A family of WZ pairs and q-identities

  • Yan-Ping MuEmail author


We observe that \((F(n+k+1,k)+G(n+k,k), G(n+k,k))\) is a WZ pair provided that (F(nk), G(nk)) 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 



  1. 1.
    Amdeberhan, T., Zeilberger, D.: Hypergeometric series acceleration via the WZ method. Electron. J. Combin. 4(2), R3 (1997)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andrews, G.: Two theorems of Gauss and allied identities proved arithmetically. Pac. J. Math. 41(3), 563–578 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, W.Y.C., Xia, E.X.W.: The \(q\)-WZ method for infinite series. J. Symb. Comput. 44(8), 960–971 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Fine, N.J.: Basic Hypergeometric Series and Applications. American Mathematical Society, Providence (1988)CrossRefGoogle Scholar
  5. 5.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  6. 6.
    Gessel, I.M.: Finding identities with the WZ method. J. Symb. Comput. 20(5), 537–566 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Habil, E.D.: Double sequences and double series. Islamic Univ. J. Series Nat. Stud. Eng. 14(1), 1–32 (2006)Google Scholar
  8. 8.
    Koornwinder, T.H.: On Zeilbergers algorithm and its \(q\)-analogue. J. Comput. Appl. Math. 48, 91–111 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. Lond. Math. Soc. 2(1), 315–336 (1917)CrossRefGoogle Scholar
  10. 10.
    Wilf, H.S., Zeilberger, D.: Rational functions certify combinatorial identities. J. Am. Math. Soc. 3(1), 147–158 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zeilberger, D.: Closed form (pun intended!). Contemp. Math. 143, 579–579 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of ScienceTianjin University of TechnologyTianjinPeople’s Republic of China

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