On Some Double Integrals Involving H-Function of Two Variables and Spheroidal Functions

The present paper evaluates certain double integrals involving H -function of two variables [21] and Spherodial functions [23]. These double integrals are of most general character known so far and can be suitably specialized to yield a number of known or new integral formulae of much interest to mathematical analysis which are likely to prove quite useful to solve some typical boundary value problems.


Introduction
The H -function occurring in the paper is defined and represented by Buschman and Srivastava [3] as follows : ,, ( ; ; ) ,( ; ) ,, ( , ) ,( , ; ) where Which contains fractional powers of the gamma functions.Here, and throughout the paper ( 1,..., ) The following sufficient condition for the absolute convergence of the defining integral for the H -function given by equation (1.1) have been given by (Buschman and Srivastava).

The H -function of two variables
The H -function of two variables introduced by Singh and Mandia [21] will be defined and represented in the following manner: , : , , : , ; , , ; , Where x and y are not equal to zero (real or complex), and an empty product is interpreted as unity , , , p q n m are non-negative integers such that 0 , ( 1, 2,3; 2,3) ( 1, 2,..., ), ( 1, 2,..., ), ( 1, 2,..., ), ( 1, 2,..., ), to the left of the contour.The functions defined in (1.7) is an analytic function of x and y , if 12) The integral in (1.7) converges under the following set of conditions: The behavior of the H -function of two variables for small values of || z follows as: Where (1.17)   is defined and investigated by Stratton [] and   later by Chu and Stratton [4] are those solutions of the differential equation: That remains finite at the singular points 1 z  .The spheroidal function can be expanded in Bessel function on ( , )   ([19], p. 190) If () cz is real and finite If 0 c  and z such that cz remains finite and the normalization is chosen to be such that The function The following results ( [5], p. 172; [9], p. 145; [13], p. 226) in the sequel will be used during the proof of our main results in a little simplification: min , , , , Re( ), Re( ) 0 , ; , , , / ; , , / ,, ,, , , , ; , / , , / ; 1 ;0

     
Where for convenience, 31 44 ,, are all positive quantities; W is not equal to zero; and the conditions (1.11) to (1.15) with x replaced by M and y replaced by v are satisfied.

Proof of (2.1):
To prove the integral relation (2.1), we first replace 1 H by its Mellin-Barnes double contour integral from (1.7) with 1 0 m  .On inverting the order of integration, which is justified due to absolute convergence of the integrals involved in the process, we obtain . Now, by virtue of the familiar result (1.23) and subsequently to the condition (1.7), the right hand side of (2.1) is readily verified.
The importance of the result (2.1) lies in the fact that many more interesting double integrals can be evaluated easily by choosing () fzin convenient form as shown below.
Proof of (2.2):In order to prove the result (2.2), we first set () In the equation (2.1) and evaluate the resulting integral as follows: First express 1 H in the double contour integral form (1.7), interchange the order integration which is permissible due to absolute convergence of the integrals thus involved in the process, and evaluate the z - integral with the help of (1.24) after using the property (1.26).On interpreting the result thus obtained by virtue of (1.7) we arrive at the right hand side of (2.2).Proof of (2.3):In order to prove the result (2.3), we first set In the result (2.1) and evaluate the resulting integral as follows: First express the spheroidal function in the expanded form (1.21), change the order of integration and summation which is justified due to the uniform convergence of the series representing spheroidal functions.By expressing the Bessel function thus involved in the form of H -function using the result (1.25), we interpret the 1 H -function in the contour integral form (1.7).Again, change the order of integration by virtue of De La Valle Poussin's well known theorem ( [2],p.504)due to absolute convergence of the integrals thus involved.Then, evaluating the inner integral by virtue of result (1.24) and inverting the double contour integrals by definition (1.7), we get the required result (2.3).
Regarding the convergence of the series on the right hand side of (2.3) it would be worth mentioning that the ratio , and the ratio of gammas involving r (even or odd) ([6],p.47( 4)),hence the series is uniformly and absolutely convergent by M-test.

Special Cases
Since the H -function of two variables and spheroidal functions emerge many higher transcendental functions and polynomials, a large number of new and interesting results follow as special cases but we record here only a few of them, for the lack of space.
, the poles of gamma functions of the numerator in (1.10) are converted to the branch points.
the H -function of two variables reduces to H -function of two variables due to [15].

1 
 , whereas the prime over the summation sigh indicates that the summation is taken over only even or odd values of k as n is even or odd.A recursion relationship ([23], (9)) for determining the , k as in (1.21) and the eigen-values () hence it follows by M-test that the series in (1.21) is absolutely and uniformly convergent.Moreover it represents a continuous function for all cz .
and the function f is so prescribed that the such that () cz remains finite, (2.3) reduces to the following new result by virtue of (1.22):