Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T05:19:16.779Z Has data issue: false hasContentIssue false
  • English
  • Français

A new proof of the Macdonald identities for An−1

Part of: Combinatorics Basic hypergeometric functions

Published online by Cambridge University Press:  09 April 2009

Shaun Cooper
Shaun Cooper
Affiliation:
Department of Mathematics Massey UniversityAlbany AucklandNew Zealand e-mail: s.cooper@massey.ac.nz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new, elementary proof of the Macdonald identities for An−1 using induction on n is given. Specifically, the Macdonald identity for An is deduced by multiplying the Macdonald identity for An−1 and n Jacobi triple product identities together.

MSC classification

Secondary: 33D70: Other basic hypergeometric functions and integrals in several variables 05A19: Combinatorial identities, bijective combinatorics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 3 , June 1997 , pp. 345 - 360
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Aitken, A. C., Determinants and matrices, 9th edition (Oliver and Boyd, New York, 1959).Google Scholar
[2]Andrews, G. E., ‘A simple proof of Jacobi's triple product identity’, Proc. Amer. Math. Soc. 16 (1965), 333–334.Google Scholar
[3]Andrews, G. E., q-series: their development and applications in analysis, number theory, combinatorics, physics and computer algebra, CBMS Regional Conf. Ser. in Math. 66 (American Math. Soc., Providence, 1986).Google Scholar
[4]Carlitz, L. and Subbarao, M. V., ‘A simple proof of the quintuple product identity’, Proc. Amer. Math. Soc. 32 (1972), 42–44.CrossRefGoogle Scholar
[5]Andrews, G. E., ‘On a combinatorial proof of Winquist and its generalization’, Duke Math. J. 39 (1972), 165–172.Google Scholar
[6]Cooper, S., On the Macdonald identities, a conjecture of Forrester and a functional equation (Ph. D. thesis, University of Wisconsin-Madison, 1995).Google Scholar
[7]Dyson, F. J., ‘Missed Opportunities’, Bull. Amer. Math. Soc. 78 (1972), 635–652.CrossRefGoogle Scholar
[8]Garvan, F. G., ‘The crank of partitions mod 8, 9 and 10’, Trans. Amer. Math. Soc. 322 (1990), 79–94.Google Scholar
[9]Gasper, G. and Rahman, M., Basic hypergeometric series (Cambridge University Press, Cambridge, 1990).Google Scholar
[10]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 5th. edition (Clarendon Press, Oxford, 1979).Google Scholar
[11]Hirschhorn, M. D., ‘A generalisation of Winquist's identity and a conjecture of Ramanujan’, J. Indian Math. Soc. 51 (1987), 49–55.Google Scholar
[12]Hirschhorn, M. D., ‘A generalisation of the quintuple product identity’, J. Austral. Math. Soc. (Series A) 44 (1988), 42–45.CrossRefGoogle Scholar
[13]Macdonald, I. G., ‘Affine root systems and Dedekind's ηfunction’, Invent. Math. 15 (1972), 91–143.CrossRefGoogle Scholar
[14]Milne, S. C., ‘An elementary proof of the Macdonald identities for A l(1)’, Adv. Math. 57 (1985), 34–70.CrossRefGoogle Scholar
[15]Stanton, D., ‘An elementary approach to the Macdonald identities’, in: q-series and partitions, IMA Vol. Math. Appl, 18 (Springer, New York, 1989) pp. 139–149.CrossRefGoogle Scholar