Two-parameter Turan inequalities for ultraspherical and Laguerre polynomials

Abstract

It is known that the classical orthogonal polynomials satisfy inequalities of the form U n 2( x) − U n + 1 ( x) U n − 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 P n λ ( x) and L n α ( x) are the ultraspherical and Laguerre polynomials and F n λ(x) = P n λ(x) P n λ(1) , G n α(x) = L n α(x) L n α(0) , then F n α(x) F n β(x) − F n + 1 α(x) F n − 1 β(x) > 0, − 1 < x < 1, − 1 2 < α ⩽ β ⩽ α + 1 and G n α(x) G n β(x) − G n + 1 α(x) G n − 1 β(x) > 0, x > 0, 0 < α ⩽ β ⩽ α + 1 . We also prove the inequality (n + 1) F n α(x) F n β(x) − nF n + 1 α(x) F n − 1 β(x) > A n[F n α(x)] 2, −1 < x < 1, − 1 2 < α ⩽ β < α + 1, where A n is a positive constant depending on α and β.