“FORMATION OF UNIFIED EULERIAN INTEGRALS OFHYPERGEOMETRIC FUNCTION WITH STATISTICAL DISTRIBUTION”

“FORMATION OF UNIFIED EULERIAN INTEGRALS

OFHYPERGEOMETRIC FUNCTION WITH STATISTICAL

DISTRIBUTION”

1

Ms. Samta Gulia ,2 Prof. (Dr.) R.S.Verma and 3Prof. (Dr.) Harish Singh

1Research Scholar, NIMS University, Jaipur, Rajasthan, India

2

Head of Department of Mathematics NIMS University, Jaipur, Rajasthan, India

3

Professor, Maharaja Surajmal Institute, Affiliated to Guru Gobind Singh Indraprastha University, New Delhi, India

INTRODUCTION

In this paper, we evaluate three unified Eulerian integrals. First integral involve the general class polynomials S_VU x and the generalized sequence of function S_n,,O x . The

second integral involve the multivariable polynomials by generalized the general class of polynomial 1 r

r 1

,...U U

,...V V

S [x1,...,xr] which was giving by Srivastava where as the last third

integral contain the product of another class of multivariable polynomials by generalized the general class of polynomial U1,...U r

V

S [x1,...,xr] which was introduced by Srivastava

ABSTRACT

In this research paper we proved three Eulerian Integrals. The results of these Integrals

provide interesting unifications and extensions of new and known results. Our integrals

involve simpler polynomials like as , Jacobi polynomials, generalized Lauricella

and Garg (1987) and all the above three integrals used the multivariable Hypergeometric function which was introduced by Srivastava and Daoust (1969) in their integrals.

1.1 Theorem:1 Second Eulerian Integrals Involving U

V

S [x] and Sn, ,0

 

[x]:

Prove that



1

0

xm(a-1)(1-xm)b-1 U

V

S



yx (1 xm)v



μ m

 , ,0

n

S 









) x (1 hx m  m

1 n n 1

;...q q : p

;...m 1;m

F

 

 

 

 

λ m λ m λ

m λ m )

n ( q q

p

) n ( m m

1

) x -(1 x n z ,...., ) x -(1 x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1

 dx

= m

1) -m

1 Γ(

) ( 1) -m

1 Γ(

 

 

 

 





) U V (

0 k

1 (k, v, u, e, p)

n 1

n 1

;...q q : λ 2 p

;...m m λ; 2 1

F







     

 

     

 

  

   



 _{ }

 



   

 

4 z ,...., 4 z ); (b ; );... (b

: λ

2) -m

1 ( ,..., λ

1 -m

1 : ) a (

); ( β ...; . ); ( β λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

n 1

) n ( q q

p

) n ( m m

1

n 1

n 1

(1.1) Then p.d.f form of (1.1),we have

F(x) =

] [ p)F

e, u, v, (k, 1)

-1 Γ(

) Γ( 1) -m

1 Γ(

] [ F

] ) x (1 hx [ S ] ) x (1 yx [ S ) x (1 x

2 n ;...q 1 :q λ 2 p

n ;...m 1 m λ; 2 1 1

) U V (

0 k

1 n ;...q 1 p:q

n ;...m 1 m 1; m m

,0 , n m u m U V 1 b m 1) (a m

 

 

 



 

 



 

 

 

 

 



m

v

Where

1 =

   

   

λ m λ m λ

m λ m ) n ( q q

p

) n ( m m

1

) x -(1 x n z ,...., ) x -(1 x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1



and

     

 

     

 

  

   



 _{ }

  

   

  

4 z ,...., 4 z );

(b );...; (b : λ

2) -m

1 (

.., ,... λ

1) -m

1 (

: ) a (

); ( β ...; . ); ( β λ

2) -m

1 (

,..., λ

1) -m

1 (

: ) (

n 1 )

n ( n q 1

q p

) n ( n m 1

m 1

2

In this integral, we use the definitions of the general class of polynomials U V

S [x], the

generalized sequence of function , ,0 n

S  [x] and the multivariable Hypergeometric function

and we calculated the equation (1.1) and make its pdf form (1.2). Here we take xm instead of x in the above definitions and we find a new results shown in proof:

Where

1 (k, v, u, e, p) =

K!

p)h e, u, (v, K) A(V, V)

( _UK ₁ R

yk (1.3)

α = a + k + R and β = b +  + R (1.4) Now assumed the following conditions are to be satisfied.

