# Onk-Gamma and k-Beta Distributions and Moment Generating Functions

Authors: Gauhar Rahman; Shahid Mubeen; Abdur Rehman; Mammona Naz
Journal: Journal of Probability and Statistics
 (2014)

---

## Abstract

The main objective of the present paper is to definek-gamma and k-beta distributions and moments generating function for the said distributions in terms of a new parameter k>0. Also, the authors prove some properties of these newly defined distributions.


---

## Body

## 1. Basic Definitions

In this section we give some definitions which provide a base for our main results. The definitions (1.1–1.3) are given in [1] while (1.4–1.6) are introduced in [2]. Also, we have taken some statistics related definitions (1.7–1.11) from [3–5].


### 1.1. Pochhmmer Symbol

The factorial function is denoted and defined by(1)(a)n={a(a+1)(a+2)⋯(a+n-1);forn≥1,a≠0,1;ifn=0.
The function (a)n defined in relation (1) is also known as Pochhmmer symbol.


### 1.2. Gamma Function

Letz∈C; the Euler gamma function is defined by
(2)Γ(z)=lim⁡n→∞n!nz-1(z)n
and the integral form of gamma function is given by
(3)Γ(z)=∫0∞‍tz-1e-tdt,R(z)>0.
From the relation (3), using integration by parts, we can easily show that
(4)Γ(z+1)=zΓ(z).
The relation between Pochhammer symbol and gamma function is given by
(5)(z)n=Γ(z+n)Γ(z).


### 1.3. Beta Function

The beta function of two variables is defined as(6)B(x,y)=∫01‍tx-1(1-t)y-1dt,Re(x),Re(y)>0
and, in terms of gamma function, it is written as
(7)B(x,y)=Γ(x)Γ(y)Γ(x+y).


### 1.4. Pochhammerk-Symbol

Fork>0, the Pochhammer k-symbol is denoted and defined by
(8)(a)n,k={a(a+k)(a+2k)⋯(a+(n-1)k);forn≥1,a≠0,1;ifn=0.


### 1.5.k-Gamma Function

Fork>0 and z∈C, the k-gamma function is defined as
(9)Γk(z)=lim⁡n→∞n!kn(nk)z/k-1(z)n,k
and the integral representation of k-gamma function is
(10)Γk(z)=∫0∞‍tz-1e-tk/kdt.


### 1.6.k-Beta Function

ForRe(x),Re(y)>0, the k-beta function of two variables is defined by
(11)Bk(x,y)=1k∫0∞‍tx/k-1(1-t)y/k-1dt
and, in terms of k-gamma function, k-beta function is defined as
(12)Bk(x,y)=Γk(x)Γk(y)Γk(x+y).Also, the researchers [6–10] have worked on the generalized k-gamma and k-beta functions and discussed the following properties:
(13)Γk(x+k)=xΓk(x),(14)(x)n,k=Γk(x+nk)Γk(x),(15)Γk(k)=1,k>0.
Using the above relations, we see that, for x,y>0 and k>0, the following properties of k-beta function are satisfied by authors (see [6, 7, 11]):
(16)βk(x+k,y)=xx+yβk(x,y),(17)βk(x,y+k)=yx+yβk(x,y),(18)βk(xk,yk)=1kβ(x,y),(19)βk(x,k)=1x,βk(k,y)=1y.
Note that when k→1, βk(x,y)→β(x,y).For more details about the theory ofk-special functions like k-gamma function, k-beta function, k-hypergeometric functions, solutions of k-hypergeometric differential equations, contegious functions relations, inequalities with applications and integral representations with applications involving k-gamma and k-beta functions and so forth. (See [12–17].)


### 1.7. Probability Distribution and Expected Values

In a random experiment withn outcomes, suppose a variable X assumes the values x1,x2,x3,…,xn with corresponding probabilities P1,P2,P3,…,Pn; then this collection is called probability distribution and Σpi=1 (in case of discrete distributions). Also, if f(x) is a continuous probability distribution function defined on an interval [a,b], then ∫ab‍f(x)dx=1.In statistics, there are three types of moments which are (i) moments about any pointx=a, (ii) moments about x=0, and (iii) moments about mean position of the given data. Also, expected value of the variate is defined as the first moment of the probability distribution about x=0 and the rth moment about mean of the probability distribution is defined as E(xi-x¯)r where x¯ is the mean of the distribution.Also,E(x) shows the expected value of the variate x and is defined as the first moment of the probability distribution about x=0; that is,
(20)μ1′=E(x)=∫ab‍xf(x)dx.


