# Note on Extensions of The Beta function

Mehar Chand  Department of Applied Sciences, Guru Kashi University,
Bathinda-1513002 (India<EMAIL_ADDRESS>
###### Abstract.

The classical beta function $B(x,y)$ is one of the most fundamental special
functions, due to its important role in various fields in the mathematical,
physical, engineering and statistical sciences. Useful extensions of the
classical Beta function has been considered by many authors. In the present
paper, our main objective is to study the convergence of extensions of
classical beta function and introduce modified extension of classical beta
function. It is interpreted numerically and geometrically in the view of
convergence, further properties and integral presentations are established.

###### Key words and phrases:

Pochhemmer symbol; gamma function; Beta function; hypergeometric function

###### 2010 Mathematics Subject Classification:

26A33, 33C45, 33C60, 33C70

##  1\. Introduction and Preliminaries

In many areas of applied mathematics, various types of special functions
become essential tools for scientists and engineers. The (Euler’s) classical
beta function $B(x,y)$ is one of the most fundamental special functions,
because of its important role in various fields in the mathematical, physical,
engineering and statistical sciences. During last four decades or so, several
interesting and useful extensions of the familiar special functions (such as
the Gamma and Beta functions, the Gauss hypergeometric function, and so on)
have been considered by many authors. In the present work, we are concerned
with various generalizations of the classical beta function, which can be
found in the literature (see [1, 2, 3, 4]).

Recently the generalized beta function
$B_{p}^{\delta,\zeta;\kappa,\mu}(\alpha,\beta)$ is introduced by Srivastava et
al., which is the most generalized extension of classical beta function and is
defined as(see[1]):

$\displaystyle\begin{aligned}
&B_{p}^{(\delta,\zeta;\kappa,\mu)}(\alpha,\beta)\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}{}_{1}F_{1}\left(\delta;\zeta;-\frac{p}{t^{\kappa}(1-t)^{\mu}}\right)dt,\end{aligned}$
(1.1)

where $\Re(p)\geq
0;\min[\Re(\alpha),\Re(\beta),\Re(\delta),\Re(\zeta)]>0;\min[\Re(\kappa),\Re(\mu)]>0$
and ${}_{1}F_{1}(.)$ is confluent hypergeometric function, which is special
case of the well known generalized hypergeometric series ${}_{p}F_{q}(.)$.

The generalized hypergeometric series ${}_{p}F_{q}(p,q\in\mathbb{N})$ is
defined as (see [5, p.73]) and [6, pp. 71-75]:

$\displaystyle\begin{aligned}
&{}_{p}F_{q}\left[\begin{array}[]{cc}\alpha_{1},\ldots,\alpha_{p};\\\
\beta_{1},\ldots,\beta_{q};\end{array}z\right]=\sum_{n=0}^{\infty}\,\frac{(\alpha_{1})_{n}\cdots(\alpha_{p})_{n}}{(\beta_{1})_{n}\cdots(\beta_{q})_{n}}\frac{z^{n}}{n!}\\\
&\hskip
11.38109pt={}_{p}F_{q}(\alpha_{1},\,\ldots,\,\alpha_{p};\,\beta_{1},\,\ldots,\,\beta_{q};\,z),\end{aligned}$
(1.2)

where $(\lambda)_{n}$ is the Pochhammer symbol defined (for
$\lambda\in\mathbb{C}$) by (see[6, p.2 and p.5]):

$\displaystyle\begin{aligned}
&(\lambda)_{n}:=\left\\{\begin{array}[]{cc}1\hskip 108.12047pt(n=0)\\\
\lambda(\lambda+1)\ldots(\lambda+n-1)\quad(n\in{\mathbb{N}})\end{array}\right.\end{aligned}$
(1.3) $\displaystyle\begin{aligned}
&=\frac{\Gamma(\lambda+n)}{\Gamma(\lambda)}\quad(\lambda\in{\mathbb{C}}\setminus{\mathbb{Z}}_{0}^{-}),\end{aligned}$
(1.4)

and ${\mathbb{Z}}_{0}^{-}$ denotes the set of Non-positive integers and
$\Gamma(\lambda)$ is familiar Gamma function.

When $\kappa=\mu$, (1.1) reduces to the generalized extended beta function
$B_{p}^{(\delta,\zeta;\mu)}(\alpha,\beta)$ defined by (see [2, p. 37])

$\displaystyle\begin{aligned} &B_{p}^{(\delta,\zeta;\mu)}(\alpha,\beta)\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}{}_{1}F_{1}\left(\delta;\zeta;-\frac{p}{t^{\mu}(1-t)^{\mu}}\right)dt,\end{aligned}$
(1.5)

where $\Re(p)\geq
0;\min[\Re(\alpha),\Re(\beta),\Re(\delta),\Re(\zeta)]>0;\Re(\mu)>0$.

The special case of (1.5), when $\mu=1$ reduces to the generalized beta type
function as follows (see[3, p.4602])

$\displaystyle\begin{aligned}
&B_{p}^{(\delta,\zeta)}(\alpha,\beta)=B_{p}^{(\delta,\zeta;1)}(\alpha,\beta)\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}{}_{1}F_{1}\left(\delta;\zeta;-\frac{p}{t(1-t)}\right)dt,\end{aligned}$
(1.6)

where $\Re(p)\geq 0;\min[\Re(\alpha),\Re(\beta),\Re(\delta),\Re(\zeta)]>0$.

