Orthogonal polynomials of a generalized-even weight (Q1603082)

A nonnegative function \(\rho\) integrable on a system of two symmetric intervals \(E=[-\delta, -\gamma]\cup [\gamma, \delta]\) with \(0\leq \gamma< \delta\leq\infty\) is called of generalized-even weight if \[ \rho(-x)=\frac{x+\lambda}{x-\lambda} \rho(x); \qquad \rho(x)=0,\;x\notin E, \] for some \(\lambda\in [-\gamma, \gamma]\). \(\rho\) is an even function when \(\lambda=0\). The system of monic orthogonal polynomials with respect to a generalized-even weight is the main object under investigation. Under additional assumption (the logarithmic derivative of \(\rho\) is a rational function) some functional, algebraic and differential properties of such a system are obtained. The results are illustrated on two basic examples \[ \rho(x; \alpha,\beta,\gamma,\lambda)= |x+\lambda|(1-x^2)^\alpha (x^2-\gamma^2)^\beta, \qquad \alpha, \beta>-1, \;0\leq\gamma<1, \] on the set \(E=[-1, -\gamma]\cup [\gamma, 1]\), and \[ \rho(x; \alpha,\gamma,\lambda)= |x+\lambda|(x^2-\gamma^2)^\alpha e^{-(x^2-\gamma^2)}, \qquad \alpha>-1, \;0\leq\gamma<\infty. \] In particular, the coefficients of the three-term recurrence relations are computed explicitly.