Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T07:00:38.005Z Has data issue: false hasContentIssue false
  • English
  • Français

A Family of Combinatorial Identities

Published online by Cambridge University Press:  20 November 2018

G. E. Andrews ,
M. V. Subbarao  and
M. Vidyasagar
G. E. Andrews
Affiliation:
Pennsylvania State University, University park, Pennsylvania
M. V. Subbarao
Affiliation:
Pennsylvania State University, University park, Pennsylvania
M. Vidyasagar
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts
Rights & Permissions [Opens in a new window]

Extract

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.

In a recent paper, Murray Eden [5] generalized the simple identity for the Eulerian product,

1.1

and obtained the following infinite family of identities:

For A= 1,2, 3,…, let

1.2

where we assume throughout that |x| < 1, empty products equal unity and empty sums equal zero; then

1.3

As Eden noted, Fh(b;x) is the generating function of ph(m, n) which denotes the number of partitions of n into m parts, in which the largest part appears exactly h times and all other parts are distinct:

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 11 - 18
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Andrews, G. E., Generalizations of the Durfee Square, J. London Math. Soc. (to appear).Google Scholar
2. Andrews, G. E., q-identities of Auluck, Carlitz, and Rodgers, Duke Math. J. 33 (1966), 575-582.Google Scholar
3. Bellman, R., A brief introduction to theta functions, Holt, New York, 1961.Google Scholar
4. Cauchy, , Collected works (1) VIII, p. 48.Google Scholar
5. Eden, Murray, A note on a new family of identities, J. Comb. Theory 5 (1968), 210-211.Google Scholar
6. Hardy, and Ramanujan, , Asymptotic formula in combinatorial analysis, Proc. Londo. Math. Soc. 2 XVII (1918), 75-115; No. 36, Collected Papers of S. Ramanujan, Cambridge Univ. Press.Google Scholar
7. Hardy, and Wright, , Theory of numbers, Oxford Univ. Press, London, 4th ed., 1965.Google Scholar
8. Heine, E., Handbuch der Kugelfunktionen, (1) Berlin, 1878.Google Scholar
9. Slater, L. J., Generalized hypergeometric functions, Cambridge Univ. Press, New York, 1966.Google Scholar
10. Rademacher, H., Lectures on analytic number theory, Tata Institute of Fundamental Research, Bombay, 1954-55.Google Scholar