### 1.8. Gamma Distribution

A continuous random variableZ is said to have a gamma distribution with parameter m>0, if its probability distribution function is defined by
(21)f(z)={1Γ(m)zm-1e-z,0≦z<∞,0,elsewhere
and its distribution function F(z) is defined by
(22)F(z)={∫0z‍1Γ(m)zm-1e-zdz,z≥0,0,z<0,
which is also called the incomplete gamma function.


### 1.9. Moment Generating Function of Gamma Distribution

The moment generating function ofZ is defined by
(23)M0(t)=E(etZ)=∫0∞‍etZf(z)dz=∫0∞‍1Γ(m)zm-1e-z(1-t)dz.


### 1.10. Beta Distribution of the First Kind

A continuous random variableZ is said to have a beta distribution with two parameters m and n, if its probability distribution function is defined by
(24)f(z)={1B(m,n)zm-1(1-z)n-1,0≦z≦1;m,n>00,elsewhere.
This distribution is known as a beta distribution of the first kind and a beta variable of the first kind is referred to as β1(m,n). Its distribution function F(z) is given by
(25)F(z)={0,z<0,∫0z‍1B(m,n)zm-1(1-z)n-1dz,0≦z≦1;m,n>0,0,z>1.


### 1.11. Beta Distribution of the Second Kind

A continuous random variableZ is said to have a beta distribution of the second kind with parameters m and n, if its probability distribution function is defined by
(26)f(z)={1β(m,n)zm-1(1+z)m+n,0≦z<∞;m,n>0,0,otherwise
and its probability distribution function is given by
(27)F(z)=∫0∞‍1β(m,n)zm-1(1+z)m+ndz,0≦z<∞;m,n>0.


## 1.1. Pochhmmer Symbol

The factorial function is denoted and defined by(1)(a)n={a(a+1)(a+2)⋯(a+n-1);forn≥1,a≠0,1;ifn=0.
The function (a)n defined in relation (1) is also known as Pochhmmer symbol.


## 1.2. Gamma Function

Letz∈C; the Euler gamma function is defined by
(2)Γ(z)=lim⁡n→∞n!nz-1(z)n
and the integral form of gamma function is given by
(3)Γ(z)=∫0∞‍tz-1e-tdt,R(z)>0.
From the relation (3), using integration by parts, we can easily show that
(4)Γ(z+1)=zΓ(z).
The relation between Pochhammer symbol and gamma function is given by
(5)(z)n=Γ(z+n)Γ(z).


## 1.3. Beta Function

The beta function of two variables is defined as(6)B(x,y)=∫01‍tx-1(1-t)y-1dt,Re(x),Re(y)>0
and, in terms of gamma function, it is written as
(7)B(x,y)=Γ(x)Γ(y)Γ(x+y).


## 1.4. Pochhammerk-Symbol

Fork>0, the Pochhammer k-symbol is denoted and defined by
(8)(a)n,k={a(a+k)(a+2k)⋯(a+(n-1)k);forn≥1,a≠0,1;ifn=0.


## 1.5.k-Gamma Function

Fork>0 and z∈C, the k-gamma function is defined as
(9)Γk(z)=lim⁡n→∞n!kn(nk)z/k-1(z)n,k
and the integral representation of k-gamma function is
(10)Γk(z)=∫0∞‍tz-1e-tk/kdt.


## 1.6.k-Beta Function

ForRe(x),Re(y)>0, the k-beta function of two variables is defined by
(11)Bk(x,y)=1k∫0∞‍tx/k-1(1-t)y/k-1dt
and, in terms of k-gamma function, k-beta function is defined as
(12)Bk(x,y)=Γk(x)Γk(y)Γk(x+y).Also, the researchers [6–10] have worked on the generalized k-gamma and k-beta functions and discussed the following properties:
(13)Γk(x+k)=xΓk(x),(14)(x)n,k=Γk(x+nk)Γk(x),(15)Γk(k)=1,k>0.
Using the above relations, we see that, for x,y>0 and k>0, the following properties of k-beta function are satisfied by authors (see [6, 7, 11]):
(16)βk(x+k,y)=xx+yβk(x,y),(17)βk(x,y+k)=yx+yβk(x,y),(18)βk(xk,yk)=1kβ(x,y),(19)βk(x,k)=1x,βk(k,y)=1y.
Note that when k→1, βk(x,y)→β(x,y).For more details about the theory ofk-special functions like k-gamma function, k-beta function, k-hypergeometric functions, solutions of k-hypergeometric differential equations, contegious functions relations, inequalities with applications and integral representations with applications involving k-gamma and k-beta functions and so forth. (See [12–17].)