(iv) Re (a, b, r) > 0, Re (α, β) > 0

(v) Min (, p, q) > 0 (j = 1, 2, ...,n) (not all zero simultaneously) (vi) (a)  > 0, v > 0, 1 > 0, 2 > 0

(b)  < 0, v < 0, 1 +  [V/U] > 0, 2 + v [V/U] > 0

(c)  > 0, v  0, 1 > 0, 2 + v [V/U] > 0

where

1 = Re (a) +  {(1(m + n) + p + 



n

1 j

j

q

2 = Re (a) +  {(1(m + n) + p + 



n

1 j

j

q

Proof:- To find the first integral (1.1), We first express the general class of polynomials

U V

S [x] and the generalized sequence of function S_n,,0 [x] which was introduced and studies by various research workers such as Singh (1981),Singh and Srivastava (1964), Krall and Frink (1949), Gould and Hopper (1962), Dhilion (1989), etc as defined below: The generalized sequence of function is

0 , , n

S  [x] =



p e, u, v,

1

θ (v, u, e, p) xR ... (1.5)

The general class of polynomials is introduced as

U V

S [x] =





) U V (

0

K K!

K) A(V, V) ( _UK

xK N= 0,1,2... ... (1.6)

By putting the value from (1.5), (1.6) in (1.1) by taking xm instead of x in the above formulas, while (1.6) is certainly useful generalization, it seems that for some purposes, multiple Hypergeometric function which are of a less general nature are of more immediate value. To this end, we consider the generalized Kamp’ ede Fe’riet function first given by Karlson (1973) which we define later..



1

0

xm(a-1)(1-xm)b-1 U

V

S



yxmμ(1xm)v



S_n,,0









n 1

n 1

;...q q : p

;...m 1;m

F

 

 

 

 

 

 z1x (1 - x) ,...., znx (1 -x)

); (b ...; );... (b : ) a (

); (β ; ... . ); (β : ) (

λ λ

) n ( n q 1

q p

) n ( n m 1

m 1

dx (1.7)



1

0

xm(a-1) (1-xm)b-1





) U V (

0

k K!

K) A(V, V) ( _UK

ykxmk (1-xm)vk





) U V (

0 k

θ1 (k, v, u, e, p)hRxmR

(1-xm)ζR 1 n n 1

;...q q : p

;...m 1;m

F _

 

 

 

 

 z1x (1 -x) ,...., znx (1 -x)

); (b ...; );... (b : ) a (

); ( β ; ... . ); ( β : ) (

λ λ

) n ( n q 1

q p

) n ( n m 1

m 1

dx (1.8)

By using multivariable Hypergeometric function which is giving by Srivastava and Daoust (1969) have giving the multiple series as:

(n) ;...q 1 :q p

(n) ;...m 1 m 1;

F

    

 

    

 

n

z z

: :

1

 1 (n)

(n) 1

;...q q : p

;...m m 1;

F

 

 

 

 

n z ,...., 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

) n ( q q

p

) n ( m m

1

n 1

n 1



=





0 s ,..., s1 n

A(s1,...,sn)

! s z

1 s 1

1

...

! s z

n s n

n

(1.9)

Where

A(s1,...,sn)













 



 

 



 

n

n 1

1 n

1

(n)

n 1

1 n

1

m

1 j

s (n) j m

1 j

... s j 1

1 j

s ... s j

q

1 j

s (n) j q

1 j

... s j p

1 j

s ... s j

) ( β )

( β )

(

) (b )

(b )

(a



Putting value F1;p:qm11;...q;...m n_n





λ λ

x n z ,...., x 1



1

0

xm(a-1) (1-xm)b-1



 ) U V ( 0 k K! K) A(V, V) ( _UK

ykxmk (1-xm)vk



 ) U V ( 0 k

1 (v, u, e, p)hRxmR

×(1-xm)ζR





0 m ,..., m_{1 1}

(m1, ..., mn)

  ! m 1 x z 1 1 m 1 m λ 1 m 1  x  ...   ! m 1 x z 1 1 m 1 m λ n m n  x  dx Where

(m1, ..., mn)=













            (n) (n) j n j 1 (n) j n j 1 (n) (n) j 1 j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a     =



 ) U V ( 0 k K! K) A(V, V) ( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR



