Abstract :
We study ratio asymptotics, that is, existence of the limit of Pn+1(z)/Pn(z) (Pn= monic orthogonal polynomial) and the existence of weak limits of pn2 dμ (pn=Pn/||Pn||) as n→∞ for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z0 with Im(z0)≠0 implies dμ is in a Nevai class (i.e., an→a and bn→b where an,bn are the off-diagonal and diagonal Jacobi parameters). For μʹs with bounded support, we prove pn2 dμ has a weak limit if and only if lim bn, lim a2n, and lim a2n+1 all exist. In both cases, we write down the limits explicitly.
