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1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 1 Issue 2 ǁ December. 2013ǁ PP-25-30
Multidimensional Mellin Transforms Involving I-Functions of
Several Complex Variables
1,
Prathima J , 2,T.M.Vasudevan Nambisan
Research scholar, SCSVMV, Kanchipuram & Department of Mathematics, Manipal Institute of Technology,
Manipal, Karnataka, India
2,
Department of Mathematics, College of Engineering, Trikaripur, Kerala, INDIA.
ABSTRACT : This paper provides certain multiple Mellin transforms of the product of two I-functions of r
variables with different arguements. In the first instance, a basic integral formula for multiple Mellin transform
of I-function of “r” variables is obtained. Thendouble Mellin transform due to Reed has been employed in
evaluating the integrals involving the product of two I- functions of r variables. This formula is very much
useful in the evaluation of some integrals and integral equations involving I-functions.
2010 Mathematics Subject Classification: 33E20, 44A20
KEYWORDS: I-function, Mellin Barnes contour integral, H-function, Mellin transform.
I.
THE I-FUNCTION OF SEVERAL COMPLEX VARIABLES
The generalized Fox’s H-function, namely I-function of ‘r’ variables is defined and represented in the
following manner [3].
0,n:m ,n ;...;mr ,nr
1 1
r ,qr
I[z1,…,zr]= I p,q:p 1,q 1;...;p
=
1
2πi 
r
z
 1

z
r



1
a j;  j
b j;  j
 
 

 
 

 

1 1 1
 r  r  r
; A j : c j , j ; Cj
; ...; c j ,  j ; C j
r

1

  
r
1
1 1
; Bj : d j , j ; Dj
, ...,  j
, ...,  j
  
; ...; d j
r
 r
, j
; Dj
r

s
s
 ...  θ1 s1 ...θ r s r  s1 ,...,s r z11 ...z r r ds1 ...ds r
L1 Lr
(1.1)
where φ s1,...,s r  and θi si  , I = 1,…,r are given by



n Aj
r i 
 Γ
1-a j +  α j s i
j=1
i=1
 s1 ,...,s r =
q B
p
r i 
r i
Aj
j
 Γ
 Γ
1-b j +  β j s i
a j -  α j si
j=1
i=1
j=n+1
i=1

(1.2)




i
i 
mi D 
 i   i  ni C j
i i
j
 Γ
d j -δ j s i  Γ
1-c j +γ j s i
j=1
j=1
θi si =
i
i
qi
pi
Dj
Cj
i i
i i
 Γ
 Γ
1-d j +δ j s i
c j -γ j s i
j=mi +1
j=n i +1
 








(1.3)
i=1, 2,…,r
Also
 z i  0 for I = 1,…,r;



i  1
;
an empty product is interpreted as unity;
the parameters mj, nj,pj, qj (j=1,…,r), n, p, q qj are non-negetive integers such that
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2. Multidimensional Mellin Transforms Involving…
0  n  p , q  0, 0  n j  p j , 0  m j  q j
i

a j  j=1,...,p  , b j  j=1,...,q  , c
j

j
i 
(j=1,…,r) (not all zero simultaneously).
 j=1,...,pi , i 1,...,r  and d j   j=1,...,qi , i 1,...,r  are complex numbers.
i
 j=1,...,p, i=1,...,r  ,  j   j=1,...,q, i=1,...,r  ,  j   j=1,...,pi , i 1,...,r  and     j=1,...,qi , i 1,...,r  are
j
i
i
i
assumed
to be positive quantities for standardisation purpose.
i
Aj  j=1,...,p  , B j  j=1,...,q  , C
j
 j=1,...,pi , i 1,...,r  and D j   j=1,...,qi , i 1,...,r  of
i

The

gamma functions involved in (1.2) and (1.3) may take non-integer values.
The contour Li in the complex si - plane is of Mellin-Barnes type which runs from
exponents
with indentation, if necessary, in such a manner that all singularities of
C j 
i
the right and



i 
i 
1  c j   j si , j  1, ..., n
i
lie to the left of
Li

i 
Dj
c  i
 
to
c  i
various
,(c real),

i
i 
d j   j si , j  1, ..., m
i
lie to
.
Following the results of Braaksma[1] the I-function of ‘r’ variables is analytic if
i 
p
pi
i 
q
qi
i  i 
i  i 
i   A j j   B j  j   C j  j   D j  j
j 1
j 1
j 1
j 1
II.
 0, i  1, ..., r
(1.4)
CONVERGENCE CONDITIONS
The integral (1.1) converges absolutely if
 
arg z k
where
1
2
 k  , k  1, ..., r
(2.1)
qk
pk
k  q
 k  mk  k   k 
 k   k  nk  k   k 
k  k  
 p
 Cj  j
 k    A j j   B j  j   D j  j  
Dj  j   Cj  j 
 0
 j 1
j 1
j 1
j mk 1
j 1
j nk 1


