Asymptotics for thenth-degree Laguerre polynomial evaluated atn

We investigate the asymptotic behaviour of ? n (n),n?? where ? n (x) denotes the Laguerre polynomial of degreen. Our results give a partial answer to the conjecture ?? n (n)>1 forn>6, made in 1984 by van Iseghem. We also show the connection between this conjecture and the continued fraction approximants of $$6\sqrt {{3 \mathord{\left/ {\vphantom {3 \pi }} \right. \kern-\nulldelimiterspace} \pi }} $$ .

Further monotonicity and convexity properties of the zeros of cylinder functions

AbstractLet cvk be the kth positive zero of the cylinder function Cv(x,α)=Jv(x) cos α−Yv sin α, 0⩽α<π, where Jv(x) and Yv(x) are the Bessel functions of the first and the second kind, respectively. We prove that the function v(d2cvkddv2+δ)cvk increases with v⩾0 for suitable values of δ and k−απ⩾ 0.7070… . From this result under the same conditions we deduce, among other things, that cvk+12δv2 is convex as a function of v⩾0. Moreover, we show some monotonicity properties of the function c2vkv. Our results improve known results.

On the zeros of Jacobi polynomials