### A confluent hypergeometric integral equation

E. R. Love, T. R. Prabhakar, N. K. Kashyap
1982 Glasgow Mathematical Journal
Introduction. Recently there have appeared papers ([7], [8]; also see [9] ) in which integral equations with kernels involving the confluent hypergeometric function i F i ( a ; c ; z ) = Aw:^' where {a)n= have been studied. These equations are mainly Volterra equations of the first kind except that they have infinite domain (0, °°). The rest are of the related type with integrals over (x, °°) instead of (0, x); and all are convolution equations. The equation solved in this paper is a Fredholm
more » ... per is a Fredholm equation of the first kind except for infinite domain: where / is the unknown function and the parameters a and c have positive real parts. Formally the relationship of this equation to those in [7] and [8] is similar to that of the equation in [5] to those in [3] and [4]. However, the equations in [3], [4] and [5] have Gauss's hypergeometric function 2 Fi in place of the confluent function. Preliminary work on the Weyl fractional integral and derivative is set out in § §2 and 3. This augments the treatments given in [4] and [6], neither of which is adequate for the present purpose. Weyl Fractional Integrals. We use the customary definition D n J"f{x) = J v D n f{x). Proof, (i) For fixed [a, ft]c(0, oo) ; f is continuous in [a, b + l]; so Ir-VCx + OI^Mr 1 -1 for a<x<b and 0 < f < l . Glasgow Math. J. 23 (1982) 31^0.