Orthogonality of Jacobi and Laguerre polynomials for general parameters via the Hadamard finite part

TL;DR

This paper extends the orthogonality properties of Jacobi and Laguerre polynomials to complex parameters using Hadamard finite parts, and explores their associated Riemann-Hilbert problems, generalizing classical results.

Contribution

It introduces a novel approach to establish orthogonality for complex parameters via Hadamard finite parts and derives the related Riemann-Hilbert problems.

Findings

Orthogonality established for complex parameters a,b.

Riemann-Hilbert problems for these polynomials are derived.

Results generalize classical orthogonality conditions.

Abstract

Orthogonality of the Jacobi and of Laguerre polynomials, P_n^(a,b) and L_n^(a), is established for a,b complex (a,b not negative integers and a+b different from -2,-3,...) using the Hadamard finite part of the integral which gives their orthogonality in the classical cases. Riemann-Hilbert problems that these polynomials satisfy are found. The results are formally similar to the ones in the classical case (when the real parts of a,b are greater than -1)