1

0

xm(a-1)(1-xm)b-1xmuk(1-xm)vkxmR(1-xm)ζR

                   (n) D 1 j (n) j n m (n) j D 1 j ... j 1 m j C 1 j (n) j ψ n m ... j ψ 1 m j (n) B 1 j (n) j 1 m (n) j B 1 j ... j 1 m j A 1 j (n) j θ n m ... j θ 1 m j ) (d ) (d ) (c ) (b ) (b ) (a     _{ } ! m 1 x z 1 1 m 1 m λ 1 m 1  x  ...   ! m 1 x z 1 1 m 1 m λ n m n  x  dx =



 ) U V ( 0 k K! K) A(V, V) ( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR



1

0

(xm)a+k+R-1(1-xm)b+vR+ζR-1













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j n j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a       ! m 1 x z 1 1 m 1 m λ 1 m 1  x  ...   ! m 1 x z 1 1 m 1 m λ n m n  x  dx =



 ) U V ( 0 k K! K) A(V, V) ( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR



1

0













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j 1 j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a       ! m 1 x z 1 1 m 1 m λ 1 m 1  x  ...   ! m 1 x z 1 1 m 1 m λ n m n  x  dx (1.10) Where α = a + k + R and β = b + k + R

=



 ) U V ( 0 k K! K) A(V, V) ( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j 1 j 1 (n) j n j 1 D 1 j m (n) j D 1 j . ... m j C 1 j ψ m ... ... ψ m j B 1 j m (n) j B 1 j ... .... m j A 1 j θ m ... ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a    



 1 0 1 m ... ... m λ 1) 1

m ( 1

x     _n m  

(1-xm)β-1  

! m 1 z 1 1 m 1 m 1  x  ...   ! m 1 z n 1 m n m n  x 

dx (1.11)

Where x x dx x x dx x m xm dx

m m m m m m 1 1 0 1 ) 1 1 ( 1 1 0 ) 1 1 ( 1 1 0 ) 1 ( ) 1 ( ) 1 ( ) 1 (               







     

By using beta integral now applying Euler Integrals



01 x

a-1

(1-x)b-1f(x) dx =

) Γ(a ) ( ) Γ( b b a  



 n n n n ) ( C ) ( b a   Above formula can be transformed into the following as

dx x

x m m

m 1 1 0 1 ) 1 1 ( ) 1 (     

   = m

=



 ) U V ( 0 k K! K) A(V, V) ( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR













             (n) (n) j n j 1 (n) j n j 1 (n) (n) j n j 1 (n) j n j 1 D 1 j m (n) j D 1 j ... m j C 1 j ψ m ... ψ m j B 1 j m (n) j B 1 j ... m j A 1 j θ m ... θ m j ) (d ) (d ) (c ) (b ) (b ) (a     × m 1) -m 1 Γ( ) 1) Γ) -m 1 Γ(        n 1 n 1 n 1 n 1 λm ... λm λm ... λm λm ... λm λm ... λm ) m 1 ( ) 1 -m 1 ( ) ( ) 1 -m 1 (                    = m 1) -m 1 Γ( ) 1) Γ) -m 1 Γ(       



 ) U V ( 0 k

ɸ1 (k,v, u, e, p)×

n ;...q 1 q : λ 2 p n ;...m 1 m λ; 2 1 F                                     2 n 2 1 ) n ( n q 1 q p ) n ( n m 1 m 1 2 z ,...., 2 z ); (b ...; );... (b : λ 1 -λ 1) -m 1 ( ,..., λ 1 -m 1 : ) a ( ); ( β ; ... . ); ( β λ 1 -λ 1) -m 1 ( ,..., λ 1) -m 1 ( : ) (

Where



 ) U V ( 0 k

1 (k,v, u, e, p) =



 ) U V ( 0 k K! K) A(V, V) ( _UK

yk



 ) U V ( 0 k

1 (v, u, e, p)hR

(1.13) = m 1 ) 1) Γ) -m 1

Γ(  



) U V (

ɸ1 (k,v, u, e, p)

                                   4 z ,...., 4 z ); (b ...; );... (b : λ 2) -m 1 ( ,..., λ 1 -m 1 : ) a ( ); ( β ; ... . ); ( β ; λ 2) -m 1 ( ,..., λ 1) -m 1 ( : ) ( n 1 ) n ( n q 1 q p ) n ( n m 1 m 1 (1.14) Hence First integral is proved

