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Orthogonal polynomials and Smith normal form

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Abstract

Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt–Stanley results for the Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers, q-Schröder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q-double factorials, as well as generating functions for permutations with eight statistics.

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Author notes

  1. Alexander R. Miller

    Present address: Fakultät für Mathematik, Universität Wien, Vienna, Austria

Authors and Affiliations

  1. Centre Émile Borel, Institut Henri Poincaré, Paris, France

    Alexander R. Miller

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Dennis Stanton

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  1. Alexander R. Miller
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  2. Dennis Stanton
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Correspondence to Dennis Stanton.

Additional information

Communicated by A. Constantin.

A. R. Miller was supported in part by the Fondation Sciences Mathématiques de Paris.

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Miller, A.R., Stanton, D. Orthogonal polynomials and Smith normal form. Monatsh Math 187, 125–145 (2018). https://doi.org/10.1007/s00605-017-1082-6

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