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Two Enumerative Proofs of an Identity of Jacobi

Published online by Cambridge University Press:  20 January 2009

C. Sudler Jr
C. Sudler Jr
Affiliation:
Los Angeles, California
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In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namely

Here, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 1 , June 1966 , pp. 67 - 71
Copyright
Copyright © Edinburgh Mathematical Society 1966

References

REFERENCES

(1)Dickson, L. E., History of the Theory of Numbers, II, New York (Chelsea, 1952).Google Scholar
(2)Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (3rd editionOxford, 1954)Google Scholar
(3)Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers (4th editionOxford, 1960)Google Scholar
(4)Macmahon, P. A.Combinatory Analysis, II (Cambridge, 1916).Google Scholar
(5)Sylvester, J. J.A constructive theory of partitions, Collected Papers, IV (Cambridge, 1912), 1–83.Google Scholar
(6)Vahlen, K. T.Beitrage zu einer additiven Zahlentheorie, J. fiir Math. 112 (1893), 1–36.Google Scholar
(7)Wright, E. M., An enumerative proof of an identity of Jacobi, J. London Math. Soc. 40 (1965), 55–57.CrossRefGoogle Scholar