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An algebraic interpretation of the multivariate q-Krawtchouk polynomials

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Abstract

The multivariate quantum q-Krawtchouk polynomials are shown to arise as matrix elements of “q-rotations” acting on the state vectors of many q-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations, and recurrence relations. The extension to an arbitrary number of variables is presented.

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Notes

  1. The polynomials \(\kappa _{n}(x;p,N)\) agree with \(K_n(x;p,N)\) in [19, Sect. 9.11] up to a normalization factor. In this paper, the uppercase K are reserved for the multivariate polynomials.

  2. Let us note that in [7, 11], a different definition of Tratnik’s two-variable Krawtchouk polynomials was used. The aforementioned algebraic interpretation is valid for both definitions.

  3. The polynomials \(k_{n}(x; p,N;q)\) agree with \(K_n^{qtm}(q^x; p, N, q)\) of [19, Sect. 14.14] up to a normalization factor.

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Acknowledgments

The authors would like to thank A. Zhedanov for stimulating discussions.

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Correspondence to Vincent X. Genest.

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VXG is supported by a postdoctoral fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC). The research of LV is supported in part by NSERC.

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Genest, V.X., Post, S. & Vinet, L. An algebraic interpretation of the multivariate q-Krawtchouk polynomials. Ramanujan J 43, 415–445 (2017). https://doi.org/10.1007/s11139-016-9776-2

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