Expansion of Analytic Functions in Jacobi Series

A classical theorem on expansion of an analytic function in a series of Jacobi polynomials Pₙ(α ,β ) is extended so that α and β, instead of being assumed real, may have any complex values provided α + β ≠ - 2, - 3, - 4, ⋯. The function to be expanded may be analytic inside an arbitrary ellipse in the complex plane, since we use notation which does not fix the foci at $ -1 $ and 1. If the ellipse is a circle, Jacobi’s series reduces to Taylor’s series as a special case, not a limiting case. Cauchy’s inequality for the coefficients of Taylor’s series has an analogue for Jacobi's series. The coefficients are determined uniquely by the analytic function. Some special series are listed.