Jacobi series for general parameters via Hadamard finite part and application to nonhomogeneous hypergeometric equations
Summary: The representation of analytic functions as convergent series in Jacobi polynomials \(P_n^{(\alpha, \beta)}\) is reformulated using the Hadamard principal part of integrals for all \(\alpha\), \(\beta\in\mathbb{C}\backslash \{0,-1,-2,\dots\}\), \(\alpha + \beta\neq -2,-3,\dots\). The coefficients of the series are given as usual integrals in the classical case (when \(\mathfrak{R} \alpha, \mathfrak{R} \beta > - 1)\) 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 \(\mathbb{C}\).
scientific article