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. Then\double 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

   

    

     

r

1 1 1  r  r  r ; A j : c j , j ; Cj ; ...; c j ,  j ; C j

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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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   j=1,...,qi , i 1,...,r  are 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

 a j  A j  j1  12    b j  B j  jk1  12   c jk   C jk   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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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 , j ;D j  j    q r 1 m r m 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  1 qr 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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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 j      ,  i min     s i  i min   i i     1 j mi    1 j n 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   r    i βj , 1 βj ,..., r βj ; Bj   1-bj -i=1 i 1 r  q 1

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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   r    i αj , 1 αj ,..., r αj ; Aj   1-aj -i=1 i 1 r p 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i +1 

, 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   r    i  j , 1  j ,..., r  j ; Aj   aj  i=1 i 1 r  p 1

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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i +1 

i 1, ..., r

,

, q

r    i   1   r    i  j , 1  j ,..., r  j ; Bj   bj  i=1 i 1 r q  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i +1 

, 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  - i min   j   <  si  and - i min    i    1 j mi   δ i    1 j n i      j    j 

  d  i     1-c i   j   i min   j  , i=1,…,r < i min   i    1 j ni   γ i   1 j mi      j    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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