Orthonormal polynomials with generalized Freud-type weights Original Research Article

We consider a certain generalized Freud-type weight WrQ2(x)=|x|2r exp(−2Q(x)), where r>−12 and Q : R→R is even and continuous, Q′ is continuous, Q′>0 in (0,∞), and Q″ is continuous in (0,∞). Furthermore, Q satisfies further conditions. Recently, Levin and Lubinsky have studied the sequence of orthonormal polynomials {Pn(WQ2;x)}n=0∞ with the Freud weight WQ2(x)=exp(−2Q(x)) on R, and then they have obtained many interesting properties of Pn(WQ2;x) [LL1]. We investigate the properties of Pn(WrQ2;x), which contain extensions of Levin and Lubinskyʹs results and improvements of Bauldryʹs results [Ba1,LL1].