The further special case of (1.6) when $\delta=\zeta$ is given due to
Choudhary et. al.[4] by

$\displaystyle\begin{aligned}
&B_{p}(\alpha,\beta)=B(\alpha,\beta;p)=B_{p}^{(\delta,\delta)}(\alpha,\beta)\\\
&\hskip
5.69054pt=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp\left(-\frac{p}{t(1-t)}\right)dt,\\\
&\hskip 99.58464pt(\Re(p)\geq 0).\end{aligned}$ (1.7)

When we choose $p=0$, all the above extensions reduces to the classical beta
function $B(x,y)$, which is defined by

$\displaystyle\begin{aligned}
&B(\alpha,\beta)=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}dt,\\\ &\hskip
85.35826pt(\Re(\alpha)>0,\Re(\beta)>0).\end{aligned}$ (1.8)

It is clear to see the following relationship between the classical beta
function $B(\alpha,\beta)$ and its extensions:

$\displaystyle\begin{aligned}
&B(\alpha,\beta)=B_{0}(\alpha,\beta)=B_{0}^{(\delta,\zeta)}(\alpha,\beta)\\\
&\hskip
56.9055pt=B_{0}^{(\delta,\zeta;1)}(\alpha,\beta)=B_{0}^{(\delta,\zeta;1,1)}(\alpha,\beta).\end{aligned}$
(1.9)

In particular, Euler’s beta function $B(x,y)$ has a close relationship to his
gamma function,

$\displaystyle\begin{aligned}
&B(\alpha,\beta)=B(\beta,\alpha)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}.\end{aligned}$
(1.10)

##  2\. Extensions of Classical Beta Function do not convergent

We claim that the above extensions of classical beta function in equations
(1.1), (1.5), (1.6), and (1.7) are not convergent. We prove it mathematically
as well as numerically tested in Section 4.

First we prove that the extension in equation (1.7) is not convergent as
follows.

######  Claim 1.

If $\Re(\alpha)>0,\Re(\beta)>0,\Re(p)>0$, then the extensions of classical
beta function in equation (1.7) does not convergent.

###### Proof.

To prove our claim, we write the exponential function in series form, the
equation (1.7) reduces to the following form

$\displaystyle\begin{aligned}
&B(\alpha,\beta;p)=\sum_{n=0}^{\infty}\frac{(-p)^{n}}{n!}\int_{0}^{\infty}t^{\alpha-n-1}(1-t)^{\beta-n-1}dt,\end{aligned}$
(2.1)

further using the definition of the beta function in the above equation (2.1),
we have

$\displaystyle\begin{aligned}
&B(\alpha,\beta;p)=\sum_{n=0}^{\infty}\frac{(-p)^{n}}{n!}B(\alpha-n,\beta-n)\\\
&=B(\alpha,\beta)+\frac{-p}{1!}B(\alpha-1,\beta-1)+...\\\
&+\frac{(-p)^{n}}{n!}B(\alpha-n,\beta-n)+...\rm{to~{}infinity~{}terms}.\end{aligned}$
(2.2)

In the above equation (2.2), $B(\alpha,\beta;p)$ is an power series involving
$B(\alpha-n,\beta-n)$ (where $n=0,1,2,...$). The series $B(\alpha,\beta;p)$
will be convergent if $B(\alpha-n,\beta-n)$ is convergent and
$B(\alpha-n,\beta-n)$ will be convergent only if $\Re(\alpha-n)>0$ and
$\Re(\beta-n)>0$. But this is not possible as $\alpha$ and $\beta$ are finite
and $n\rightarrow\infty$. The series $B(\alpha,\beta;p)$ contains many terms,
which does not exist. Moreover $B(\alpha,\beta;p)$ is the sum of terms
involving $B(\alpha-n,\beta-n)$, which are not convergent, which implies that
$B(\alpha,\beta;p)$ is not convergent. ∎

######  Example 1.

If we choose $\alpha=5,\beta=7,p=3$, then from equation (2.2), we have

$\displaystyle\begin{aligned}
&B(5,7;3)=\sum_{n=0}^{\infty}\frac{(-3)^{n}}{n!}B(5-n,7-n),\end{aligned}$
(2.3)

The above series $B(5,7;3)$ is the sum of terms involving the beta function
$B(5-n,7-n)$ and $B(5-n,7-n)$ does not exit for $n>5$. Therefor $B(5,7;3)$ is
not convergent.

In the following section, we introduce modified extension of classical beta
function in equation (3.1). Further, the extension of classical beta function
(1.7), Modified extension of classical beta function (3.1) and the classical
beta function are compared and tested numerically to test the convergence of
extensions of the classical beta function.

##  3\. Modified extension of beta function

In this section, we introduce modified extension of classical beta function
(MECBF). Its convergence is proved mathematically, then numerical results are
established and compared the results with that of the classical beta function
and extension of classical beta function.

We introduce modified extension of classical beta function as follows

$\displaystyle\begin{aligned} &B_{m}(\alpha,\beta)=B(\alpha,\beta;m)\\\
&\hskip
28.45274pt=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}e^{mt(1-t)}dt,\end{aligned}$
(3.1)

where $\Re(\alpha)>0,\Re(\beta)>0,m\in\mathbb{C};|m|<M$ (where $M$ is positive
finite real number).

