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Franklin’s argument proves an identity of Zagier. (English) Zbl 0962.05011

Electron. J. Comb. 7, No. 1, Research paper R54, 5 p. (2000); printed version J. Comb. 7, No. 2 (2000).
The following remarkable identity of D. Zagier (to appear in Topology) \[ \sum^\infty_{n= 0} [(q)_\infty- (q)_n]= (q)_\infty \sum^\infty_{k= 1}{q^k\over 1- q^k}+ \sum^\infty_{r= 1}(-1)^r [(3r- 1) q^{r(3r- 1)/2}+ 3rq^{r(3r+ 1)/2}] \] is proved by a slight modification of Franklin’s well-known involution proof of Euler’s pentagonal number theorem.

MSC:

05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions
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Coefficients of q in series expansion of Zagier’s identity.