(2.2)
 
And if
arg z k
conditions.
(i)
k

1
2
 k  , and  k  0, k  1, ..., r
then integral (1.1) converges absolutely under the following
 k  0,  k   1 where  k is given by (1.4) and
p 1


j 1  2

k
 a j  A j  j1  1    b j  B j  j1  1   c jk   C jk  



 2
 2





p
q

qk
 1   d k 

j
j 1  2




k 
D j , k  1, ..., r
(2.3)
(ii)


 k  0 with sk   k  itk  k and tk are real, k 1, ..., r ,  k are
so
chosen
that
for
tk   we
have
 k   k k   1
We have discussed the proof of convergent conditions in our earlier paper [3].
Remark 1
i 
If D j  1  j  1, ..., mi , i  1, ..., r  in (1.1), then the function will be denoted by

I z1 , ..., z r

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3. Multidimensional Mellin Transforms Involving…



1
r
 z1  a j ; j ,..., j ;A j  :

p
1
0,n:m1 ,n1 ;...;m r ,n r 
I
 
p,q:p1 ,q1 ;...;p r ,q r


 1 1 1 
1
 r    1 1  ,
z r
, j ;D j  ;...;
 b j ; j ,...,  j ;B j  :  d j , j ;1 
d

q 1 
m1 m1 1 j
q1
1

1

1
1 1
cj , j ;Cj

; ...;
p1
1

cj
r
r
, j
r 
;Cj

pr
 d  r  ,  r  ;1
 r r r  
,
 j

 d , j ;D j 
j

 m r m r 1  j
q r
1
=
1
2πi 
r
 
  







s
s
 ...  θ1 s1 ...θr s r  s1 ,...,s r z11 ...z r r ds1 ...ds r
L1 Lr
(2.4)
where
i
mi
 i   i  ni C j
i  i 
 Γ d j -δ j s i  Γ
1-c j +γ j s i
j=1
j=1
θi s i =
, i  1, ..., r
i 
i
qi
pi
Dj
Cj
i  i 
i  i 
 Γ
 Γ
1-d j +δ j s i
c j -γ j s i
j=mi +1
j=n i +1


 






(2.5)
Remark 2
i 
i 
If C j  1  j  1, ..., ni , i  1, ..., r  , D j  1  j  1, ..., mi , i  1, ..., r  and n=0 in (1.1), then the function will be denoted
by
I

z , ..., z r
1 1



 1 1 1 
1
 r    1 1 
 z1  a j ; j ,..., j ;A j  :  c j , j ;1  ,
 c , j ;C j  ;...;
 p 1
n1 n11 j
p1

1
0,0:m1 ,n1 ;...;m r ,n r 
I

p,q:p1 ,q1 ;...;p r ,q r 


 1 1 1 
1
 r    1 1 
z r
 b j ; j ,...,  j ;B j  :  d j , j ;1  ,
 d , j ;D j  ;...;

q 1 
m1 m11 j
q1
1

1


c  ,   ;C  

n r n r 1
pr 

 d  r  ,  r  ;1 ,
 d  r  ,  r  ;D r  

 j


j
j
j 
m r m r 1  j
q r 
1

r
r 
c j ,  j ;1
r
,
r
j
j
r
j
(2.6)
where


1 s1 ,...,s r =

1


q B
p
r i 
r i 
Aj
j
 Γ
 Γ
1-b j +  β j s i
a j -  α j si
j=1
i=1
j=n+1
i=1


 



mi
 i   i  ni
i  i 
 Γ d j -δ j si  Γ 1-c j +γ j si
j=1
j=1
θi s i =
i
i
qi
pi
Dj
Cj
i i
i  i 
 Γ
 Γ
1-d j +δ j s i
c j -γ j s i
j=mi +1
j=ni +1
 

(2.7)

