Fractional Calculus and Analytic Continuation of the Complex Fourier-Jacobi Transform

Abstract

By using the Riemann-Liouville type fractional integral operators we shall reduce the complex Fourier-Jacobi transforms of even functions on $\mathbf{R}$ to the Euclidean Fourier transforms. As an application of the reduction formula, Parseval's formula and an inversion formula of the complex Jacobi transform are easily obtained. Moreover, we shall introduce a class of even functions, not $C^\infty$ and not compactly supported on $\mathbf{R},$ whose transforms have meromorphic extensions on the upper half plane.