aa r X i v : . [ m a t h . C A ] M a r Properties of Ultra Gamma Function
Kuldeep Singh GehlotSeptember 12, 2018
Government College Jodhpur,JNV University Jodhpur, Rajasthan, India-306401.Email: drksgehlot@rediffmail.com
Abstract
In this paper we study the integral of type δ,a Γ ρ,b ( x ) = Γ( δ, a ; ρ, b )( x ) = Z ∞ t x − e − tδa − t − ρb dt. Different authors called this integral by different names like ultra gamma function, generalizedgamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral,Bessel integral etc. We prove several identities and recurrence relation of above said integral,we called this integral as Four Parameter Gamma Function. Also we evaluate relation betweenFour Parameter Gamma Function, p-k Gamma Function and Classical Gamma Function. Withsome conditions we can evaluate Four Parameter Gamma Function in term of Hypergeometricfunction.
Mathematics Subject Classification :
Keywords:
Four Parameter Gamma Function, Ultra Gamma Function, Two Parameter GammaFunction, Two Parameter Pochhammer Symbol.
The main aim of this paper is to introduce Four Parameter Gamma Function in the form, δ,a Γ ρ,b ( x ) = Γ( δ, a ; ρ, b )( x ) = Z ∞ t x − e − tδa − t − ρb dt. (1.1)where x ∈ C/δZ − ; δ, ρ, a, b ∈ R + − { } and Re ( x − ρn ) > , n ∈ N. Four Parameter Gamma Function is the deformation of the two parameter Gamma Functiondefined by [4], such that δ,a Γ ρ,b ( x ) ⇒ k,p Γ ,b ( x ) = e − b p Γ k ( x ), as δ = k, a = p, ρ = 0 . And this Four Parameter Gamma Function is the deformation of the k-Gamma Function definedby [1], such that δ,a Γ ρ,b ( x ) ⇒ k,k Γ ,b ( x ) = e − b Γ k ( x ), as δ = a = k, ρ = 0 . Also the Four Parameter Gamma Function is the deformation of the classical Gamma Function,such that δ,a Γ ρ,b ( x ) ⇒ , Γ ,b ( x ) = e − b Γ( x ), as δ = a = 1 , ρ = 0 . Throughout this paper Let
C, R + , Re () , Z − and N be the sets of complex numbers, positive realnumbers, real part of complex number, negative integer and natural numbers respectively. Weuse the notation and terminology of [2] and [3].1he p - k Gamma Function (i.e. Two Parameter Gamma Function), p Γ k ( x ) is given by [4], For x ∈ C/kZ − ; k, p ∈ R + − { } and Re ( x ) > , n ∈ N, is p Γ k ( x ) = 1 k lim n →∞ n ! p n +1 ( np ) xk p ( x ) n +1 ,k . (1.2)or p Γ k ( x ) = 1 k lim n →∞ n ! p n +1 ( np ) xk − p ( x ) n,k . (1.3)And the integral representation of p - k Gamma Function is given by p Γ k ( x ) = Z ∞ e − tkp t x − dt. (1.4) δ,a Γ ρ,b ( x ) or Γ( δ, a ; ρ, b )( x ) . Theorem 2.1
The relation between Four Parameter Gamma Function, p - k Gamma Functionand Classical Gamma Function is given byΓ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a Γ δ ( x − ρn ) , (2.1)Γ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n δ lim m →∞ m ! a m +1 ( ma ) x − ρnδ − a ( x − ρn ) m,δ , (2.2)And Γ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a ( x − ρnδ ) δ Γ( x − ρnδ ) . (2.3)where x ∈ C/δZ − ; δ, ρ, a, b ∈ R + − { } and Re ( x − ρn ) > , n ∈ N. Proof: Using the definition (1.1), we haveΓ( δ, a ; ρ, b )( x ) = Z ∞ t x − e − tδa − t − ρb dt, Γ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n Z ∞ t x − ρn − e − tδa dt, using [4], equation (2.14), we haveΓ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a Γ δ ( x − ρn ) , And by using [4], theorem (2.9), we haveΓ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a ( x − ρnδ ) δ Γ( x − ρnδ ) . By using equation (1.3) in (2.1), we get the result (2.2).This completes the proof.
