N- FOURIER SERIES EQUATIONS INVOLVING JACOBI POLYNOMIALS OF DIFFERENT INDICES

: In this paper, we have considered the N-Fourier series equations involving Jacobi polynomials of different indices of the first and second kind and solved the two sets of series equations.


INTRODUCTION
Dwivedi and Trivedi [2] Considered quadruple series equations involving Jacobi polynomials of the same indices which are orthogonal to the weight function in the interval (0,1) Szego [9] later on standardized the notation for the Jacobi polynomials. Askey [1] remarked that the dual series equations involving Jacobi polynomial of different indices cannot be solved. Later on Dwivedi and Gupta [4] obtained the solution of such quadruple series equations which include dual and triple series as particular cases. If we review the literature then we observe that the existing solutions on series equations are derived only from dual to six Fourier series equations, no further generalizations are available till date. This tempted us to find the solution of n-Fourier series equations involving Jacobi polynomials of different indices and in this paper we have obtained certain results. By considering the special values of n = 2,3,4,5,6 we shall be able to derive solutions of dual, triple,quadruple,5-tuple and 6-tuple Fourier series equations involvingjacobi polynomials of different indices [5], [6]. (1) where, and =0.
(2) where, and = ∞ Here n is taken as an even number. If n is odd then the equations will be =0.

2) N-series equations of the second kind
N-series equations of the second kind involving jacobi polynomials of different indices are as follows : (5) where, (6) where, Here also n is taken as an even number. If n is odd then the equations will be (7) where, and =0.
is an arbitrary non-negative integer.
, where i = 1,3,5,....,n-1 and where are prescribed functions. and are unknown coefficients, are determined and the parameters satisfy the conditions , .
Here we solve only equations (1),(2)of first kind and equations (5),(6) of the first kind and equations (7),(8) of the second kind will follow easily.

PRELIMINARY RESULTS
In the course of analysis, we shall use the following results: (i) The orthogonality relation for the Jacobi polynomials (9) where is the Kronecker delta, (ii) The series , (10) where, and (11) It is assumed that parameters are so constrained that is independent of n, this is of course possible when, for instance = ν = λμ` and = .
where, and where are unspecified functions. Using orthogonality relation it follows from equations (1) and (12) .
Substituting this value of in equation (2) This equation is an Abel type integralequation and its solution is given by where (20) Changing the order of integration of the last integral of equation (26), we get . (21) Using these equations, And (23) Equation (24) is also Abel type integral equation. Therefore its solution is given by for all Therefore, for all Applying the above result in equation (24) and also applying the Leibnitz theorem we get  . .
Substituting this value of in equation (5) and interchanging the order of integration and summation, we get Equation (45) is also Abel type integral equation. Therefore its solution is given by for all Therefore, for all Applying the above result in equation (45) and also applying the Leibnitz theorem, we get This equation can be written as, (t, y) dt (49) Where, (t, y) = . (50) Substituting in equation (49) we will get n/2 simultaneous Fredholm Integral equations of the second kind. With the help of these n/2 simultaneous equations we can calculate and then the values of can be determined .After all these calculations we can compute the coefficient with the help of equation (33).