## 1.7. Probability Distribution and Expected Values

In a random experiment withn outcomes, suppose a variable X assumes the values x1,x2,x3,…,xn with corresponding probabilities P1,P2,P3,…,Pn; then this collection is called probability distribution and Σpi=1 (in case of discrete distributions). Also, if f(x) is a continuous probability distribution function defined on an interval [a,b], then ∫ab‍f(x)dx=1.In statistics, there are three types of moments which are (i) moments about any pointx=a, (ii) moments about x=0, and (iii) moments about mean position of the given data. Also, expected value of the variate is defined as the first moment of the probability distribution about x=0 and the rth moment about mean of the probability distribution is defined as E(xi-x¯)r where x¯ is the mean of the distribution.Also,E(x) shows the expected value of the variate x and is defined as the first moment of the probability distribution about x=0; that is,
(20)μ1′=E(x)=∫ab‍xf(x)dx.


## 1.8. Gamma Distribution

A continuous random variableZ is said to have a gamma distribution with parameter m>0, if its probability distribution function is defined by
(21)f(z)={1Γ(m)zm-1e-z,0≦z<∞,0,elsewhere
and its distribution function F(z) is defined by
(22)F(z)={∫0z‍1Γ(m)zm-1e-zdz,z≥0,0,z<0,
which is also called the incomplete gamma function.


## 1.9. Moment Generating Function of Gamma Distribution

The moment generating function ofZ is defined by
(23)M0(t)=E(etZ)=∫0∞‍etZf(z)dz=∫0∞‍1Γ(m)zm-1e-z(1-t)dz.


## 1.10. Beta Distribution of the First Kind

A continuous random variableZ is said to have a beta distribution with two parameters m and n, if its probability distribution function is defined by
(24)f(z)={1B(m,n)zm-1(1-z)n-1,0≦z≦1;m,n>00,elsewhere.
This distribution is known as a beta distribution of the first kind and a beta variable of the first kind is referred to as β1(m,n). Its distribution function F(z) is given by
(25)F(z)={0,z<0,∫0z‍1B(m,n)zm-1(1-z)n-1dz,0≦z≦1;m,n>0,0,z>1.


## 1.11. Beta Distribution of the Second Kind

A continuous random variableZ is said to have a beta distribution of the second kind with parameters m and n, if its probability distribution function is defined by
(26)f(z)={1β(m,n)zm-1(1+z)m+n,0≦z<∞;m,n>0,0,otherwise
and its probability distribution function is given by
(27)F(z)=∫0∞‍1β(m,n)zm-1(1+z)m+ndz,0≦z<∞;m,n>0.


## 2. Main Results:k-Gamma and k-Beta Distributions

In this section, we define gamma and beta distributions in terms of a new parameterk>0 and discuss some properties of these distributions in terms of k.Definition 1.
LetZ be a continuous random variable; then it is said to have a k-gamma distribution with parameters m>0 and k>0, if its probability density function is defined by
(28)fk(z)={1Γk(m)zm-1e-zk/k,0≦z<∞,k>0,0,elsewhere
and its distribution function Fk(z) is defined by
(29)Fk(z)={∫0z‍1Γk(m)zm-1e-zk/kdz,z>0,0,z<0.Proposition 2.
The newly definedΓk(m) distribution satisfies the following properties.(i)
Thek-gamma distribution is the probability distribution that is area under the curve is unity.(ii)
The mean ofk-gamma distribution is equal to a parameter m.(iii)
The variance ofk-gamma distribution is equal to the product of two parameters mk.Proof of (i).
Using the definition ofk-gamma distribution along with the relation (10), we have
(30)∫0∞‍fk(z)dz=1Γk(m)∫0∞‍zm-1e-zk/kdz=Γk(m)Γk(m)=1.Proof of (ii).
As mean of a distribution is the expected value of the variate, so the mean of thek-gamma distribution is given by
(31)z¯=Ek(Z)=1Γk(m)∫0∞‍z·zm-1e-zk/kdz.
Using the definition of k-gamma function and the relation (13), we have
(32)z¯=1Γk(m)∫0∞‍zme-zk/kdz=Γk(m+k)Γk(m)=mΓk(m)Γk(m)=m.Proof of (iii).
As variance of a distribution is equal toE(x2)-(E(x))2, so the variance of k-gamma distribution is calculated as
(33)Var⁡k(Z)=Ek(Z2)-(Ek(Z))2.
Now, we have to find Ek(Z2), which is given by
(34)Ek(Z2)=1Γk(m)∫0∞‍z2·zm-1e-zk/kdz=1Γk(m)∫0∞‍zm+1e-zk/kdz=Γk(m+2k)Γk(m)=(m+k)mΓk(m)Γk(m)=m(m+k).
Thus we obtain the variance of k-gamma distribution as
(35)σk2=m(m+k)-m2=mk,
where σk2 is the notation of variance present in the literature.