(2.8)
i=1,2,…,r
Definition
Mellin transform of a function f(x) is defined as


  x  ; s 0 x s1 f  x dx
(2.9)
M f
III.
RESULT USED
We shall use the following result given by Reed [4]
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4. Multidimensional Mellin Transforms Involving…
  s 1 t 1  1
y
  x

2
0 0

 2 i
 
1  i   2  i 


1  i   2  i 
 
g s, t x

 
 s t
y dsdt  dxdy  g s , t


(2.10)
Basic formula for the multiple Mellin transform of the I-function of r variables
  s 1 s 1


 ...  x11 ...x r r I1 t1x11 ,...,t r x r r  dx1...dx r


0 0
-s1
-sr
 -s1   -s r   -s1 -s r  1 r
=
θ1 
 ...θr  σ  1   ,...,   t1 ...t r
1...1  σ1 
 r  1
r 
1
(3.1)
 -s
-s
where 1  1 ,..., r
 1
r

 and

 -si 
, i 1, 2, ..., r are given by (2.7) and (2.8) respectively. The condition given
 σi 
θi 
by (1.4) and the convergence conditions given in section 2 are assumed to be satisfied and
  d  i  
  1 - c  i  
  j    s i  i min   j  ,
 i min
1 j mi     i  
1 j n i     i  
  j 
  j 
i  1, 2, ..., r,
Proof
Express the I-function of ‘r’ variables involved in the left hand side of (3.1) in terms of Mellin-Barnes
type contour integral (1.1), i i = -u i ,i=1,2,...,r and interpret the resulting integral with the help of Reeds theorem
(2.10) to get the required result.
Special cases
i
When all the exponents Aj  j=1,...,p  , B j  j=1,...,q  , C j
 j= ni +1,...,pi , i 1,...,r  and D j   j= mi +1,...,qi , i 1,...,r  are
i
equal to unity in (3.1), we get the multiple Mellin transform of H-function of several variables where H-function
of several variables is defined by Srivastava and Panda [6]. If we put r = 2 in (3.1) we get double Mellin
transform of I-function of two variables where I-function of two variables is defined by Shantha,Nambisan and
Rathie[5]. Further putting all exponents unity, it reduces to the double Mellin transform of H-function of two
variables
where H-function of two variables is defined by Mittal and Gupta [2] .
IV.
MULTIPLE MELLIN TRANSFORM OF THE PRODUCT OF TWO I-FUNCTIONS OF
‘R’ VARIABLES
   1  1


 ...  x1 1 ...x r r I1 s1x11 ,...,s r x r r

0 0
1
=
1 ... r

- 1
1
t1

- r
r
...t r
I




I1  t1x1 1 ,...,t r x r r 



dx1…dxr


0, 0 : m1  n1 , m1  n1 ; ...; m r  nr , mr  n r
 
p  q, p  q : p1  q1 , p1  q1 ; ...; pr  qr , pr  q r
 - 1
s t 1
 1 1
 r
 - r
s r t r
C : C1 ;...;C r
D : D1 ;...;D r






(4.1)
where
C
1

1  r 
a j ;α j ,...,α j ; A j

,
p
r 

 i  1 1 r  r  
 i
 1-bj -i=1  βj ,  βj ,...,  βj ; Bj 
 q
i
1
r
1
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5. Multidimensional Mellin Transforms Involving…
Ci 
1
      
      
i
i
c j ,γ j ;1
ni +1
D
1
Di 

1
ni
 i   i 
1-dj
,
-δ j
1
i
i
i
c j ,γ j ; C j
1  r 
b j ;β j ,...,β j ; B j

mi +1
i
i

;1 
mi
,
  i   i   i i  i i  
 1-dj -δj  ,δj  ; Dj 
qi
i
i
m+1 
i 1, ..., r
,
,
q
r 

 i  1 1 r  r  
 i
 1-aj -i=1  αj ,  αj ,...,  αj ; Aj 
p
i
1
r
1
1-cj
,
mi
i
i 
,δj
pi
        
      
i
i
d j ,δ j ;1
i
i
i 
-γj
1
i
i
i
d j ,δ j ; D j
i 
i
,γj
i
i
i

;1 
ni
,
 i  i   i i  i i  
 1-cj -γj  ,γj  ; Cj  ,
pi
i
i
n +1 
i
, i  1, ..., r
qi
The integral (4.1) is valid under the following sets of conditions.
(a) The conditions, modified appropriately(if necessary), given in section 3 are satisfied by both the I-functions
of r
variables in the integrand of (4.1).
(b) i > 0, i >0, i=1,2,...,r
   i  
  i   
dj
d
 - i min   j   <  si 
and - i min   
i
i 
1 j mi
  δ j 

