A simple combinatorial method for proving Jacobi's identity and its generalizations
In his work on theta functions, Jacobi found the famous triple product identity \[ (*)\quad \prod^{\infty}_{n=1}(1-q^{2n})(1+tq^{2n- 1})(1+t^{-1} q^{2n-1})=\sum^{\infty}_{n=-\infty}t^ n q^{n^ 2}, \] where q and t are complex numbers with \(| q| <1\) and \(t\neq 0\). Several proofs of (*) are known some of which are purely combinatorial, e.g. those by \textit{E. M. Wright} [J. London Math. Soc. 40, 55-57 (1965; Zbl 0125.025)], \textit{C. Sudler} [Proc. Edinb. Math. Soc., II. Ser. 15, 67-71 (1966; Zbl 0143.024)], \textit{J. Zolnowsky} [Discrete Math. 9, 293-298 (1974; Zbl 0292.10014)], and \textit{R. P. Lewis} [Am. Math. Mon. 91, 420-423 (1984; Zbl 0551.05016)]. In this paper another and very short proof of (*) is presented. The key result reads as follows. Let p(N) be the number of partitions of the nonnegative integer N, let \(Q_ r\) denote the set of pairs (\(\alpha\),\(\beta)\) of decreasing sequences \(\alpha =(\alpha_ 1,...,\alpha_ s)\), \(\beta =(\beta_ 1,...,\beta_ t)\) of odd positive integers such that \(s-t=r\), and let q(N,r) be the number of integers k with \(k=2N+r^ 2=\sum \alpha_ i+\sum \beta_ j\) for any arbitrary pair \((\alpha,\beta)\in Q_ r\). Then we have \(p(N)=q(N,r)\) for any N and r. This theorem is proved by means of a parametrization method the explanation of which will be omitted here. Surprisingly, this method enables an immediate generalization of (*) to an identity for formal power series over a noncommutative ring.
scientific article; zbMATH DE number 3974963