2.1 Theorem 2: Second Eulerian Integral Involving 1 r

r 1 ;...U U ;...V V

S [x1....xr]

Prove that



1

0

xm(a-1) (1-xm)b-1 1 r

r 1 ;...U U ;...V V

S



v vr



) x (1 x y ,..., ) x (1 x y m μ m r m μ m 1 r 1 1   n 1 n 1 ;...q q : p ;...m 1;m F         λ m λ m λ m λ m ) n ( q q p ) n ( m m 1 ) x -(1 x n z ,...., ) x -(1 x 1 z ); (b ...; );... (b : ) a ( ); ( β ; ... . ); ( β : ) ( n 1 n 1  dx = 1) -m 1 Γ( ) 1) Γ -m 1 Γ( 1 1 1 1        ( m



 ) U V ( 0 k 1 1 1 …...



 ) U V ( 0 k r r r r r 1

1K ... r U K

U

t) ( v )

v

(  A(V1, k1 ; …...Vr, kr)



          r 1 i 1 k i ! k y i 

 _ 1 n

F(x) = ] [ F ! k k y ) k , ...v . ; k , A(v ) v ( . ) v ( ... 1) -m 1 Γ( ) ( 1) -m 1 Γ( ] [ F ] ) x (1 x y ... ) x (1 x y [ S ) x (1 x 4 n ;...q 1 :q λ p n ;...m 1 m λ; 1 r 1 i ₁ i i r r 1 1 r k r v r ) r U r V ( 0 r k ... 1 k 1 u r ) 1 U 1 V ( 0 1 k 1 1 1 1 3 n ;...q 1 p:q n ;...m 1 m 1; m r u m r 1 m u m 1 r ;...v 1 u r ;...v 1 v 1 b m 1) (a m                                m r v v

= 0 elsewhere (2.2) Where

3 =

        λ m λ m λ m λ m ) n ( q q p ) n ( m m 1 ) x -(1 x n z ,...., ) x -(1 x 1 z ); (b ...; );... (b : ) a ( ); ( β ; ... . ); ( β : ) ( n 1 n 1  and                          _{ }           4 z ,...., 4 z ); (b ; );... (b ; λ 2) -m 1 ( ,..., λ 1 -m 1 : ) a ( ); ( β ...; . ); ( β ; λ 2) -m 1 ( ,..., λ 1) -m 1 ( : ) ( n 1 ) n ( n q 1 q 1 1 p ) n ( n m 1 m 1 1 1 1 1 4

To prove this integral we define as following the multivariable Hypergeometric Function by generalized the general class of polynomial 1 r

r 1 ;...U U ;...V V

S [x1....xr] .

r 1 r 1 U ... U V ... V

S [x1, ..., xr] =



 ) U V ( 0 k 1 1 1



 ) U V ( 0 k r r r ) ....k k A(V, ) V ( ) V

( _{1 U k ... r U k ...U k 1 r}

1 r r 1

1  

 _ ! k x ... ! k x r k r 1 k 1 r 1 (2.3)

Where Vi = 0, 1,2 ... (i = 1, ..., r), U1,...Ur are arbitrary positive integers, The

Where

α1 = a + 



n

1 j

j kj and β1 = b + 



n

1 j

vj kj (2.4)

Now assumed the following conditions are to be satisfied. (x) Re (a, b, r) > 0, Re (α1, β1) > 0

(xi) Min (, p, q) > 0 (j = 1, 2, ..., n) (not all zero simultaneously) (xii) (a) 1 > 0, 2 > 0

(b) 1 + [V/U] > 0, 2 + [V/U] > 0

(c) 1 > 0, 2 + [V/U] > 0

(d) 1 + [V/U] > 0 and 2 > 0

Where

1 = Re (a) + p +

 n 1 j

j

q and 2 = Re (a) + p + 



n

1 j

j

q (2.5)

3.1 Theorem 3: Third Eulerian Integrals Involving

r ..U ;... 1 U V

S [x1…….xr] :

Prove that



1

0

xm(a-1) (1-xm)b-1 U1;...U r

V

S



v vr



) x (1 x y ,..., ) x (1 x

y m

μ m r m

μ m 1

1 



n 1

n 1

;...q q : p

;...m 1;m

F

 

 

 

 

