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\(q\)-Rotations and Krawtchouk polynomials

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Abstract

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.

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Acknowledgments

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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Correspondence to Vincent X. Genest.

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V. X. Genest and L. Vinet were supported in part by NSERC.

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Genest, V.X., Post, S., Vinet, L. et al. \(q\)-Rotations and Krawtchouk polynomials. Ramanujan J 40, 335–357 (2016). https://doi.org/10.1007/s11139-015-9681-0

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