Inequalities concerning bessel functions and orthogonal polynomials

A n (x ) = P . ' ( x ) P,~-I ( x ) P,~+I ( x ) > 0 , n > 1, 1 < x < + 1 (1.1) Szeg6 remarks further that inequalities analogous to (1.1) hold also for the ultraspherical, Laguerre and Hermite polynomials. Thus for the ultraspherical polynomials P,,,~ (x), the analogue of (1.1) reads: &,,,a (x) ----F,,,x' (x) -F,,-1,a (x) F.+1,x (x) _>0, A> --1⁄2, n > 1, l < x ~ + 1, (1.2) where F~.~(x)----P,,.a(x): P,,x(1). Subsequently, Madhava Rao and Thiruvenkatachar 2 showed that an elementary proof of (1.1) may be obtained by merely finding lX,," (x), which is given by the elegant formula 2 A" (x) ---n (n + 1) P''~ (x). (1.3)