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XXV.—A Surface Integral and Symmetric Functions.

Published online by Cambridge University Press:  14 February 2012

Henry Jack
Henry Jack
Affiliation:
University of Dundee
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Synopsis

In a previous paper (Jack 1970, § 5) the pre-zonal polynomials were defined by the surface integral (1) below. In that paper they were evaluated using a series of differential type operators acting on the sums of powers of the roots of a matrix.

The present paper, although it is a continuation of Jack (1970), is self-contained and consists of three parts. The first part, §§ 3–7, shows how the pre-zonal polynomials may be calculated, certainly for low degree, using only the first of the differential operators. The next part, §§ 8–11, is devoted to establishing various recurrence relations between the pre-zonal polynomials and to conjecturing similar recurrence relations between the zonal polynomials (James 1964, p. 478). In the final part, §§ 12–14 the pre-zonal polynomials are evaluated directly from the integral (1) below, in terms of the augmented monomial symmetric functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 69 , Issue 4 , 1972 , pp. 347 - 364
Copyright
Copyright © Royal Society of Edinburgh 1972

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References

References to Literature

David, F. N., Kendall, M. G. and Barton, D. E., 1966. Symmetric Function and Allied Tables. Cambridge U.P.Google Scholar
Jack, H., 1965. ‘A generalisation of Dirichlet's multiple integral’, Proc. Edinb. Math. Soc, 14, 233–237.CrossRefGoogle Scholar
Jack, H., 1970. ‘A class of symmetric polynomials with a parameter’, Proc. Roy. Soc. Edinb., 69A, 1–18.Google Scholar
James, A. T., 1964. ‘Distributions of matrix variâtes and latent roots derived from normal samples’, Ann. Math. Statist., 35, 475–501.CrossRefGoogle Scholar
Littlewood, D. E., 1950. The Theory of Group Characters and Matrix Representation of Groups, 2nd Ed.Oxford U.P.Google Scholar
Muir, T., 1920. The Theory of Determinants, III. London: Macmillan.Google Scholar
Ostrowski, A., 1956–1957. ‘Uber die Darstellung von Symmetrischen Funktionen durch Potenzsummen’, Math. Annin, 132, 362–372.CrossRefGoogle Scholar