1153
On The Applications Of Aleph
()-Function In A
Slightly Different Type-1 Beta Density Model
Yashwant Singh
Abstract: In the present paper, the author has studied about the structures which are the products and ratios of statistically
independently distributed positive real scalar random variables.The author has derived the exact density of a slightly different Type-1 beta density by the Mellin Transform and Hankel Tranform of the unknown density and afterthat the unknown density has been derived in terms of Aleph functions by taking the inverse Mellin transform and Inverse Hankel Transform .A more
general structure of Type-1 beta density has also been discussed. Some special cases in terms of H -function are also given.
Key words: Type-1 Beta density, Aleph Function, I -function, H -function, Mellin Transform, Inverse Mellin Transform, Hankel
Transform, Inverse Hankel Transform. (2010 Mathematics Subject Classification: 33CXX, 44A15, 82XX)
————————————————————
1. Introduction
The I -function introduced by Saxena [4] will be represented and defined in slightly different manner as follows:
, , :
( ) ( )
i i m n p q R
I z I z 1 , 1 ,
1 , 1 ,
, , ,
,
, : , , ,
1
( ) 2
j j n j i j i n p i i i j j m j i j im
q i
a a
m n s
p q R b b
L
I z s z d s
i
(1)Where i 1 , z 0and zs e x p [s i n |z |ia r g z]where |z |represents the natural logarithm of |z |and a rg zis not
the principal value. Here
1 1
1 1
1
1 ( )
1
m n
j j j j
j j
R q p
ji ji ji ji
j m j n
i
b B s a A s
s
b B s a A s
(2)
For R 1, the I -function reduces to the H -function.
The - function introduced by Suland et.al. [6] defined and represented in the following form:
,, : : [ ] i i i m n
p q r
z z
, 1 , 1 ,
, ; :
1 , 1 ,
( , ) , [ ( , ) ] ( , ) , [ ( , ) ]
i
i i
i
j j n i j i j i n p m n
p q i r
j j m i j i j i m q
a a
z
b b
1
( ) 2
s
L
s z d s
(3)
Where 1;
1 1
1 1 1
( ) (1 )
( )
(1 ) (
i i
m n
j j j j
j j
q p
r
i j i j i j i j i
i j m j n
b s a s
s
b s a s
(4)
*
1 , 1 ,
, , [ , ]
i
j j n i ji ji n p
A a a , *
1 ,1 ,
, , [ , ]
i
j j m i ji ji m q
B b b
The Mellin transform of f( )x denoted by M
f x( );s
or F s( )is given by
10
( ) ; s ( )
M f x s x f x d x
(5)The Hankel transform of f( )x denoted by H
f( );x p
or Fv(p)is given by
0
( ) ; ( ) ( )
v v
H f x p x J p x f x d x
(6)A real scalar random variable x is said to have a real type-1 generalized beta distribution,if the density is of the following form
([1], p.121, eq. (4.8)):
1 1
( )
(1 ) ; 0 1 , 0 , 0 ( ) ( )
0 , ( )
x x x
e l s e w h e r e
f x
(7)
Where the parameters and are real. The following discussion holds even when and are complex quantities. In this
case, the conditions become R e () 0 , R e () 0 where Re(.)means the real part of (.).
2. General structures
A real scalar random variable x is said to have a real generalized type-1 generalized beta distribution,if the density is of the
following form:
1 1 2 1
(1 ) ( , ; , ; ; )
0 , ( )
tm
t
t m A
x A x F a b c w p x t m
e ls e w h e r e
f x
(8)
For
1
0 x A , 0 , 0 ,A 0 ,a 0 ,b 0 ,c 0 ,1 A x 0
.
Where the parameters and are real. The following discussion holds even when and are complex quantities. In this
case, the conditions become R e () 0 , R e () 0 where Re(.)means the real part of (.).
The ht h-moment of x, when x has the density in (8), is given by
h ht m h t m
E x
t m h
A
(9)
1155
When and hare real, the moments can exist for some values of halso such that h 0.
The Mellin transform of f( )x is obtained from (8) as:
11 ( )
1
s
t m s t m
M f x
t m s
A
(10)
For R e ( tm s 1) 0 , R e ( tm ) 0 ,s v2r 2 0 , 0.
The unknown density f( )x is obtained in terms of -function by taking the inverse Mellin transform of (10). That is
1
1 1
1
, 1 , 0
1 ,1: : 1 1
, ( )
iR t m
t m A
f x A x
t m
(11)
The Hankel transform of f( )x is obtained from (9) as:
11
( ) ( )
1
v s
t m s t m
H f x J p
t m s
A
(12)
For R e ( tm s 1) 0 , R e ( tm) 0 ,s v 2r2 0 , 0.
