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The positive-definiteness of the complete symmetric functions of even order

Published online by Cambridge University Press:  24 October 2008

D. B. Hunter
D. B. Hunter
Affiliation:
University of Bradford
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Extract

An important role in the classical theory of symmetric functions of a real n-tuple x = (x1, x2, …, xn) is played by the complete symmetric functions or homogeneous product sums hr defined by the generating function

(see Littlewood (5), p. 82). In an earlier paper (4) I conjectured that h2r is positive definite. The main object of the present paper is to prove this conjecture in a rather sharper form.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 2 , September 1977 , pp. 255 - 258
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Chisholm, J. S. R. Convergence properties of Padé approximants. Proceedings of a conference on Padé approximants and their applications. University of Kent, 1972.Google Scholar
(2)Flatto, L. and Gross, B.On the group of linear transformations leaving a polynomial invariant. Amer. Math. Monthly 74 (1967), 942–947.CrossRefGoogle Scholar
(3)Gillespie, R. P.Partial differentiation (Edinburgh: Oliver and Boyd, 1951).Google Scholar
(4)Hunter, D. B.Some properties cf orthogonal polynomials. Math. Comp. 29 (1975), 559–565.CrossRefGoogle Scholar
(5)Littlewood, D. E.A University algebra, 2nd edn. (London: Heinemann, 1958).Google Scholar
(6)Marcus, M. and Pierce, S.Symmetric positive definite multilinear functionals with a given automorphism. Pacific J. Math. 31 (1969), 119–132.CrossRefGoogle Scholar
(7)Panik, M. J.Classical optimization: foundations and extensions (Amsterdam: North-Holland Publishing Co., 1976).Google Scholar
(8)Pierce, S.Orthogonal groups of positive definite multilinear functionals. Pacific J. Math. 33 (1970), 183–189.CrossRefGoogle Scholar