In our investigation to test the convergence of the above extension of the
classical beta function the following definitions are required.

######  Definition 1 (Ratio Test[9]).

Let $\sum_{n=1}^{\infty}p_{n}$ be a series of positive terms, and suppose that
$\displaystyle\frac{p_{n+1}}{p_{n}}\rightarrow{\rm a\,\,limit\,\,}l$ as
$n\rightarrow\infty$. Then (i) If $l<1$ the series $\sum p_{n}$ is convergent,
(ii) If $l>1$, then the series is divergent. If $l=1$, the ratio test fails,
and the question of convergence of $\sum p_{n}$ must be investigated by some
other methods.

######  Definition 2 (Leibniz’s Test or Alternating Series Test[9]).

The series $\sum_{n=1}^{\infty}(-1)^{n+1}a_{n}$ (where $a_{n}>0$) is
convergent provided (i) $a_{n}$ is a decreasing sequence, (ii)
$a_{n}\rightarrow 0$ as $n\rightarrow\infty$.

######  Theorem 1.

If $\Re(\alpha)>0,\Re(\beta)>0,m\in\mathbb{C};|m|<M$ (where $M$ is positive
finite real number), then the modified extension of the classical beta
function in equation (3.1) is convergent.

###### Proof.

We can write the above equation (3.1) as follows

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\sum_{n=0}^{\infty}\frac{[mt(1-t)]^{n}}{n!}dt\\\
&\hskip
28.45274pt=\sum_{n=0}^{\infty}\frac{(m)^{n}}{n!}\int_{0}^{1}t^{\alpha+n-1}(1-t)^{\beta+n-1}dt,\end{aligned}$
(3.2)

further, using the definition of beta function, the above equation (3.2)
reduces to

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\sum_{n=0}^{\infty}\frac{(m)^{n}}{n!}B(\alpha+n,\beta+n).\end{aligned}$
(3.3)

In the above equation, $B(\alpha,\beta;m)$ is in series form involving
$B(\alpha+n,\beta+n)$ (where $n=0,1,2,...$) and in each term of the series,
$B(\alpha+n,\beta+n)$ is convergent since $\Re(\alpha+n)>0$ and
$\Re(\beta+n)>0$ for $\Re(\alpha),\Re(\beta)>0$, which implies that each term
of the series (3.3) exist.

Now we shall prove that $B(\alpha,\beta;m)$ is convergent. $m$ may be greater
than or less than $0$, so there are two cases as follows.

######  Case 1.

If $m>0$, then to prove $B(\alpha,\beta;m)$ is convergent.

The equation (3.3) can be written as

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\sum_{n=0}^{\infty}a_{n},\,\,\\\ &\hskip 56.9055pt{\rm
where}\,\,\displaystyle
a_{n}=\frac{(m)^{n}}{n!}B(\alpha+n,\beta+n).\end{aligned}$ (3.4)

Further

$\displaystyle\begin{aligned}
&\lim_{n\rightarrow\infty}\frac{a_{n}}{a_{n+1}}=\infty>1.\end{aligned}$ (3.5)

By ratio test, $B(\alpha,\beta;m)$ is convergent for $m>0.$

######  Case 2.

If $m<0$, then to prove that the extension of classical beta function
$B(\alpha,\beta;m)$ is convergent.

To prove this case, let $m=-p$ (where $p>0$), then equation (3.3)

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\sum_{n=0}^{\infty}\frac{(-p)^{n}}{n!}B(\alpha+n,\beta+n),\end{aligned}$
(3.6)

the above equation (3.6) can be written as

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=-\sum_{n=0}^{\infty}(-1)^{n-1}b_{n},\,\,\\\ &\hskip
56.9055pt{\rm where}\,\,\displaystyle
b_{n}=\frac{(p)^{n}}{n!}B(\alpha+n,\beta+n).\end{aligned}$ (3.7)

The series (3.7) is an alternating series, therefor

  1. (1)

$a_{n}>0$, $\forall p>0,\Re(\alpha,\beta)>0$

  2. (2)

$\displaystyle
a_{n}-a_{n+1}=\frac{p^{n}}{n!}B(\alpha+n,\beta+n)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\times\left[1-\frac{(\alpha+n)(\beta+n)}{(\alpha+\beta+2n)(\alpha+\beta+2n+1)}\right]>0$
$\Rightarrow a_{n}$ is decreasing

  3. (3)

$\displaystyle\lim_{n\rightarrow\infty}a_{n}=0$ if $p\leq 2$
($\displaystyle{p^{n}}/{n!}\rightarrow 0$ as $n\rightarrow\infty$ only if
$p\leq 2$ and $B(\alpha+n,\beta+n)\rightarrow 0$ as $n\rightarrow\infty$)

All the conditions of Leibniz’s test have been satisfied, therefore
$B(\alpha,\beta;m)$ is convergent for $0<p\leq 2\,\,\,i.e.\,-2\leq m<0$.