### 2.1.k-Beta Distribution of First Kind

LetZ be a continuous random variable; then it is said to have a k-beta distribution of the first kind with two parameters m and n, if its probability distribution function is defined by
(36)fk(z)={1kBk(m,n)zm/k-1(1-z)n/k-1,0≦z≦1;m,n,k>0,0,elsewhere.
In the above distribution, the beta variable of the first kind is referred to as β1,k(m,n) and its distribution function Fk(z) is given by
(37)Fk(z)={0,z<0,∫0z‍1kBk(m,n)zm/k-1(1-z)n/k-1dz,0≦z≦1;m,n>0,0,z>1.Proposition 3.
Thek-beta distribution β1,k(m,n) satisfies the following basic properties.(i)
k-beta distribution is the probability distribution that is the area of β1,k(m,n) under a curve fk(z) is unity.(ii)
The mean of this distribution ism/(m+n).(iii)
The variance ofβ1,k(m,n) is mnk/((m+n)2(m+n+k)).Proof of (i).
By using the above definition ofk-beta distribution, we have
(38)∫0z‍Fk(z)dz=∫0z‍1kBk(m,n)zm/k-1(1-z)n/k-1dz,hhhhhhhhhhhh0≦z≦1;m,n>0.
By the relation (11), we get
(39)∫0z‍Fk(z)dz=∫01‍1kBk(m,n)zm/k-1(1-z)n/k-1dz=Bk(m,n)Bk(m,n)=1.Proof of (ii).
The mean of the distribution,μ1,k′, is given by
(40)μ1,k′=Ek(Z)=∫0z‍zFk(z)dz=∫0z‍1kBk(m,n)z·zm/k-1(1-z)n/k-1dz,hhhhhhhhhhhhhh0≦z≦1;m,n>0.
Using the relations (12), (13), and (16), we have
(41)μ1,k′=∫01‍1kBk(m,n)zm/k(1-z)n/k-1dz=Bk(m+k,n)Bk(m,n)=Γk(m+k)Γk(n)Γk(m+n)Γk(m)Γk(n)Γk(m+n+k)=mm+n.Proof of (iii).
The variance ofβ1,k(m,n) is given by
(42)σk2=(Var⁡)k=Ek(Z2)-(Ek(Z))2,(43)Ek(Z2)=∫01‍1kBk(m,n)zm/k+1(1-z)n/k-1dz=Bk(m+2k,n)Bk(m,n)=Γk(m+2k)Γk(n)Γk(m+n)Γk(m)Γk(n)Γk(m+n+2k)=m(m+k)(m+n)(m+n+k).
Thus substituting the values of Ek(Z2) and Ek(Z) in (42) along with some algebraic calculations we have the desired result.


### 2.2.k-Beta Distribution of the Second Kind

A continuous random variableZ is said to have a k-beta distribution of the second kind with parameters m and n, if its probability distribution function is defined by
(44)fk(z)={1kβk(m,n)zm/k-1(1+z)(m+n)/k,0≦z<∞;m,n,k>0,0,otherwise.Note. The k-beta distribution of the second kind is denoted by β2,k(m,n).Theorem 4.
Thek-beta function of the second kind represents a probability distribution function that is
(45)∫0∞‍fk(z)dz=1.Proof.
We observe that(46)∫0∞‍fk(z)dz=∫0∞‍1kβk(m,n)zm/k-1(1+z)(m+n)/kdz.
Let 1+z=1/y, so that dz=-dy/y2; thus by using the relation (11), the above equation gives
(47)=1βk(m,n)1k∫01‍yn/k-1(1-y)m/k-1dy=βk(m,n)βk(m,n)=1.


