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A family of WZ pairs and q-identities

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Abstract

We observe that \((F(n+k+1,k)+G(n+k,k), G(n+k,k))\) is a WZ pair provided that (F(n, k), G(n, k)) is a WZ pair. This observation enables us to construct a bilateral sequence of WZ pairs starting from a single WZ pair. As an application, we give a one-line proof of the Rogers–Fine identity. Moreover, combing this observation and the q-WZ method for infinite series, we are able to derive a series of identities from a single q-identity. We illustrate this approach by Euler’s identity and the q-Gauss sum.

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  1. College of Science, Tianjin University of Technology, Tianjin, 300384, People’s Republic of China

    Yan-Ping Mu

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  1. Yan-Ping Mu
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Correspondence to Yan-Ping Mu.

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Supported by the National Natural Science Foundation of China (Grants 11471244 and 11771330).

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Mu, YP. A family of WZ pairs and q-identities. Ramanujan J 49, 97–104 (2019). https://doi.org/10.1007/s11139-018-0077-9

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  • DOI: https://doi.org/10.1007/s11139-018-0077-9

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