Orthogonal Polynomials Associated with Singular Integral Equations Having a Cauchy Kernel

Abstract

The Chebyshev polynomials of both the first and second kind are of fundamental importance when considering the particular case of the singular integral equation $a(t)\phi (t) + ({{b(t)} / \pi })\lambda \int_{ - 1}^1 {({{\phi (\tau )} / {(\tau - t)}})} d\tau = f(t), - 1 < t < 1$, in which $a \equiv 0$ and $b \equiv - 1$ on $[ - 1,1]$. We identify the two sets of orthogonal polynomials which play a corresponding role for the singular integral equation with general a, b and consider some of the relationships between these two sets of polynomials.