From the above two cases 1 and 2 implies that the power series in equation
(3.3) is convergent. ∎

##  4\. Comparison of numerical values of extended beta function, modified
extension of the classical beta function and classical beta function

In Table 1, the columns $\rm B1,B2$ are the values of the classical beta
function $B(x,y)$; the columns $\rm EB1,$ $\rm EB2,$ $\rm EB3,$ $\rm EB4$ are
the values of the extension of classical beta

Table 1. Comparison of extensions of classical beta function

Table 2. Numerical Values of MECBF

function; the columns $\rm MB1,$ $\rm MB2,$ $\rm MB3,$ $\rm MB4$ are values of
modified extension of the classical beta function, where $\rm B1=$ $\rm
B(x,y)$, $\rm EB1=$ $\rm B(x,y;0.01)$, $\rm MB1=B(x,y;0.01)$, $\rm B2=$ $\rm
B(x,0.25)$, $\rm EB2$ $\rm=B(x,0.25;0.00)$, $\rm MB2=$ $\rm B(x,0.25;0.00)$,
$\rm EB3=$ $\rm B(x,0.25;0.01)$, $\rm MB3=$ $\rm B(x,0.25;0.01)$, $\rm EB4=$
$\rm B(x,y;0.00)$; $\rm MB4=$ $\rm B(x,y;0.00)$ and $\rm x,y=0:1:10$.

These values are computed by employing Matlab (R2012a). The values of the
extension of the classical beta function are computed by using the equation
(2.2) and those of modified extension of the classical beta function from
equation (3.3). Both the extension in equations (2.2) and (3.3) are in series
form involving beta functions $B(x-n,y-n)$ (exist for $\Re(x-n>0,y-n)>0$) and
$B(x+n,y+n)$ (exist for $\Re(x+n>0,y+n)>0$) respectively, and the values can
be computed easily. Therefor the values of extension of the classical beta
function (2.2) are computed by taking sum of the first five terms of the
series in the view of existing values of $B(x-n,y-n)$ to find some numerical
values and in the case of modified extension of classical beta function, sum
of first 1000 terms have been taken for $x,y=0:1:10$. The numerical results of
the same are established in Table 1.

From Table 1, it is easily observed that the values of extended beta function
in columns $\rm EB1,EB4$ does not exist for $x,y=0:1:5$ (since
$\Re(x-n<0,y-n)<0$); rest values of the same exist for $x,y=6:1:10$ (since
$\Re(x-n)>0,\Re(y-n)<0$). In columns $\rm EB2,EB3$ the values of extended beta
function do not exist since in these cases we choose $y=0.25$ therefor
$\Re(y-n)$ becomes less than zero, which implies if $0<x,y<1$ the values of
the extended beta function do not exist even if sum of first two terms of the
series (2.2) to be taken. If we take large sum of the terms of the series
(2.2) (say $n=1000$) the extended beta function will not converge for any
value of $x,y\in[0,1000]$ from which we conclude that the extended beta
function does not exits for any value of $x$ and $y$, since it is series
having infinite terms and $\Re(x),\Re(y)$ can not be infinite.

In the case of modified extension of classical beta function, the columns $\rm
B1,MB4$ are identical, which implies that if we choose $m=0$, we have the
values of classical beta function from the modified extension of the classical
beta function. Column $\rm B2,MB2$ are also identical since $m=0$ and at the
fix value of $y=0.25$. If we compare the values of $\rm B1$ with $\rm MB1$ and
those of $\rm B2$ with $\rm MB3$, we can see the impact of the value of $m$ on
the classical beta function.

######  Note 1.

It is also noted that if $p=0$, the extension of classical beta function
should be reduced to the classical beta function $i.e.$ $B(\alpha,\beta;0)$
$=B(\alpha,\beta)$. But the same not depict in the graph cited in [4, Fig. 1,
p. 31]. The graph depicts the values of extension of the classical beta
function $B(1,2.25;0)=$ $B(1,0.25)=13.75$ when $p=0,x=1,$ $y=0.25$ and
$B(10,2.25;0)=$ $B(10,0.25)=2.18$(approximately) when $p=0,$ $x=10,$ $y=0.25$,
which are not the correct values of the classical beta function. The correct
values of the classical beta function at these point are $B(1,0.25)=4$ and
$B(10,0.25)$ $=2.0582$. It also can be easily observed from the figure that
when $p\neq 0$ the graph of extended beta function totally different from that
of the classical beta function. $i.e.$ the behavior and nature of the curves
differ from that of classical beta function.

######  Note 2.

From the above discussion, it is very clear that the extension of classical
beta function in equation (1.7) do not convergent. The extensions in equations
(1.1), (1.5) and (1.6) are further extension of the extended beta function
(1.7), therefor all these extensions also can not be convergent.

######  Note 3.

Due to the lack of the convergence of the extension of the classical beta
function, all the associated results established by Chaudhary $et\,\,al.$ [4]
do not convergent.

##  5\. Numerical results and their interpretation of modified extension of
the classical beta function

The numerical results of modified extension of classical beta function (MECBF)
have been calculated in this section by employing the Matlab. We choose the
values of variable $x,y$ and parameter $m$ as $x,y\in[0,10]$ and
$m\in[-2.0335,2.0335]$. All the numerical values of MECBF are presented in
Table 1, from which we can easily observe that $B(x,y;m)$ does not exist at
$x=y=0$ and it is also tested that $B(x,y;m)$ does not exist for $m<-2.0335$
and $m>2.0335$. It can be easily investigated that $B(x,y;m)\rightarrow\infty$
as $x,y\rightarrow 0$ and $B(x,y;m)\rightarrow 0$ as $x,y\rightarrow\infty$,
which implies that the behavior of modified extension of classical beta
function is the same as that of classical beta function, which can be observed
from Table 2 and Figure 1. Also $B(x,y;0)$ represents the values of classical
beta function $B(x,y)$.

