Two-parameter Turan inequalities for ultraspherical and Laguerre polynomials

AbstractIt is known that the classical orthogonal polynomials satisfy inequalities of the form Un2(x) − Un + 1(x) Un − 1(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if Pnλ(x) and Lnα(x) are the ultraspherical and Laguerre polynomials and Fnλ(x) = Pnλ(x)Pnλ(1), Gnα(x) = Lnα(x)Lnα(0), then Fnα(x) Fnβ(x) − Fn + 1α(x) Fn − 1β(x) > 0, − 1 < x < 1, −12 < α ⩽ β ⩽ α + 1 and Gnα(x) Gnβ(x) − Gn + 1α(x) Gn − 1β(x) > 0, x > 0, 0 < α ⩽ β ⩽ α + 1. We also prove the inequality (n + 1) Fnα(x) Fnβ(x) − nFn + 1α(x) Fn − 1β(x) > An[Fnα(x)]2, −1 < x < 1, −12 < α ⩽ β < α + 1, where An is a positive constant depending on α and β