Jacobi series for general parameters and applications

TL;DR

This paper develops a unified Jacobi series representation for analytic functions with complex parameters, extending classical integral formulas and applying the results to ensure unique analytic solutions of hypergeometric equations at singular points.

Contribution

It introduces a unified approach to Jacobi series for complex parameters and applies it to establish the existence and uniqueness of analytic solutions for hypergeometric equations.

Findings

Unified Jacobi series representation for complex parameters

Integral and Hadamard principal part formulas for coefficients

Existence and uniqueness of analytic solutions to hypergeometric equations

Abstract

Representation of analytic functions as convergent series in Jacobi polynomials $P_n^{(a,b)}$ is reformulated using a unified approach for almost all complex $a, b$. The coefficients of the series are given as usual integrals in the classical case (when $\Re a, \Re b >-1$), or by the Hadamard principal part of these integrals when they diverge. As an application it is shown that inhomogeneous hypergeometric equations do generically have a unique solution which is analytic at both singular points in the complex plane.