Orthogonal polynomials and Smith normal form

Alexander R. Miller, Dennis Stanton

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)125-145
Number of pages21
JournalMonatshefte fur Mathematik
Issue number1
StatePublished - Sep 1 2018

Bibliographical note

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

Publisher Copyright:
© 2017, Springer-Verlag GmbH Austria.


  • Orthogonal polynomials
  • Smith normal form


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