bb,c b n!. n F α,β a,b;c;z } ab c ab c Ψ B α,β b+n+,c b z n a+ n Bb+,c b n [ Ψ F α,β a+,b+;c+;z ]. Recursive alication of this rocedure gives us the general form: d n which comletes the roof. F α,β dz n a,b;c;z } a nb n c n 8 [ Ψ F α,β a+n,b+n;c+n;z ]

9 Theorem 3.4. For Ψ-confluent hyergeometric function, we have the following differentiation formula: d n Proof. Taking the derivative of Ψ Φ α,β Φ α,β dz n b;c;z } b n [ Ψ Φ α,β b+n;c+n;z ]. c n d dz Relacing n n+, we get d dz Φ α,β b;c;z } d dz b;c;z with resect to z, we obtain { } Ψ B α,β b+n,c b z n Bb,c b n Φ α,β b;c;z } b c b c n Ψ B α,β n b+n,c b Bb,c b z n n!. Ψ B α,β b+n+,c b z n Bb+,c b [ Ψ Φ α,β b+;c+;z ]. Recursive alication of this rocedure gives us the general form: which comletes the roof. d n Φ α,β dz n b;c;z } b n [ Ψ Φ α,β b+n;c+n;z ] c n 3.3 Mellin transform reresentations In this section, we obtain the Mellin transform reresentations of the Ψ-Gauss and Ψ-confluent hyergeometric functions. Theorem 3.5. For the Ψ-Gauss hyergeometric function, we have the following Mellin transform reresentation: M [Ψ F α,β a,b;c;z : s ] Ψ Γ α,β sbb+s,c+s b F a,b+s;c+s;z. Bb,c b Proof. To obtain the Mellin transform, we multily both sides of by s and integrate with resect to over the interval [,. Thus, we get M [Ψ F α,β a,b;c;z : s ] s Bb,c b n Ψ B α,β s [Ψ F α,β a,b;c;z ] d b+n,c b Bb,c b a n z n d t b t c b zt a s Ψ α,β; d. t t 9

10 Since substituting u in the above equation, t t s Ψ α,β; d t s t sψ Γ α,β s. t t Thus, we get M [Ψ F α,β a,b;c;z : s ] Ψ Γ α,β sbb+s,c+s b F a,b+s;c+s;z. Bb,c b Corollary 3.6. By the Mellin inversion formula, we have the following comlex integral reresentation for Ψ F α,β a,b;c;z: Ψ F α,β a,b;c;z +i πi i Ψ Γ α,β sbb+s,c+s b F a,b+s;c+s;z s ds. Bb,c b Theorem 3.7. For the Ψ-confluent hyergeometric function, we have the following Mellin transform reresentation: M [Ψ Φ α,β b;c;z : s ] Ψ Γ α,β sbb+s,c+s b Φb+s;c+s;z. Bb,c b Proof. To obtain the Mellin transform, we multily both sides of by s and integrate with resect to over the interval [,. Thus, we get M [Ψ Φ α,β b;c;z : s ] s Bb,c b n Ψ B α,β s [Ψ Φ α,β b;c;z ] d b+n,c b Bb,c b z n d t b t c b e zt [ s Ψ α,β; ] d. t t Since substituting u in the above equation, t t s Ψ α,β; d t s t sψ Γ α,β s. t t Thus, we get M [Ψ Φ α,β b;c;z : s ] Ψ Γ α,β sbb+s,c+s b Φb+s;c+s;z. Bb,c b Corollary 3.8. By the Mellin inversion formula, we have the following comlex integral reresentation for Ψ Φ α,β b;c;z: Ψ Φ α,β b;c;z +i πi i Ψ Γ α,β sbb+s,c+s b Φb+s;c+s;z s ds. Bb,c b

11 3.4 Transformation formulas Theorem 3.9. For the Ψ-Gauss hyergeometric function, we have the following transformation formula: [ ] Ψ F α,β a,b;c;z z a Ψ F α,β z a,c b;b;, z where arg z < π. Proof. By writing [ z t] a z a + zt a z and relacing t t in, we obtain Ψ F α,β a,b;c;z z a Bb,c b t c b t b zt a Ψ α,β; z t t Hence, Re > ; and z < π; Rec > Reb >. Ψ F α,β a,b;c;z z a [ Ψ F α,β a,c b;b; ] z z which comletes the roof. Theorem 3.. For the Ψ-confluent hyergeometric function, we have the following transformation formula: Proof. Ψ Φ α,β b;c;z Ψ Φ α,β b;c;z e z[ψ Φ α,β c b;b; z ]. n Ψ B α,β b+n,c b z n Bb,c b Bb,c b Relacing t η in, we obtain which comletes the roof. t b+n t c b e zt Ψ α,β; t t Ψ Φ α,β b;c;z e z[ψ Φ α,β c b;b; z ].

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