Interlacing of zeros of orthogonal polynomials under modification of the measure (Q741094)

The authors investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure \(d \mu(x)\), supported on the interval \((a,b)\) and the other with respect to the measure \(| x - c |^\tau | x - d |^\gamma d\mu(x)\), where \(c\) and \(d\) are outside \((a,b)\). They prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either \(0<\tau\), \(\gamma\leq 1\) or \(\gamma=0\) and \(0<\tau\leq 2\). This result is inspired by an open question of Richard Askey and it generalizes recent results on some families of orthogonal polynomials. Moreover, the authors obtain further statements on the interlacing of the zeros of Jacobi, Laguerre, Meixner, Hahn, Askey-Wilson orthogonal polynomials.