ON THE SINGULARITIES OF THE CONTINUOUS JACOBI TRANSFORM WHEN a + ß = 0

Let a,ß > -1 and Pla'3)(x) = (l-x)a(l+x)ßp[a'a){x), where Pjf (x) is the Jacobi function of the first kind, A > —(a + ß + l)/2, and -1 < i < 1. Let r the integral exists. It is known that for a + •>-Ä4r'<",')(*-i),£Ä<-> *J o whenever ß = 0, we have /(*) = lim 4 f fC« (x l) PÍÍ$A-*)\ . . r2(A + i/2) X sin 7rA^-:-L_¡-d\ w in J, r(A-ra + l/2)F(A + ^ + l/2) almost everywhere in [—1,1], In this paper, we devise a technique to continue f(x) analytically to the complex 2-plane and locate the singularities of f(z) by relating them to the singularities of (¿A lit) -r. Jo -\tF(a,0), T(A + a + 1) ' However, this will be done in the more general case where the limit in (*) exists in the sense of Schwartz distributions and defines a generalized function f(x). In this case, we pass from f(x) to its analytic representation /(*) = 7T~ (/Mi-)> ^^supp/, 27TI \ X — Z I and then relate the singularities of f(z) to those of g(t).