## 2.1.k-Beta Distribution of First Kind

LetZ be a continuous random variable; then it is said to have a k-beta distribution of the first kind with two parameters m and n, if its probability distribution function is defined by
(36)fk(z)={1kBk(m,n)zm/k-1(1-z)n/k-1,0≦z≦1;m,n,k>0,0,elsewhere.
In the above distribution, the beta variable of the first kind is referred to as β1,k(m,n) and its distribution function Fk(z) is given by
(37)Fk(z)={0,z<0,∫0z‍1kBk(m,n)zm/k-1(1-z)n/k-1dz,0≦z≦1;m,n>0,0,z>1.Proposition 3.
Thek-beta distribution β1,k(m,n) satisfies the following basic properties.(i)
k-beta distribution is the probability distribution that is the area of β1,k(m,n) under a curve fk(z) is unity.(ii)
The mean of this distribution ism/(m+n).(iii)
The variance ofβ1,k(m,n) is mnk/((m+n)2(m+n+k)).Proof of (i).
By using the above definition ofk-beta distribution, we have
(38)∫0z‍Fk(z)dz=∫0z‍1kBk(m,n)zm/k-1(1-z)n/k-1dz,hhhhhhhhhhhh0≦z≦1;m,n>0.
By the relation (11), we get
(39)∫0z‍Fk(z)dz=∫01‍1kBk(m,n)zm/k-1(1-z)n/k-1dz=Bk(m,n)Bk(m,n)=1.Proof of (ii).
The mean of the distribution,μ1,k′, is given by
(40)μ1,k′=Ek(Z)=∫0z‍zFk(z)dz=∫0z‍1kBk(m,n)z·zm/k-1(1-z)n/k-1dz,hhhhhhhhhhhhhh0≦z≦1;m,n>0.
Using the relations (12), (13), and (16), we have
(41)μ1,k′=∫01‍1kBk(m,n)zm/k(1-z)n/k-1dz=Bk(m+k,n)Bk(m,n)=Γk(m+k)Γk(n)Γk(m+n)Γk(m)Γk(n)Γk(m+n+k)=mm+n.Proof of (iii).
The variance ofβ1,k(m,n) is given by
(42)σk2=(Var⁡)k=Ek(Z2)-(Ek(Z))2,(43)Ek(Z2)=∫01‍1kBk(m,n)zm/k+1(1-z)n/k-1dz=Bk(m+2k,n)Bk(m,n)=Γk(m+2k)Γk(n)Γk(m+n)Γk(m)Γk(n)Γk(m+n+2k)=m(m+k)(m+n)(m+n+k).
Thus substituting the values of Ek(Z2) and Ek(Z) in (42) along with some algebraic calculations we have the desired result.


## 2.2.k-Beta Distribution of the Second Kind

A continuous random variableZ is said to have a k-beta distribution of the second kind with parameters m and n, if its probability distribution function is defined by
(44)fk(z)={1kβk(m,n)zm/k-1(1+z)(m+n)/k,0≦z<∞;m,n,k>0,0,otherwise.Note. The k-beta distribution of the second kind is denoted by β2,k(m,n).Theorem 4.
Thek-beta function of the second kind represents a probability distribution function that is
(45)∫0∞‍fk(z)dz=1.Proof.
We observe that(46)∫0∞‍fk(z)dz=∫0∞‍1kβk(m,n)zm/k-1(1+z)(m+n)/kdz.
Let 1+z=1/y, so that dz=-dy/y2; thus by using the relation (11), the above equation gives
(47)=1βk(m,n)1k∫01‍yn/k-1(1-y)m/k-1dy=βk(m,n)βk(m,n)=1.


## 3. Moment Generating Function ofk-Gamma Distribution

In this section, we derive the moment generating function of continuous random variableZ of newly defined k-gamma distribution in terms of a new parameter k>0, which is illustrated as
(48)M0,k(t)=Ek(etZk)=∫0∞‍1Γk(m)etzkzm-1e-zk/kdz=1Γk(m)∫0∞‍zm-1e(-zk/k)(1-kt)dz.
Let u=z(1-kt)1/k, so that z=u/(1-kt)1/k and dz=du/(1-kt)1/k. Then substituting these values in (48), we obtain
(49)M0,k(t)=1(1-kt)(m-1)/kΓk(m)∫0∞‍um-1e-uk/kdu(1-kt)1/k=1(1-kt)m/kΓk(m)∫0∞‍um-1e-uk/kdu=Γk(m)(1-kt)m/kΓk(m)=(1-kt)-m/k,|kt|<1.
Now differentiating r times M0,k(t) with respect to t and putting t=0, we get
(50)μr,k′=m(m+k)(m+2k)⋯(m+(r-1)k).
Thus when r=1, we obtain μ1,k′=m, when r=2, μ2,k′=m(m+k), and hence μ2,k=μ1,k′2-μ2,k′=mk = variance of the k-gamma distribution proved in Proposition 2.


