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.
WZ pair Basic hypergeometric series Rogers–Fine identity Euler’s identity q-Gauss sum
Mathematics Subject Classification
33F10 33D15 65B10
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Amdeberhan, T., Zeilberger, D.: Hypergeometric series acceleration via the WZ method. Electron. J. Combin. 4(2), R3 (1997)MathSciNetzbMATHGoogle Scholar