λ m λ m λ

m λ m )

n ( q q

p

) n ( m m

1

) x -(1 x n z ,...., ) x -(1 x 1 z ); (b ...; );... (b

: ) a (

); ( β ; ... . ); ( β : ) (

n 1

n 1

 dx

= m

1 ) 1) Γ -m

1

Γ(₂  (₂



 ...Uk v k

U1 1 r r

1 1K ...U K U

v)

n 1 n 1 .q ;... q : λ 2 p .m ;... m λ; 2 1

F_

                         _{ }          4 z ,...., 4 z ); (b ; );... (b : λ 2) -m 1 ( ,..., λ 1 -m 1 : ) a ( ); ( β ...; . ); β ( ; λ 2) -m 1 ( ,..., λ 1) -m 1 ( : ) ( n 1 ) n ( q q 2 2 p ) n ( m m 2 2 2 2 1 n 1 n 1 (3.1) Then p.d.f. of (3.1) is given by:

F(x) = ] [ F ) k , ...v . ; k , A(v ) v ( 1) -m 1 Γ( ) 1) Γ -m 1 Γ( ] [ F ] ) x (1 x y ... ) x (1 x y [ S ) x (1 x 6 n ;...q 1 :q λ p 2 ;...m 1 m λ; 1 r r 1 1 r k r v ... 1 k 1 u r v r k r U ... 1 k 1 U 0 r k ,..., 1 k 2 2 2 2 5 n ;...q 1 p:q n ;...m 1 m 1; m r u m r 1 m u m 1 r ;...v 1 u v 1 b m 1 a m                    



( m r v v (3.2) = 0, elsewhere

Where

5 =

To prove this integral we define as following the multivariable Hypergeometric Function by generalized the general class of polynomial S_VU1;...Ur [x1…….xr] as follows:

r 1...U

U V

S [x1, ..., xr] =



 



V k ....U k U

0 k ,... k

r r 1 1

r 1

) ....k k A(V, V)

( _{Uk ...U k 1 r}

1 1

1 

 

! k x ... ! k x

r k r

1 k 1

r 1

. (3.3)

where U1 ... Ur are arbitrary positive integers and the coefficient A(V; k1, ... kr)

(V, k1 > 0, i = 1, ...., r) are arbitrary constants, real or complex. By using (3.3) and the

multivariable Hypergeometric function from (1.9),we occurred (2.1) and its pdf form (3.2).

Where α2 = a + 



n

1 j

jkj and β2 = b + 



n

1 j

vjkj (3.4)

Now assumed the following condition to be satisfied. (xvi) Re (a, b, r) > 0, Re (α2, β2) > 0

(xvii) Min (, p, q) > 0 (j = 1, 2, ...,n) (not all zero simultaneously) (xviii) (a) 1 > 0, 2 > 0

(b) 1 + [V/U] > 0, 2 + [V/U] > 0

(c) 1 > 0, 2 + [V/U] > 0

(d) 1 + [V/U] > 0 and 2 > 0

Where

1 = Re (a) + p + 



n

1 j

j

q

Proofs of Theorem 2 and Theorem 3: The equations (2.1) of theorem 2 and (3.1) of theorem 3, can be derived by following the same lines as given in the proof of first integral and making appropriate modifications and changes.

References:

1. DHILLON, S.S.: A Study of Generalization of Special Function of Mathematical

Physics and Applications Ph.D. Thesis, Bundelkhand Univ. India (1989).

2. GOULD, L H.W. AND HOPPER, A.T.: Operational formulas connected with

two generalisation of Hermite polymials, Duke Math. J 29 (1962), pp 51-63.

3. KARLSONS, P.W.: Reduction of Certain generalised Kampe de feriet functions

Math, Scand. 32 (1973B), pp 265-268.

4. KRALL, H.L. AND FRINK, O.: A new class of orthogonal polynomials; the

Bessel Polynomials, Trans, Amer. Math. Soc. 65 (1949), pp 110-115.

5. SINGH. R.O. AND SHRTVASTAVA, K.N.: A note on generalistion of Lagnerre

and Humbert Polynomials. Recerca (Napoli) (2) 14, Settembre dicuembre, 11-21; Errata, Ibid, (2) (15), Maggio-agosto, 63 (1964).

6. SINGH, A.: A Study of Special Function Sof Mathematics Physics and Their

application in Combinateral Analysis. Ph.D. Thesis, Bundelkhand University India (1981).

7. SRIVASTAVA, H.M. AND DOAUST, M.C. Certain generalised Neumann