Theorem 2.2
Given x ∈ C/δZ − ; δ, ρ, k, p, a, b ∈ R + − { } and Re ( x − ρn ) > , n ∈ N, then thefollowing recurrence relations exists,Γ( δ, a ; ρ, b )( x ) = kδ Γ( k, a ; kρδ , b )( kxδ ) , (2.4)2( δ, a ; ρ, b )( x ) = kδ ( ap ) xδ Γ( k, p ; kρδ , b ( ap ) ρδ )( kxδ ) , (2.5)Γ( δ, a ; ρ, b )( x ) = ( ap ) xδ Γ( δ, p ; ρ, b ( ap ) ρδ )( x ) , (2.6)Proof: From equation (2.1), we haveΓ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a Γ δ ( x − ρn ) , using result (2.11) of [4],we haveΓ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n kδ a Γ k ( k ( x − ρn ) δ ) , using equation (2.14) of [4], we haveΓ( δ, a ; ρ, b )( x ) = kδ Z ∞ t kxδ − e − tka − t − kρδb dt, Γ( δ, a ; ρ, b )( x ) = kδ Γ( k, a ; kρδ , b )( kxδ ) . Which completes the proof of (2.4).Similarly we can prove the result (2.5) and (2.6).
Theorem 2.3
Given x ∈ C/δZ − ; δ, ρ, a, b ∈ R + − { } and Re ( x − ρn ) > , n ∈ N, thenwe can represent the Four Parameter Gamma Function in the form of series,Γ( δ, a ; ρ, b )( x ) = 1 δ ( a ) xδ ∞ X n =0 ( − n n ! ( a − ρδ b ) n ∞ Y m =1 { (1 − m ) ( x − nρδ ) (1 + x − nρmδ ) − } (2.7)Proof: From equation (2.1) we have,Γ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a Γ δ ( x − ρn ) , using equation (2.15) of [4], we haveΓ( δ, a ; ρ, b )( x ) = 1 δ ( a ) xδ ∞ X n =0 ( − n n ! ( a − ρδ b ) n ∞ Y m =1 { (1 − m ) ( x − nρδ ) (1 + x − nρmδ ) − } . Which completes the proof.
Theorem 2.4
Given x ∈ C/δZ − ; δ, ρ, a, b ∈ R + − { } and Re ( x − ρn ) > , n ∈ N, thenthe fundamental equation satisfied by Four Parameter Gamma Function is, x Γ( δ, a ; ρ, b )( x ) = δa Γ( δ, a ; ρ, b )( x + δ ) − ρb Γ( δ, a ; ρ, b )( x − ρ ) (2.8)Proof: From equation (2.1), we haveΓ( δ, a ; ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a Γ δ ( x − ρn ) , replace x by x + δ we have,Γ( δ, a ; ρ, b )( x + δ ) = ∞ X n =0 ( − n n ! b n a Γ δ ( x − ρn + δ ) , δ, a ; ρ, b )( x + δ ) = ∞ X n =0 ( − n n ! b n ( x − ρn ) aδ a Γ δ ( x − ρn ) , Γ( δ, a ; ρ, b )( x + δ ) = xaδ Γ( δ, a ; ρ, b )( x ) − ∞ X n =1 ( − n aρ ( n − b n δ a Γ δ ( x − ρn ) ,n can be replace by n + 1, we haveΓ( δ, a ; ρ, b )( x + δ ) = xaδ Γ( δ, a ; ρ, b )( x ) + aρbδ ∞ X n =0 ( − n ( n )! b n a Γ δ ( x − ρn − ρ ) , using equation (2.1),Γ( δ, a ; ρ, b )( x + δ ) = xaδ Γ( δ, a ; ρ, b )( x ) + aρbδ Γ( δ, a ; ρ, b )( x − ρ ) . Which completes the proof. δ,a Γ ρ,b ( x ) or Γ( δ, a ; ρ, b )( x ) . Theorem 3.1
Given x ∈ C/δZ − ; δ, ρ, a, b ∈ R + − { } , Re ( x − ρn ) > , n ∈ N and ρδ ∈ N, thenwe have, Γ( δ, a ; − ρ, b )( x ) = a xδ Γ( xδ ) δ ρδ F [( x − rδ − δρ ) r =1 , ,..., ρδ ; − ; − b ( aρδ ) ρδ ] . (3.1)Proof: Consider the equation (2.3),Γ( δ, a ; − ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a ( x + ρnδ ) δ Γ( x + ρnδ ) , Γ( δ, a ; − ρ, b )( x ) = ∞ X n =0 ( − n n ! b n a ( x + ρnδ ) δ Γ( x + ρnδ )Γ( xδ )Γ( xδ ) , Γ( δ, a ; − ρ, b )( x ) = Γ( xδ ) ∞ X n =0 ( − n n ! b n a ( x + ρnδ ) δ ( xδ ) ρδ n , we know the generalized pochammer symbol is given,( α ) rn = r rn r Y n =1 ( α + n − r ) n , then we have, Γ( δ, a ; − ρ, b )( x ) = Γ( xδ ) a xδ δ ∞ X n =0 n ! [ − b ( aρδ ) ρδ ] n ρδ Y r =1 ( x − rδ − δρ ) n , this will give the desired result. 4 eferenceseferences