Figure 1. Graphical presentation of of MECBF for y=1.

##  6\. Geometrical interpretation of modified extension of the classical beta
function

If $\beta=1,$ $m\in[-2,2]$ and $x\in[0,1]$, then we observe that graph of
$B(x,y;m)$ in Figure 1 is decreasing. In figure 2, we choose $m=2$ and
$x,y\in[0,25]$, which depicts that as $x,y\rightarrow 0$,
$B(x,y;m)\rightarrow\infty$ and $B(x,y;m)\rightarrow 0$ as
$\alpha,\beta\rightarrow\infty$. We also check the effect of $m$ on the
modified extension of classical beta function. For this purpose, we fix the
values of $x$ and $y$ as shown in figure 3, then we plot the graph which
depicts that $B(x,y;m)$ is an increasing function as the values of $m$
increase. It is very clear from Figure 1 that for the graph of classical beta
function and modified extension of classical beta function remains concave
upward (or convex downward) for different values of $\alpha,\beta$ and $m$.
The value of $m$ does not effect the nature of classical beta function, the
main effect of the value of $m$ is that it just push the curve up or drag down
the curve from the curve of the classical beta function as shown in the Figure
1 the same behavior can be observed from the Table 2.

Figure 2. 3D presentation of of MECBF.

From the above proof of radius of convergence of series and further more
numerically investigation of the power series in Table 1, we find that the
interval of convergence of the series is $(-2.0336,2.0336)$, which implies
that $B(x,y;m)$ is convergent for $|m|<M$ where $M$ is positive real number
slightly greater than 2.0335.

######  Note 4.

From the above discussion, it is easy to conclude that the value of $\Re(m)$
lies in the interval $[-2.0335,2.0335]$ $i.e.$ $-2.0335\leq\Re(m)\leq 2.0335$.

######  Note 5.

In the sequel of this paper, $|m|<M$ represents the circle of convergence and
$M$ is the radius of convergence of equation (3.1), where $M$ is sightly
grater than from $2.0335.$

Figure 3. Graph of MECBF for fixed value of $x,y$ and $m=-2:.5:2$.

##  7\. Integral representation of the MECBF function

The integral representation of the modified extended beta function is
important to check both the extension is natural and simple and for use later.
It is also important to investigate the relationship between the classical
beta function and the modified extension of the classical beta function. In
this connection we first provide a relationship between them. The following
integral formula is useful for further investigation [7]

$\displaystyle\begin{aligned}
&\int_{0}^{\infty}x^{m}\exp(-\beta{}x^{n})dx=\frac{\Gamma(\gamma)}{n\beta^{\gamma}},\hskip
5.69054pt\left(\gamma=\frac{m+1}{n}\right).\end{aligned}$ (7.1)

######  Theorem 2 (Relation between modified extension of the classical beta
function and the classical beta function).

If $\Re(\alpha+s)>0,$ $\Re(\beta+s)>0,$ $m\in\mathbb{C};$ $|m|<M$ (where $M$
is positive finite real number slightly greater than 2.0335), then we have the
following relation

$\displaystyle\begin{aligned}
&\int_{0}^{\infty}m^{s-1}B(\alpha,\beta;m)dm=(-1)^{s}\Gamma(s)B(\alpha+s,\beta+s).\end{aligned}$
(7.2)

###### Proof.

Multiplying both sides of equation (3.1) by $m^{s-1}$, then integrate w.r.t.
$m$ from $m=0$ to $m=\infty$, we have

$\displaystyle\begin{aligned} &\int_{0}^{\infty}m^{s-1}B(\alpha,\beta;m)dm\\\
&=\int_{0}^{\infty}m^{s-1}\left[\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp(mt(1-t))dt\right]dm,\end{aligned}$
(7.3)

interchanging the order of integration, the above equation (7.3), reduces to

$\displaystyle\begin{aligned} &\int_{0}^{\infty}m^{s-1}B(\alpha,\beta;m)dm\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\left[\int_{0}^{\infty}m^{s-1}\exp(mt(1-t))dm\right]dt,\end{aligned}$
(7.4)

further using the formula given in equation (7.1), after simplification the
above equation (7.4) reduces to

$\displaystyle\begin{aligned} &\int_{0}^{\infty}m^{s-1}B(\alpha,\beta;m)dm\\\
&=(-1)^{s}\Gamma(s)\int_{0}^{1}t^{\alpha+s-1}(1-t)^{\beta+s-1}dt,\end{aligned}$
(7.5)

using the definition of classical beta function, we have the required result.
∎

######  Remark 1.

By setting $s=1$, the result in equation (7.2) reduces to

$\displaystyle\begin{aligned}
&\int_{0}^{\infty}B(\alpha,\beta;m)dm=-B(\alpha+1,\beta+1),\\\ &\hskip
56.9055pt\Re(\alpha)>-1,\Re(\beta)>-1,\end{aligned}$ (7.6)

gives the interesting relation between classical beta function and modified
extended beta function.

