Abstract
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.
Highlights
The purpose of this note is to provide a unified and easy-touse formalism of expansion of analytic functions in Jacobi series for general parameters. Such expansions are useful in approximation theory, as well as in a variety of other areas, such as the study of differential equations and non-self-adjoint spectral problems which appear in the study of stability of nonlinear partial differential equations
We summarize below a few well known facts about Jacobi polynomials Pn(α,β)(z), n = 0, 1, 2, . . . for α, β > −1
The Jacobi polynomials are determined by the condition of mutual orthogonality on the interval [−1, 1] with respect to the weight
Summary
The purpose of this note is to provide a unified and easy-touse formalism of expansion of analytic functions in Jacobi series for general parameters. 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, .
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