The unknown density f( )x is obtained in terms of -function by taking the inverse Hankel transform of (12). That is
1
1 1
1
, 1 , 0
1 ,1: : 1 1
,
( ) ( )
i
v R t m
t m A
f x J p A x
t m
(13)
Consider a set of real scalar random variables x1, ...,xk , mutually independently distributed, where xjhas the density in (8) with
the parameters j,j; j 1, ...,k and consider the product
1 2... k
u x x x (14)
In the standard terminology in statistical literature, the ht hmoment of u , when u has the density in (8), is given by
j
j j
j j
h
j j
j
j j
t m h t m
E u
t m h
A
For R e ( j t m h) 0 ,j 0 ; j 1, ...,k
Then the Mellin transform of g u( )of u is obtained from the property of the statistical independent and is given by
1 11
( ) s ... ks
M g u M x M x (16)
11
1
( )
1 j
j j
j k
j s
j
j j
j
j j
t m s t m
M g u
t m s
A
(17)
For R e ( j t m s 1) 0 ,j 0 ,s v 2r 2
The unknown density f( )x is obtained in terms of -function by taking the inverse Mellin transform of (17). That is
1
1 1
1
, , 0
, : 1 1
1 1
, ( )
j
j j
j j
i j
j
j j
j
j
k k
j k
k k R t m
j j
j
j
t m A
g u A u
t m
(18)
Then the Hankel transform of g u( )of u is obtained from the property of the statistical independent and is given by:
1 1
2 2
[ ( )] v( ) v( ) ... k v( k)
H g u H x J p x H x J p x H x J p x (19)
11
1
( ) ( )
1 j
j j
j k
j
v j s
j j
j
j j
t m s t m
H g u J p
t m s
A
(20)
For R e ( j t m s 1) 0 ,j 0 ,s v 2r 2 0
The unknown density f( )x is obtained in terms of -function by taking the inverse Hankel transform of (20). That is
1
1 1
1
, , 0
, : : 1 1
1 1
,
( ) ( )
j
j j
j j
i j
j
j j
j
j
k k
j k
v j k k R j t m
j
j
t m A
g u J p A u
t m
(21)
If we consider more general structures in the same category. For example, consider the structure
1 2
1 1 2 ... , 0 , 1, ..., k
k j
u x x x j k (22)
1157
Then the Mellin transform of g u( 1)of u1 is given as
1( 1 ) ( 1 )1 1
( ) s ... k s
k
M g u M x M x (23)
1
1 ( 1 )( 1)
( )
( 1) j
j
j j j
j k
j j
s j
j j j
j
j j
t m s t m
M g u
t m s
A
(24)
For R e ( j t m s 1) 0 ,s v 2r 2 ,j 0.
The unknown density g u( 1)is obtained in terms of -function by taking the inverse Mellin transform of (24). That is
, , 0
1 , : : 1
1 1
, ( )
j
j
j j j
j
j j j
i j j j
j
j j
j
j
k k
j k
k k R t m
j j
j
j
t m A
g u A u
t m
(25)
For R e ( j t m s 1) 0 ,s v 2r 2 ,j 0.
The Hankel transform of g u( 1)of u1is obtained from the property of the statistical independent and is given by:
1 1 2 2
1 1 1 2 2
[ ( )] ( ) ( ) ... k ( k)
v v k v k
H g u H x J p x H x J p x H x J p x (26)
1
( 1 )1
( 1)
( ) ( )
( 1) j
j
j j j
j k
j j
v s
j
j j j
j
j j
t m s t m
H g u J p
t m s
A
(27)
For R e ( j t m s 1) 0 ,s v 2r 2 0 ,j 0.
The unknown density g u( 1)is obtained in terms of -function by taking the inverse Hankel transform of (27). That is
, , 0
1 , : : 1
1 1 ,
( ) ( )
j
j
j j j
j
j j j
i j j j
j
j j
j
j
k k
j k
v j k k R j t m
j
j
t m A
g u J p A u
t m
(28)
For R e ( j t m s 1) 0 ,s v 2r 2 0 ,j 0.
A More General Structure
1 2
1 , , ...,
, ...,
r
r k
x x x
w
x x
(29)
Wherex1, ...,xk , mutually independently distributed real random variables having the density in (8) with xjhaving parameters
, ; 1, ...,
j j j k
.