######  Remark 2.

All the derivatives of the modified extension of classical beta function
(MECBF) with respect to the parameter $m$ can be expressed in terms of the
function as

$\displaystyle\begin{aligned}
&\frac{\partial^{n}}{\partial^{n}m}B(\alpha,\beta;m)=B(\alpha+n,\beta+n),\\\
&\hskip 56.9055pt\Re(\alpha+n)>0,\Re(\beta+n)>0.\end{aligned}$ (7.7)

######  Theorem 3 (Integral representations of the modified extension of the
classical beta function).

If $\Re(x)>0,$ $\Re(y)>0,$ $m\in\mathbb{C};$ $|m|<M$ (where $M$ is positive
finite real number slightly greater than 2.0335), then we have the following
relation

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=2\int_{0}^{\frac{\pi}{2}}\cos^{2\alpha-1}\theta\sin^{2\beta-1}\theta\\\
&\hskip
56.9055pt\times\exp(m\cos^{2}\theta\sin^{2}\theta)d\theta,\end{aligned}$ (7.8)

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\int_{0}^{\infty}\frac{u^{\alpha-1}}{(1+u)^{\alpha+\beta}}\exp\left({mu}/{(1+u)^{2}}\right)du,\end{aligned}$
(7.9)

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\frac{1}{2}\int_{0}^{\infty}\frac{u^{\alpha-1}+u^{\beta-1}}{(1+u)^{\alpha+\beta}}\\\
&\hskip 56.9055pt\times\exp\left({mu}/{(1+u)^{2}}\right)du,\end{aligned}$
(7.10)

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=(c-a)^{1-\alpha-\beta}\int_{a}^{c}(u-a)^{\alpha-1}(c-u)^{\beta-1}\\\
&\hskip
56.9055pt\times\exp\left(\frac{m(u-a)(c-u)}{(c-a)^{2}}\right)du,\end{aligned}$
(7.11)

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=2^{1-\alpha-\beta}\int_{-1}^{1}(1+t)^{\alpha-1}(1-t)^{\beta-1}\\\
&\hskip 56.9055pt\times\exp\left(m(1-t^{2})/4\right)du,\end{aligned}$ (7.12)

$\displaystyle\begin{aligned} &B(\alpha,\beta;m)=2^{1-\alpha-\beta}\\\
&\times\int_{-\infty}^{\infty}\exp\left[(\alpha-\beta)x+\frac{m}{4\cosh^{2}x}\right]\frac{dx}{(\cosh
x)^{\alpha+\beta}},\end{aligned}$ (7.13)

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=2^{2-\alpha-\beta}\int_{0}^{\infty}\cosh((\alpha-\beta)x)\\\
&\hskip 56.9055pt\times\exp\left[\frac{m}{4\cosh^{2}x}\right]\frac{dx}{(\cosh
x)^{\alpha+\beta}},\end{aligned}$ (7.14)

$\displaystyle\begin{aligned} &B(\alpha,\beta;m)=2^{1-\alpha-\beta}\\\
&\times\int_{-\infty}^{\infty}\exp\left[\frac{1}{2}(\alpha-\beta)x+\frac{m}{2\cosh
x}\right]\frac{dx}{(\cosh x/2)^{\alpha+\beta}},\end{aligned}$ (7.15)

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=2^{2-\alpha-\beta}\int_{0}^{\infty}\cosh((\alpha-\beta)x/2)\\\
&\hskip 56.9055pt\times\exp\left[\frac{m}{2\cosh x}\right]\frac{dx}{(\cosh
x/2)^{\alpha+\beta}}.\end{aligned}$ (7.16)

###### Proof.

The result (7.8) can be easily obtained by setting $t=\cos^{2}\theta$, to
prove (7.9) choose $t=u/(1+u)$, (7.10) can be easily obtained by applying the
symmetric property in equation (7.9) then adding new one and (7.9), the result
in equation (7.11) is obtained by taking $t=(u-a)/(c-a)$, setting $a=-1,c=1$
in equation (7.11) gives the result in equation (7.12) and to prove the result
in equation (7.13) put $u=\tanh x$ in (7.12). The results in equation (7.14),
(7.15) and (7.16) can be easily obtained from the result (7.13). ∎

######  Remark 3 (Useful inequalities).

If $\Re(\alpha)>0,$ $\Re(\beta)>0$, then we have the following inequality

$\displaystyle\begin{aligned} &|B(\alpha,\beta;m)|\leq
1.6626B(\alpha,\beta)\end{aligned}$ (7.17)

follows from the integral representation (7.9). Since the function
$\exp(mu/(1+u)^{2})$ attains its maximum value $1.6626$ at $u=1$ and
$m=2.0335$. The equality is verified with the help of numerical results by
using Matlab.

##  8\. Properties of the modified extension of the classical beta function

######  Theorem 4 (Functional Relation).

If $\Re(\alpha)>0,$ $\Re(\beta)>0,$ $m\in\mathbb{C};$ $|m|<M$ (where $M$ is
positive finite real number slightly greater than 2.0335), then we have the
following relation

$\displaystyle\begin{aligned}
&B(\alpha,\beta+1;m)+B(\alpha+1,\beta;m)=B(\alpha,\beta;m).\end{aligned}$
(8.1)

###### Proof.

LHS of the above equation (8.1) is equal to

$\displaystyle\int_{0}^{1}\\{t^{\alpha-1}(1-t)^{\beta}+t^{\alpha}(1-t)^{\beta-1}\\}e^{mt(1-t)}dt$
(8.2)

after simplification the above equation (8.2) reduced to

$\displaystyle\int_{0}^{1}\\{t^{\alpha-1}(1-t)^{\beta-1}\\}e^{mt(1-t)}dt=B(\alpha,\beta;m).$
(8.3)

∎

If we choose $m=0$, we get the usual relation for the beta function from
(8.1).