 
1 j m
i
  δj  

 
  1-c i 
j
< i min  
i 
1 j n i   γ
  j
  1-c i  

  i min   j  , i=1,…,r
 1 j ni   γ i  

  j 
Proof
Express the Express the function

I1 s1x11 ,...,s r x r r


 involved in the integrand of (4.1) as Mellin-Barnes

type contour integral using (2.6), change the order of integrations, change the order of integrations, evaluate the
inner integrals using (3.1) and interpret the result with the help of (1.1) to obtain the right hand side of (4.1).
Special cases
   1  1


 ...  x1 1 ...x r r I1 s1x1 1 ,...,s r x r r 


0 0
1
=
1 ... r

- 1
1
t1

- r
r
...t r
I



I1  t1x1 1 ,...,t r x r r 



dx1…dxr


0, 0 : m1  m1 , n1  n1 ; ...; m r  mr , nr  n r


p  p, q  q : p1  p1 , q1  q1 ; ...; pr  pr , q r  qr
 1
s t 1
 1 1
 r
 r
sr t r
C : C1 ;...;C r
D : D1 ;...;D r






(4.2)
where
C
1

1  r 
a j ;α j ,...,α j ; A j

,
p
r 

 i  1 1 r  r  
 i
 aj  i=1   j ,   j ,...,   j ; Aj 
 p
i
1
r
1
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29 | P a g e

6. Multidimensional Mellin Transforms Involving…
Ci 
1
        
      
i
i
c j ,γ j ;1
ni +1
D
1
Di 

1
ni
i
cj
,
i
+ j
1  r 
b j ;β j ,...,β j ; B j
mi +1
i
i

;1
ni
,
  i   i   i  i  i  i  
 cj + j  , j  ; Cj 
pi
i
i
n +1 
i
i 1, ..., r
,

,
q
r 

 i  1 1 r  r  
 i
 bj  i=1   j ,   j ,...,   j ; Bj 
q 
i
1
r
1
dj
,
mi
i
, j
pi
        
      
i
i
d j ,δ j ;1
i
1
i
i
i
c j ,γ j ; C j
i
i
i
+ j
1
i
i
i
d j ,δ j ; D j
i
i
i
, j
i
i

;1 
mi
,
  i   i   i  i  i  i  
 dj + j  , j  ; Dj  ,
qi
i
i
m +1 
i
, i  1, ..., r
qi
The conditions of validity of (4.2) are
i > 0, i >0, i=1,2,...,r
  1-c i  
  d  i   
j
 -  min   j   <  s
and - i min   
 i   i 1 j mi    i     i 
1 j n i   
  j 
  δ j 
  d  i  
  1-c i  
j
   min   j  , i=1,…,r
< i min  
 i   i 1 j ni    i  
1 j mi   
  j 
  γ j 
Proof of (4.2) is similar to that of (4.1).
When all the exponents are equal to unity in (4.1)and (4.2), we get the multiple Mellin transform of product of
two H-functions of several variables. In (4.1) and (4.2) if we put r = 2 we get double Mellin transforms of
product of two I-functions of two variables.
Further putting all exponents unity it reduces to the double Mellin transforms of product of two H-functions of
two variables.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Braaksma. B. L. J., Asymptotic expansions and analytic continuations for a class of Barnes-integrals,
Compositio Math. 15,239-341, (1964)
Mittal P.K and Gupta K.C., An integral involving generalized function of two variables, Proc.Indian
Acad.Sci Sect. A 75, 117-123, (1972).
Prathima J, Vasudevan Nambisan T.M., Shantha Kumari K A study of I-function of several complex
variables, International journal of Engineering Mathematics, Hindawi Publications (In Press).
Reed I.S, The Mellin type of double integral, Duke Math.J. 11(1944), 565-575.
Shanthakumari.K,Vasudevan Nambisan T.M,Arjun K.Rathie: A study of the I-function
of two variables,Le Matematiche(In Press),(2013). Also see arXiv:1212.6717v1 [math.CV].30Dec2012
Srivastav H.M,Panda R:Some bilateral generating functions for a class of generalized
hypergeometric polynomials,J.Reine Angew.Math.17,288 ,265-274,(1976).
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