Euler’s Pentagonal Number Theorem and the Rogers-Fine Identity
George E. Andrews1 and Jordan Bell2
1Department of Mathematics, The Pennsylvania State University, University Park, State College, PA 16802, USA
2Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada
Annals of Combinatorics 16 (3) pp.311-420 September, 2012
AMS Subject Classification: 05A19, 05A30, 33D15
Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number theorem in his notebooks around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty of clearly presenting it with the notation at the time. In this paper we show that the method of Euler’s unpublished proof can be used to give a new proof of the celebrated Rogers-Fine identity.
Keywords: Rogers-Fine identity, pentagonal number theorem, q-series, Leonhard Euler

1. Andrews, G.E.: Two theorems of Gauss and allied identities proved arithmetically. Pacific J. Math. 41, 563–578 (1972)

2. Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam (1976)

3. Andrews, G.E.: Euler’s Pentagonal Number Theorem. Math. Mag. 56(5), 279–284 (1983)

4. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook: Part I. Springer, New York (2005)

5. Bell, J.: A summary of Euler’s work on the pentagonal number theorem. Arch. Hist. Exact Sci. 64(3), 301–373 (2010)

6. Bogolyubov, N.N., Mikhailov, G.K., Yushkevich, A.P. (eds.): Euler and Modern Science. The Mathematical Association of America, Washington, DC (2007)

7. Euler, L.: Principes de la trigonom´etrie sph´erique tir´es de la m´ethode des plus grands et plus petits. M´emoires de l’acad´emie des sciences de Berlin 9, 223–257 (1753)

8. Fine, N.J.: Basic Hypergeometric Series and Applications. American Mathematical Society, Providence, RI (1988)

9. Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. London Math. Soc. (2) 16, 315–336 (1917)

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