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Probabilistic proofs of asymptotic formulas for some classical polynomials

Published online by Cambridge University Press:  24 October 2008

Makoto Maejima  and
Walter Van Assche
Makoto Maejima
Affiliation:
Department of Mathematics, Keio University, Yokohama 223, Japan
Walter Van Assche
Affiliation:
Departement Wiskunde, Katholieke Universiteit Leuven, B-3030 Leuven, Belgium
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Extract

Recently Stadje [9] gave asymptotic formulas for some Bessel functions by using a probabilistic argument: some Bessel functions can be represented by the probability distribution of partial sums of symmetrized independent and identically distributed random variables (Feller [4], p. 149). Stadje used this idea and the local limit theorem, which is well known in probability theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 499 - 510
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Chihara, T. S.. An Introduction to Orthogonal Polynomials (Gordon & Breach, 1978).Google Scholar
[2]Erdélyi, A.. Asymptotic forms for Laguerre polynomials. J. Indian Math. Soc. 24 (1960), 235–250.Google Scholar
[3]Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.. Higher Transcendental Functions, vol. II (McGraw-Hill, 1953).Google Scholar
[4]Feller, W.. An Introduction to Probability Theory and its Applications, vol. II, 2nd ed. (Wiley, 1971).Google Scholar
[5]Moecklin, E.. Asymptotische Entwicklungen der Laguerreschen Polynome. Comment. Math. Helv. 7 (1934), 24–46.Google Scholar
[6]Petrov, V. V.. Sums of Independent Random Variables (Springer, 1975).Google Scholar
[7]Plancherel, M. and Rotach, W.. Sur les valeurs asymptotiques des polynomes d'Hermite. Comment. Math. Helv. 1 (1929), 227–254.Google Scholar
[8]Riordan, J.. An Introduction to Combinatorial Analysis (Wiley, 1958).Google Scholar
[9]Stadje, W.. Probabilistic proofs of some formulae for Bessel functions. Indag. Math. 45 (1983), 343–359.Google Scholar
[10]Szegö, G.. Orthogonal Polynomials. Amer. Math. Soc. Coll. Publ. 23 (1939).Google Scholar
[11]Taylor, W. C.. A complete set of asymptotic formulas for the Whittaker function and the Laguerre polynomials. J. Math. Phys. (M.I.T.) 18 (1939), 34–49.Google Scholar