Then the Mellin transform of g w( )is given by
1 1 ( 1 ) ( 1 )1 1
( ) s ... rs r s ... k s
M g w M x M x M x M x (30)
11 1
1
( )
1
j j
j j
j
k r
j j
s
j j
j j
j
j j
t m t m s
M g w
t m s
A A
1
( 1)
( 1) j
j k
j s
j r
j
j j
tm s
s A
(31)
For R e ( j t m (s 1) ) 0 ,s v 2r 2 0
The unknown density g w( )is obtained in terms of -function by taking the inverse Mellin transform of (31). That is
1
1 1
1
, , 0
, : : 1 1
1 1
, ( )
j
j j
j j
j
i j
j
j j
j
j
k k
j r
r r R t m
j j
j
j
t m A
g w A w
t m
1 1 1
1 ,
0 ,
, : : 1 1 1
1 ,
; 1, ..., j
j j
j
i j
j
j j
t m k
k r k r k r R
j r
A w j r k
(32)
Then the Hankel transform of g w( ) is given as:
1
1
1
1
1 1 1 1
[ ( ) ] v( ) ... r v( r) r v r ... k v k
H g w H x J p x H x J p x H x J p x H x J p x (33)
11 1
1
( ) ( )
1
j j
j j
j
k r
j j
v s
j j
j j
j
j j
tm tm s
H g w J p
tm s
A A
1159
1
( 1)
( 1) j
j k
j s
j r
j
j j
tm s
s A
(34)
For R e ( j t m (s 1) ) 0 ,s v 2r 2 0
The unknown density g w( )is obtained in terms of -function by taking the inverse Hankel transform of (34). That is
1
1 1
1
, , 0
, : : 1 1
1 1
,
( ) ( )
j
j j
j j
j
i j
j
j j
j
j
k k
j r
v j r r R j t m
j
j
t m A
g w J p A w
t m
1 1 1
1 ,
0 ,
, : ; 1 1 1
1 ,
; 1, ..., j
j j
j
i j
j
j j
t m k
k r k r k r R
j r
A w j r k
(35)
We can consider more general structures in the same category. For example, consider the structure
1 1
1 1
1 , ...,
, ..., r
k r
r
r k
x x
w
x x
(36)
Wherex1, ...,xk , mutually independently distributed real random variables having the density in (8) with xjhaving parameters
, ; 1, ...,
j j j k
.
Then the Mellin transform of g w( 1)is given by
1( 1 ) ( 1 ) 1( 1 ) ( 1 )1 1 1
( ) s ... r s r s ... k s
r r k
M g w M x M x M x M x
(37)
1
1 1
( 1)
( )
( 1)
j j
j j
j j j
j
k r
j j
s
j j
j j j
j
j j
t m t m s
M g w
t m s
A A
1
( 1)
( 1) j
j
j j
k
j s
j r
j j
j j
t m s
s A
(38)
The unknown density g w( 1)is obtained in terms of -function by taking the inverse Mellin transform of (38). That is
, , 0
1 , : : 1
1 1 ,
( )
j
j
j j j
j
j
j j
j
i j j j
j
j j
j
j
k r
j r
r r R t m
j j
j
j
t m A
g w A w
t m
1 ,
0 ,
, : : 1
1
1 ,
; 1, ...,
j j j
j
j j
j
i j j j
j
j j
t m k
k r k r k r R
j r
A w j r k
(39)
For R e ( j t m (s 1) ) 0 ,s v 2r 2 0 ,j 0.
Then the Hankel transform of g w( 1) is given as:
1 1
1 1 1
[ ( )] ( ) ... r ( r)
v r v r
H g w H x J p x H x J p x
1
1
1 1 ...
k k
r r
r v r k v k
H x J p x H x J p x
(40)
1
1 1
( 1)
( ) ( )
( 1)
j j
j j
j j j
j
k r
j j
v s
j j
j j j
j
j j
t m t m s
H g w J p
t m s
A A
1
( 1)
( 1) j
j
j j
k
j s
j r
j j
j j
t m s
s A
(41)
For R e ( j t m (s 1) ) 0 ,j 0 ,s v 2r 2 0
The unknown density g w( 1)is obtained in terms of -function by taking the inverse Hankel transform of (41). That is
, , 0
1 , : : 1
1 1
,
( ) ( )
j
j
j j j
j
j
j j
j
i j j j
j
j j
j
j
k r
j r
v j r r R j t m
j
j
t m A
g w J p A w
t m
1161
1 ,
0 ,
, : : 1
1 1 ,
; 1, ...,
j j j
j
j j
j
i j j j
j
j j
t m k
k r k r k r R
j r
A w j r k
(42)
For R e ( j t m (s 1) ) 0 ,j 0 ,s v 2r 2 0.