######  Theorem 5 (Symmetry).

If $\Re(\alpha)>0,$ $\Re(\beta)>0,$ $m\in\mathbb{C};$ $|m|<M$ (where $M$ is
positive finite real number slightly greater than 2.0335), then we have the
following relation

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=B(\beta,\alpha;m).\end{aligned}$ (8.4)

###### Proof.

From equation (3.3) we have

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\sum_{n=0}^{\infty}\frac{(m)^{n}}{n!}B(\alpha+n,\beta+n)\end{aligned}$
(8.5)

Since usual beta function is symmetric $i.e.$
$B(\alpha,\beta)=B(\beta,\alpha)$. Using this property in the right hand side
of the equation (8.5), then we have

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\sum_{n=0}^{\infty}\frac{(m)^{n}}{n!}B(\beta+n,\alpha+n)=B(\beta,\alpha;m).\end{aligned}$
(8.6)

∎

######  Theorem 6 (First Summation Relation).

If $\Re(\alpha)>0,$ $\Re(1-\beta)>0,$ $m\in\mathbb{C};$ $|m|<M$ (where $M$ is
positive finite real number slightly greater than 2.0335), then we have the
following relation

$\displaystyle\begin{aligned}
&B(\alpha,1-\beta;m)=\sum_{n=0}^{\infty}\frac{(\beta)_{n}}{n!}B(\alpha+n,1;m).\end{aligned}$
(8.7)

###### Proof.

The LHS of the equation (8.7) can be written as

$\displaystyle\begin{aligned}
&B(\alpha,1-\beta;m)=\int_{0}^{1}t^{\alpha-1}(1-t)^{-\beta}\exp(mt(1-t))dt,\end{aligned}$
(8.8)

using the binomial series expansion $\displaystyle(1-t)^{-\beta}=$
$\displaystyle\sum_{n=0}^{\infty}\frac{(\beta)_{n}}{n!}t^{n}$ in the above
equation (8.8) and then interchanging the order of summation and integration,
the above result (8.8) reduced to the following form

$\displaystyle\begin{aligned} &B(\alpha,1-\beta;m)\\\
&=\sum_{n=0}^{\infty}\frac{(\beta)_{n}}{n!}\int_{0}^{1}t^{\alpha+n-1}\exp(mt(1-t))dt\end{aligned}$
(8.9)

$\displaystyle\begin{aligned} &\Rightarrow
B(\alpha,1-\beta;m)=\sum_{n=0}^{\infty}\frac{(\beta)_{n}}{n!}B(\alpha+n,1;m).\end{aligned}$
(8.10)

∎

######  Theorem 7 (Second Summation Relation).

If $\Re(\alpha)>0,\Re(\beta)>0,m\in\mathbb{C};|m|<M$ (where $M$ is positive
finite real number slightly greater than 2.0335), then we have the following
relation

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\sum_{n=0}^{\infty}B(\alpha+n,\beta+1;m).\end{aligned}$
(8.11)

###### Proof.

The LHS of the equation (8.7) can be written as

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp(mt(1-t))dt,\end{aligned}$
(8.12)

using the binomial series expansion
$\displaystyle(1-t)^{\beta-1}=(1-t)^{\alpha}\sum_{n=0}^{\infty}t^{n},\hskip
5.69054pt(|t|<1)$ and interchanging the order of summation and integration,
the above equation (8.12) reduces to

$\displaystyle\begin{aligned}
&B(\alpha,\beta;m)=\sum_{n=0}^{\infty}\int_{0}^{1}t^{\alpha+n-1}(1-t)^{\beta}\exp(mt(1-t))dt\end{aligned}$
(8.13)

$\displaystyle\begin{aligned} &\Rightarrow
B(\alpha,\beta;m)=\sum_{n=0}^{\infty}B(\alpha+n,\beta+1;m).\end{aligned}$
(8.14)

∎

######  Theorem 8 (Separation).

If $\Re(\alpha)>0,$ $\Re(\beta)>0,$ $m\in\mathbb{C};$ $|m|<M$ (where $M$ is
positive finite real number slightly greater than 2.0335), then
$B(\alpha,\beta;m)$ can can be separated into real and imaginary part of $m$
as follows

$\displaystyle\begin{aligned} &B(\alpha,\beta;r\cos\theta)\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp(rt(1-t)\cos\theta)dt,\end{aligned}$
(8.15) $\displaystyle\begin{aligned} &B(\alpha,\beta;r\sin\theta)\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp(rt(1-t)\sin\theta)dt,\end{aligned}$
(8.16)

where $r=\sqrt{x^{2}+y^{2}}=|m|<M$ and $\theta=\tan^{-1}y/x.$

###### Proof.

Since $m\in\mathbb{C}$, so let $m=x+iy$ where $x,y\in\mathbb{R}$ and also let
$x+iy=r\cos\theta+ir\sin\theta$ $\Rightarrow$ $r=\sqrt{x^{2}+y^{2}}$ and
$\theta=\tan^{-1}y/x$, then from equation (3.1), we have

