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Jacobi’s triple product identity and the quintuple product identity. (English) Zbl 1183.33030

Summary: The simplest proof of Jacobi’s triple product identity originally due to Cauchy (1843) and Gauss (1866) is reviewed. In the same spirit, we prove by means of induction principle and finite difference method, a finite form of the quintuple product identity. Similarly, the induction principle will be used to give a new proof of another algebraic identity due to V. J. W. Guo and J. Zeng [J. Math. Anal. Appl. 327, No. 1, 310–325 (2007; Zbl 1106.33017)], which can be considered as another finite form of the quintuple product identity.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11B65 Binomial coefficients; factorials; \(q\)-identities

Citations:

Zbl 1106.33017
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