Jacobi Series for General Parameters via Hadamard Finite Part and Application to Nonhomogeneous Hypergeometric Equations

Abstract EN

The representation of analytic functions as convergent series in Jacobi polynomials Pn(α,β) is reformulated using the Hadamard principal part of integrals for all α,β∈C∖{0,-1,-2,…}, α+β≠-2,-3,….

The coefficients of the series are given as usual integrals in the classical case (when Rα,Rβ>-1) or by their Hadamard principal part when they diverge.

As an application it is shown that nonhomogeneous differential equations of hypergeometric type do generically have a unique solution which is analytic at both singular points in C.