# A bound on the Laguerre polynomials

Studia Mathematica (1991)

• Volume: 100, Issue: 2, page 169-181
• ISSN: 0039-3223

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## Abstract

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We give the following bounds on Laguerre polynomials and their derivatives (α ≥ 0): $\genfrac{}{}{0pt}{}{|{t}^{k}{d}^{p}\left({L}_{n}^{\alpha }\left(t\right){e}^{-t/2}\right)|\le {2}^{-min\left(\alpha ,k\right)}{4}^{k}\left(n+1\right)...\left(n+k\right)\left(n+p+max\left(\alpha -k,0\right)}{n\right)}$ 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.

## How to cite

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Duran, Antonio. "A bound on the Laguerre polynomials." Studia Mathematica 100.2 (1991): 169-181. <http://eudml.org/doc/215880>.

@article{Duran1991,
abstract = {We give the following bounds on Laguerre polynomials and their derivatives (α ≥ 0): $|t^k d^p (L_n^α(t) e^\{-t/2\})| ≤ 2^\{-min(α,k)\} 4^k(n + 1)...(n + k) (\{n + p + max(α - k, 0)\} \atop \{n\})$ 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.},
author = {Duran, Antonio},
journal = {Studia Mathematica},
keywords = {Laguerre polynomials; bounds for polynomials},
language = {eng},
number = {2},
pages = {169-181},
title = {A bound on the Laguerre polynomials},
url = {http://eudml.org/doc/215880},
volume = {100},
year = {1991},
}

TY - JOUR
AU - Duran, Antonio
TI - A bound on the Laguerre polynomials
JO - Studia Mathematica
PY - 1991
VL - 100
IS - 2
SP - 169
EP - 181
AB - We give the following bounds on Laguerre polynomials and their derivatives (α ≥ 0): $|t^k d^p (L_n^α(t) e^{-t/2})| ≤ 2^{-min(α,k)} 4^k(n + 1)...(n + k) ({n + p + max(α - k, 0)} \atop {n})$ 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.
LA - eng
KW - Laguerre polynomials; bounds for polynomials
UR - http://eudml.org/doc/215880
ER -

## References

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1. [1] A. J. Duran, The analytic functionals in the lower half plane as a Gel'fand-Shilov space, Math. Nachr. (to appear). Zbl0796.46026
2. [2] A. J. Duran, The Stieltjes moments problem for rapidly decreasing functions, Proc. Amer. Math. Soc. 107 (1989), 731-741. Zbl0676.44007
3. [3] A. J. Duran, Laguerre expansions of tempered distributions and generalized functions, J. Math. Anal. Appl. 150 (1990), 166-180. Zbl0715.46017
4. [4] A. Erdélyi (ed.), Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York 1953.
5. [5] A. Erdélyi (ed.), Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York 1953. Zbl0051.30303
6. [6] I. M. Gel'fand et G. E. Shilov, Les distributions, Vol. 2, Dunod, Paris 1964.
7. [7] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, New York 1959.

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