A new proof of Euler’s pentagonal number theorem

A. David Christopher
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 1, Pages 49–52
DOI: 10.7546/nntdm.2018.24.1.49-52
Full paper (PDF, 163 Kb)

Details

Authors and affiliations

A. David Christopher
Department of Mathematics, The American College
Tamil Nadu – 625002, India

Abstract

A new proof of Euler’s pentagonal number theorem is obtained.

Keywords

• Partitions
• Euler’s pentagonal number theorem
• Jacobi’s triple product identity

2010 Mathematics Subject Classification

• Primary 05A17
• Secondary 11P81

References

1. Andrews, G. E. (1965) A Simple proof of Jacobi’s Triple Product Identity, Proceedings of the American Mathematical Society, 16, 2, 333–334.
2. Andrews, G. E. (1984) Generalised Frobenius partitions, Mem. Amer. Math Soc., 49, 301, iv+44 pp.
3. Cheema, M. S. (1964) Vector partitions and Combinatorial identities, Math. Comp., 18, 414–420.
4. Dyson, F. J. (1969) A new symmetry of partitions, J. Combin. Theory, 7, 56–61.
5. Euler, L. (1780) Evolution producti infiniti (1 – x)(1 – x²)(1 – x³)(1 – x⁴)(1 – x⁵)  etc. in seriem simplicem, Acta Academiae Sci entarum Imperialis petropolitinae 1780, 1783, 47–55.
6. Ewell, J. A. (1983) A Simple Proof of Fermat’s Two-Square Theorem, Amer. Math. Monthly, 90, 9, 635–637.
7. Franklin, F. (1881) Surle d´evelopment du produit infini (1 – x)(1 – x²)(1 – x³)(1 – x⁴)…, Comptes Rendus Acad. Sci. Paris, 92, 448–450.
8. Hirschhorn, M. D. (1985) A Simple Proof of Jacobi’s Two-Square Theorem, Amer. Math. Monthly, 92, 8, 579–580.
9. Sudler, C. (1966) Two enumerative proofs of an identity of Jacobi, Proc. Edinburgh. Math. Soc., 15, 67–71.
10. Wei, C., Gong, D. (2011) Euler’s pentagon number theorem implies Jacobi triple product identity, Integers, 11, 6, 811–814.
11. Wright, E. M. (1965) An enumerative proof of an identity of Jacobi, J. London Math. Soc., 40, 55–57.