0000000000411983
AUTHOR
Andrea Laforgia
Monotonicity properties of zeros of generalized Airy functions
We show, among other things, that the positive zeros of a solution ofy ″+x α y=0,y(0)=0 decrease to 1 asα increases, 0〈α〈∞.
Upper bounds for the zeros of ultraspherical polynomials
AbstractFor k = 1, 2, …, [n2] let xnk(λ) denote the Kth positive zero in decreasing order of the ultraspherical polynomial Pn(λ)(x). We establish upper bounds for xnk(λ). All the bounds become exact when λ = 0 and, in some cases (see case (iii) of Theorem 3.1), also when λ = 1. As a consequence of our results, we obtain for the largest zero xn1(λ)0.. We point out that our results remain useful for large values of λ. Numerical examples show that our upper bounds are quite sharp.
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 $$6\sqrt {{3 \mathord{\left/ {\vphantom {3 \pi }} \right. \kern-\nulldelimiterspace} \pi }} $$ .
Bounds for Bessel functions
We establish lower and upper bounds for the Bessel functionJ v (x) and the modified Bessel functionI v(x) of the first kind. Our chief tool is the differential equation satisfied by these functions.