Orthogonal polynomials and Smith normal form

作者：Miller Alexander R; Stanton Dennis^*

来源：Monatshefte fur Mathematik, 2018, 187(1): 125-145.

DOI：10.1007/s00605-017-1082-6

• Orthogonal polynomials
• Smith normal form

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-Schroder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q-double factorials, as well as generating functions for permutations with eight statistics.

• 出版日期2018-9