A bound on the Laguerre polynomials

Antonio J. Duran

ArticleOriginal scientific text

Title

A bound on the Laguerre polynomials

Authors ¹

Affiliations

Departamento de Análisis Matemático, Universidad de Sevilla, Apartado 1160, 41080 Sevilla, Spain

Abstract

We give the following bounds on Laguerre polynomials and their derivatives (α ≥ 0):

| t^{k} d^{p} (L_{n}^{α} (t) e^{- \frac{t}{2}}) | \leq 2^{- min (α, k)} 4^{k} (n + 1) ... (n + k) ({n + p + max (α - k, 0)} a ⊤ {n})

for all natural numbers k, p, n ≥ 0 and t ≥ 0. Also, we give (as the main result of this paper) a technique to estimate the order in k and p in bounds similar to the previous ones, which will be used to see that the estimate on k and p in the previous bounds is sharp and to give an estimate on k and p in other bounds on the Laguerre polynomials proved by Szegö.

Keywords

ENG

Laguerre polynomials

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A. Erdélyi (ed.), Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York 1953.
A. Erdélyi (ed.), Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York 1953.
I. M. Gel'fand et G. E. Shilov, Les distributions, Vol. 2, Dunod, Paris 1964.
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, New York 1959.

Title

A bound on the Laguerre polynomials

Affiliations

Abstract

Keywords

Bibliography