Orthogonal polynomials on two symmetric intervals (Q1582843)

Given \(E=[-b,-a] \cup [a,b]\subset \mathbb R\) and an even weight function \(\rho\) on \(E\) satisfying a number of additional assumptions, the author establishes several algebraic properties of the corresponding orthogonal polynomials, such as a differential and an integral equation and a Rodrigues-type formula. In the particular case of \[ \rho (x)=|x|(b^2-x^2)^\alpha (x^2-a^2)^\beta , \quad \max (\alpha,\beta)\geq -1/2, \quad \alpha, \beta >-1, \] it is proved that the orthogonal polynomial of even degree attains the maximum of its absolute value at an endpoint of \(E\).