Jacobi series for general parameters and applications

2016/06/08 by Costin, Rodica D., David, Marina
#42C10. Secondary: 35A25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary: 33C45

paper · doi:10.48550/arxiv.1606.02642

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.

Related

• Explicit Formulas for the Associated Jacobi Polynomials and Some Applications
• A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the hypergeometric function
• Analogies of Jacobi's formula
• Multi-integral representations for Jacobi functions of the first and second kind
• Heine, Hilbert, Pade, Riemann, and Stieltjes: a John Nuttall's work 25 years later