3. Special Cases
If we take i 1,R 1in (11),the unknown density f( )x is obtained in terms of H -function . That is
1
1 1
1
, 1 , 0
1 ,1 1 1
, ( )
t m
t m A
f x H A x
t m
(43)
For j 1; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (13), the unknown density f( )x is obtained in terms of H -function . That is
1
1 1
1
, 1 , 0
1 ,1 1 1
, ( ) v( )
t m
t m A
f x J p H A x
t m
(44)
For j 1; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (18), the unknown density g u( ) is obtained in terms of H -function . That is
1
1 1
1
, , 0
, 1 1
1 1
, ( )
j
j j
j j
j j
j j
j
j
k k
j k
k k t m
j j
j
j
t m A
g u H A u
t m
(45)
For j 1; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (21), the unknown density g u( ) is obtained in terms of H -function . That is
1
1 1
1
, , 0
, 1 1
1 1 ,
( ) ( )
j
j j
j j
j j
j j
j
j
k k
j k
v j k k j t m
j
j
t m A
g u J p H A u
t m
(46)
If we take R 1 iin (25), the unknown density g u( 1) is obtained in terms of H -function . That is
, , 0
1 , 1
1 1
, ( )
j
j
j j j
j
j j j
j j j
j
j j
j
j
k k
j k
k k t m
j j
j
j
t m A
g u H A u
t m
(47)
For j 1 j; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (28), the unknown density g u( 1) is obtained in terms of H -function . That is
, , 0
1 , 1
1 1
,
( ) ( )
j
j
j j j
j
j
j j
j
j j j
j
j j
j
j
k k
j k
v j k k j t m
j
j
t m A
g u J p H A u
t m
(48)
For j 1 j; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (32), the unknown density g w( ) is obtained in terms of H -function . That is
1
1 1
1
, , 0
, 1 1
1 1 ,
( )
j
j j
j j
j
j j
j j
j
j
k k
j r
r r t m
j j
j
j
t m A
g w H A w
t m
1 1 1
1 ,
0 ,
, 1 1 1
1 ,
; 1, ..., j
j j
j
j j
j j
t m k
k r k r k r
j r
H A w j r k
(49)
For j 1; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (35), the unknown density g w( ) is obtained in terms of H -function . That is
1
1 1
1
, , 0
, 1 1
1 1
,
( ) ( )
j
j j
j j
j
j j
j j
j
j
k k
j r
v j r r j t m
j
j
t m A
g w J p H A w
t m
1 1 1
1 ,
0 ,
, 1 1 1
1 ,
; 1, ..., j
j j
j
j j
j j
t m k
k r k r k r
j r
H A w j r k
1163
For j 1; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (39), the unknown density g w( 1) is obtained in terms of H -function . That is
, , 0
1 , 1
1 1
, ( )
j
j
j j j
j
j
j j
j
j j j
j
j j
j
j
k r
j r
r r t m
j j
j
j
t m A
g w H A w
t m
1 ,
0 ,
, 1
1 1 ,
; 1, ...,
j j j
j
j j
j
j j j
j
j j
t m k
k r k r k r
j r
H A w j r k
(51)
For j 1 j; j 1, ...,k , the H -function reduces to the G -function.
If we take R 1 iin (42), the unknown density g w( 1) is obtained in terms of H -function . That is
, , 0
1 , 1
1 1
,
( ) ( )
j
j
j j j
j
j
j j
j
j j j
j
j j
j
j
k r
j r
v j r r j t m
j
j
t m A
g w J p H A w
t m
1 ,
0 ,
, 1
1
1 ,
; 1, ...,
j j j
j
j j
j
j j j
j
j j
t m k
k r k r k r
j r
H A w j r k
(52)
For j 1 j; j 1, ...,k , the H -function reduces to the G -function.
References
[1]. A.M. Mathai, “A pathway to matrix-variate gamma and normal densities,” Linear Alg. Appl.396 317-328, 2005.
[2]. A.M. Mathai and R.K. Saxena,“The H-function with Applications in Statistics and Other Disciplines,” Wiley Eastern, New
Delhi and Wiley Halsted, New York, 1978.
[3]. A.M. Mathai, R.K. Saxena and H.J. Haubold, “The H-function: Theory and Applications,” CRC Press, New York, 2010.
[4]. V.P. Saxena , “ Formal solution of certain new pair of dual integral equations involving H - Functions,” Proc. Nat.
Acad. India, sect. A 52 366-375, 1982.
[5]. H.M. Srivastava, K.C. Gupta and S.P. Goyal, “The H-function of One and Two Variables With Applications,” South
Asian Publisher, New Delhi, 1982.
[6]. N. Sudland, B. Baumann and T.F. Nonnenmacher, “Open problem: who knows about the Aleph ()-functions?,” Frac.