$\displaystyle\begin{aligned} &B(\alpha,\beta;x+iy)\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp(re^{i\theta}t(1-t))dt,\end{aligned}$
(8.17)

after simplification the above equation (8.17) reduces to

$\displaystyle\begin{aligned} &B(\alpha,\beta;r\cos\theta+ir\sin\theta)\\\
&=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp(rt(1-t)\cos\theta)dt\\\ &\hskip
28.45274pt+i\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\exp(rt(1-t)\sin\theta)dt,\end{aligned}$
(8.18)

equating the real and imaginary parts of $m$ only, we have the required
results. ∎

##  9\. The modified extended beta distribution

It is expected that there will be many applications of the modified extension
of the classical beta function, like there were of the generalized gamma
function. One application that springs to mind is to Statistics. For example,
the conventional beta distribution can be extended, by using our modified
extension of the classical beta function, to variables p and q with an
infinite range. It appears that such an extension may be desirable for the
project evaluation and review technique used in some special cases.

We define the extended beta distribution by

$\displaystyle\begin{aligned}
&f(t)=\left\\{\begin{array}[]{cc}\displaystyle\frac{t^{p-1}(1-t)^{q-1}\exp(mt(1-t))}{B(p,q;m)},\hskip
5.69054pt0<t<1\\\ {}\hfil\\\ 0,\hskip 56.9055pt{\rm
otherwise}.\end{array}\right.\end{aligned}$ (9.1)

A random variable $X$ with probability density function (pdf) given in
equation (9.1) will be said to have the extended beta distribution with
parameters $p$ and $q$, $-\infty<p,q<\infty$ and $|m|<M$ where $M$ is positive
real number slightly greater than 2.0335. If $\nu$ is any real number [8],
then

$\displaystyle E(X^{\nu})=\frac{B(p+\nu,q;m)}{B(p,q;m)}.$ (9.2)

In particular, for $\nu=1$,

$\displaystyle\mu=E(X)=\frac{B(p+1,q;m)}{B(p,q;m)}$ (9.3)

represents the mean of the distribution and

$\displaystyle\begin{aligned} &\sigma^{2}=E(X^{2})-(E(X))^{2}\\\
&=\frac{B(p,q;m)B(p+2,q;m)-B^{2}(p+1,q;m)}{B^{2}(p,q;m)}\end{aligned}$ (9.4)

is a variance of the distribution.

The moment of generating function of the distribution is

$\displaystyle\begin{aligned}
&M(t)=\sum_{n=0}^{\infty}\frac{t^{n}E(X^{n})}{n!}\\\
&=\frac{1}{B(p,q;m)}\sum_{n=0}^{\infty}\frac{t^{n}}{n!}B(p+n,q;m).\end{aligned}$
(9.5)

The commutative distribution of (9.1) can be written as

$\displaystyle F(x)=\frac{B_{x}(p,q;m)}{B(p,q;m)}$ (9.6)

where

$\displaystyle\begin{aligned}
&B_{x}(p,q;m)=\int_{0}^{x}t^{p-1}(1-t)^{q-1}\exp(mt(1-t))dt,\\\ &\hskip
113.81102pt|m|<M,-\infty<p,q<\infty\end{aligned}$ (9.7)

is the modified extended incomplete beta function. For $m=0$, we must have
$p,q>0$ in (9.7) for convergence, and $B_{x}(p,q;0)=B_{x}(p,q)$, where
$B_{x}(p,q)$ is the incomplete beta function [7] defined as

$\displaystyle\begin{aligned}
&B_{x}(p,q)=B_{x}(p,q;0)=\frac{x^{p}}{p}{}_{2}F_{1}(p,1-q;p=1;x).\end{aligned}$
(9.8)

It is to be noted that the problem of expressing $B_{x}(p,q,m)$ in terms of
other special functions remains open. Presumably, this distribution should be
useful in extending the statistical results for strictly positive variables to
deal with variables that can take arbitrarily large negative values as well.

##  10\. conclusion

The (Euler’s) classical beta function $B(x;y)$ play an important role in
various fields in the mathematical, physical, engineering and statistical
sciences. The extensions of the classical beta function is also important for
further investigation in the respective areas, provided these extensions of
beta function should be convergent for each parameter involving. The modified
extension of classical beta function is convergent and if we choose $m=0$, the
classical beta function is obtained. It is very clear from Figure 1 that for
the graph of classical beta function and modified extension of classical beta
function remains concave upward (or convex downward) for different values of
$\alpha,\beta$ and $m$. The value of $m$ does not effect the nature of
classical beta function, the main effect of the value of $m$ is that it just
push the curve up or drag down the curve from the curve of the classical beta
function as shown in the Figure 1 the same behavior can be observed from the
Table 2. Both modified extension of classical beta function and classical beta
function $i.e.$ $B(\alpha,\beta;m)$ $\&$ $B(\alpha,\beta)\rightarrow 0$ as
$x,y\rightarrow\infty$; $B(\alpha,\beta;m)$ $\&$
$B(\alpha,\beta)\rightarrow\infty$ as $x,y\rightarrow 0$ and both
$B(\alpha,\beta;m)$ $\&$ $B(\alpha,\beta)$ does not exist at $x=y=0$.

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