### 3.1. Higher Moment in terms ofk

Therth moment in terms of k is given by
(51)μr,k′=E(Zr)=1kBk(m,n)∫01‍zr·zm/k-1(1-z)n/k-1dz=1kBk(m,n)∫01‍zm/k+r-1(1-z)n/k-1dz=Bk(m+rk,n)Bk(m,n)=Γk(m+rk)Γk(m+n)Γk(m)Γk(m+rk+n)=m(m+k)(m+2k)⋯(m+(r-1)k)(m+n)(m+n+k)(m+n+2k)⋯(m+n+(r-1)k).Theorem 5.
The moments of the higher order ofk-beta distribution of the second kind are given as
(52)μr,k′=m(m+k)(m+2k)⋯(m+(r-1)k)(n-k)(n-2k)⋯(n-rk).Proof.
Consider(53)μr,k′=E(Zr)=∫0∞‍1kβk(m,n)zm/k-1+r(1+z)(m+n)/kdz.
Changing the variables as z=(1-y)/y⇒dz=(-1/y2)dy, above equation becomes
(54)=1kβk(m,n)∫01‍yn/k-r-1(1-y)m/k+r-1dy.
Replacing (1-y) by t, we have
(55)μr,k′=1βk(m,n)1k∫01‍tm/k+r-1(1-t)n/k-r-1dt=βk(m+rk,n-rk)βk(m,n)=Γk(m+rk)Γk(n-rk)Γk(m+n)Γk(m)Γk(n)Γk(m+n)=Γk(m+rk)Γk(n-rk)Γk(m)Γk(n).
Now using Γk(n-rk)=Γk(n)/(n-k)(n-2k)⋯(n-rk) in the above equation we get the desired result.


## 3.1. Higher Moment in terms ofk

Therth moment in terms of k is given by
(51)μr,k′=E(Zr)=1kBk(m,n)∫01‍zr·zm/k-1(1-z)n/k-1dz=1kBk(m,n)∫01‍zm/k+r-1(1-z)n/k-1dz=Bk(m+rk,n)Bk(m,n)=Γk(m+rk)Γk(m+n)Γk(m)Γk(m+rk+n)=m(m+k)(m+2k)⋯(m+(r-1)k)(m+n)(m+n+k)(m+n+2k)⋯(m+n+(r-1)k).Theorem 5.
The moments of the higher order ofk-beta distribution of the second kind are given as
(52)μr,k′=m(m+k)(m+2k)⋯(m+(r-1)k)(n-k)(n-2k)⋯(n-rk).Proof.
Consider(53)μr,k′=E(Zr)=∫0∞‍1kβk(m,n)zm/k-1+r(1+z)(m+n)/kdz.
Changing the variables as z=(1-y)/y⇒dz=(-1/y2)dy, above equation becomes
(54)=1kβk(m,n)∫01‍yn/k-r-1(1-y)m/k+r-1dy.
Replacing (1-y) by t, we have
(55)μr,k′=1βk(m,n)1k∫01‍tm/k+r-1(1-t)n/k-r-1dt=βk(m+rk,n-rk)βk(m,n)=Γk(m+rk)Γk(n-rk)Γk(m+n)Γk(m)Γk(n)Γk(m+n)=Γk(m+rk)Γk(n-rk)Γk(m)Γk(n).
Now using Γk(n-rk)=Γk(n)/(n-k)(n-2k)⋯(n-rk) in the above equation we get the desired result.


## 4. Conclusion

In this paper the authors conclude that we have the following.(i)
Ifk tends to 1, then k-gamma distribution and k-beta distribution tend to classical gamma and beta distribution.(ii)
The authors also conclude that the area ofk-gamma distribution and k-beta distribution for each positive value of k is one and their mean is equal to a parameter m and m/(m+n), respectively. The variance of k-gamma distribution for each positive value of k is equal to k times of the parameter m. In this case if k=1, then it will be equal to variance of gamma distribution. The variance of k-beta distribution for each positive value of k is also defined.(iii)
In this paper the authors introduced moments generating function and higher moments in terms of a new parameterk>0.



---

*Source: 982013-2014-07-15.xml*