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A combinatorial proof of a positivity result

Published online by Cambridge University Press:  24 October 2008

J. Gillis
Affiliation:
Weizmann Institute, Rehovot, Israel
J. Kleeman
Affiliation:
Weizmann Institute, Rehovot, Israel

Extract

We shall be concerned with the Laguerre polynomials, defined by Some time ago Szegö (8) showed that for (a, b, c = 0, 1, …). It is easily deduced from this (cf. (5)) that for all A λ ≥ 3.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Askey, R. and Gasper, G.Certain rational functions whose power series have positive coefficients. Amer. Math. Monthly 79 (1972), 327.CrossRefGoogle Scholar
(2)Askey, R., Ismail, M. E. H. and Koornwinder, T.Weighted permutation problems and Laguerre polynomials. J. Combinatorial Theory, ser. A (in the press).Google Scholar
(3)Erdelyi, A. et al. Higher transcendental functions, vol. II (New York, 1953).Google Scholar
(4)Even, S. and Gillis, J.Derangements and Laguerre polynomials. Math. Proc. Cambridge Philos. Soc. 79 (1976), 135.Google Scholar
(5)Gillis, J.Integrals of products of Laguerre polynomials, SIAM J. Math. Anal. 6 (1975), 318.CrossRefGoogle Scholar
(6)Ismail, M. E. H. and Tamhanker, M. V.A combinatorial approach to some positivity problems. SIAM J. Math. Anal. (in the Press).Google Scholar
(7)MacMahon, P. A.Combinatory analysis, vol. I (Cambridge, 1915).Google Scholar
(8)Szegö, G.Über gewisse Potenzreihen mit lauter positiven Koeffizienten, Math. Z. 37 (1933), 674.Google Scholar