A REPRESENTATION OF JACOBI FUNCTIONS

Recently, the continuous Jacobi transform and its inverse are defined and studied in [1] and [2]. In the present work, the transform is used to derive a series representation for the Jacobi functions P λ ( α , β ) ( x ) , − ½ ≤ α , β ≤ ½ , α + β = 0 , and λ ≥ − ½ . The case α = β = 0 yields a representation for the Legendre functions and has been dealt with in [3]. When λ is a positive integer n , the representation reduces to a single term, viz., the Jacobi polynomial of degree n .


Introduction.
The continuous Jacobi transform and its inverse were introduced and studied in [i] and [2] These transforms generalize the work of Butzer, Stens and Wehrens [3] on the continuous Legendre transform and the work of Debnath [4] on the discrete Jacobl transform.
In [2] an application to sampling technique was given.In the present work, the continuous Jacobl transform is used to derzve a representation of Jacobi functions P')(x).The representa- series tion includes that for the Legendre function given in [3].When 1 is a positive integer, the representation reduces to the Jacobi polynomial (see e.g.[5]) 2.
Preliminaries.In this section we review material needed in the development of the paper.

3.
Derivation of the Representation Formula.
The series representation that we will develop, in this section, (,-) for P (x) is In order to derive (3.1) and (3.2), we shall first introduce an auxillary function k(x;h), apply (2.7), (2.9) to k(x;h) and utilize the uniqueness of the Jacobi transform.
i l-t) ed f(x;) (i t x Lemma 3.2. For -% a %, (a 0) we have The series converges for all x such that l--x < i; 0 x < i.When x =0, that is, if We rewrite f(x; as f (x;) I 0 l_t) ed I I l-t e (t + (-) dt J(x;e) + f(0,e), say.