Equioscillatory property of the Laguerre polynomials

Abstract

We show that the functionis the orthonormal Laguerre polynomial of degree k and d m , d M are some approximations for the extreme zeros. As a corollary we obtain a very explicit, uniform in k and α, sharp upper bound on the Laguerre polynomials.

On the maximum value of a confluent hypergeometric function

“…When the parameters verify the so called oscillatory conditions a < 0 and b−a > 1, the estimate of 1 F 1 (a; b; x) is much more complicated and, to the author's knowledge, has not been studied in the literature, except for the case when a is a negative integer and b > 0 (see, for instance, [9,10,17,21,23], and references therein).…”

“…Suppose that a function y = y(x) satisfies on an interval I ⊂ R the differential equation (p y ) + q y = 0, (9) where p = p(x) > 0, q = q(x) > 0 and both p and q are continuous on that interval. Define Sonin's function by…”

Approximations for the Bessel and Airy functions with an explicit error term

“…For Bessel and Airy functions, as well as for Hermite polynomials (see [2]), the details of this program can be worked out in a quite routine way. For Jacobi and Laguerre polynomials it is a much more involved problem and the result is known only for oscillatory and transition regions [5][6][7]. It is worth noticing that despite the fact that it is rather a technical problem and we do have appropriate tools to tackle it (see, for example, Lemma 11 and Remark 1 below), one still needs a good deal of calculations to extend the bounds to the monotonicity region.…”

“…The sign of S = 2ab + b y 2 /b 2 depends only on a and b, which in many cases enables one to find the global maximum of |y|. The following approach was briefly described in [7]. We want to find an approximation of a solution of the differential equation…”

“…It also leads to very sharp approximation of the Airy function Ai(−x) and its zeros. We will need the following lemma given in [7].…”