# Orthogonal polynomials and Smith normal form

## TL;DR

This paper proves and generalizes Smith normal form evaluations for various Hankel, Toeplitz, and Gram matrices related to q-analog combinatorial numbers using orthogonal polynomials and matrix analysis.

## Contribution

It introduces new proofs and generalizations of Smith normal forms for matrices associated with q-analog sequences, expanding the understanding of their algebraic structure.

## Key findings

- Proved Smith normal form evaluations for Hankel matrices of q-Catalan numbers.
- Generalized results to other matrices like Toeplitz and Gram matrices.
- Provided new algebraic insights into q-analog combinatorial sequences.

## 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-Schr\"oder numbers, q-Stirling numbers, q-matching numbers, q-factorials, q-double factorials, as well as generating functions for permutations with eight statistics.