The Ramanujan Journal

, Volume 40, Issue 2, pp 335–357 | Cite as

\(q\)-Rotations and Krawtchouk polynomials

  • Vincent X. GenestEmail author
  • Sarah Post
  • Luc Vinet
  • Guo-Fu Yu
  • Alexei Zhedanov


An algebraic interpretation of the one-variable quantum \(q\)-Krawtchouk polynomials is provided in the framework of the Schwinger realization of \(\fancyscript{U}_{q}(sl_{2})\) involving two independent \(q\)-oscillators. The polynomials are shown to arise as matrix elements of unitary “\(q\)-rotation” operators expressed as \(q\)-exponentials in the \(\fancyscript{U}_{q}(sl_{2})\) generators. The properties of the polynomials (orthogonality relation, generating function, structure relations, recurrence relation, difference equation) are derived by exploiting the algebraic setting. The results are extended to another family of polynomials, the affine \(q\)-Krawtchouk polynomials, through a duality relation.


\(q\)-Krawtchouk polynomials \(\fancyscript{U}_{q}(sl_{2})\) algebra \(q\)-Oscillator algebra 

Mathematics Subject Classification

33D45 16T05 



L.V. wishes to acknowledge the hospitality of the Shanghai Jiao Tong University where this research project was initiated. V.X.G and L.V. would like to acknowledge the support provided to them by the University of Hawai’i, where this research was completed. V.X.G. holds an Alexander-Graham-Bell fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC). The research of L.V. is supported in part by NSERC.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Vincent X. Genest
    • 1
    Email author
  • Sarah Post
    • 2
  • Luc Vinet
    • 1
  • Guo-Fu Yu
    • 3
  • Alexei Zhedanov
    • 4
  1. 1.Centre de recherches mathématiquesUniversité de MontréalMontrealCanada
  2. 2.Department of MathematicsUniversity of Hawai’iHonoluluUSA
  3. 3.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  4. 4.Donetsk Institute for Physics and TechnologyDonetskUkraine

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