A note on matrices over $\mathbb{Z}$ with entries stemming from binomial coefficients and from Catalan numbers once pure and once taken modulo $2$

TL;DR

This paper explores matrices over integers derived from binomial coefficients and Catalan numbers, examining their properties and similarities to Pascal matrices, especially when entries are considered modulo 2, with implications for number theory and counterexamples in algebraic conjectures.

Contribution

It analyzes the properties of matrices formed from binomial coefficients and Catalan numbers, both pure and modulo 2, highlighting their similarities to Pascal matrices and their relevance in number theory.

Findings

Identifies common features of Pascal and Hankel matrices over integers and modulo 2.

Provides insights into matrices with 0/1 entries relevant to algebraic conjectures.

Connects properties of these matrices to applications in number theory and counterexamples.

Abstract

The Pascal matrix, which is related to Pascal's triangle, appears in many places in the theory of uniform distribution and in many other areas of mathematics. Examples are the construction of low-discrepancy sequences as well as normal numbers or the binomial transforms of Hankel matrices. Hankel matrices which are defined by Catalan numbers and related to the paperfolding sequence are interesting objects in number theory. Therefore, matrices that share many properties with the Pascal matrix or such Hankel matrices are of interest. In this note we will collect common features of the Pascal matrix and the same modulo $2$ as well as the Hankel matrix defined by Catalan numbers once pure and once modulo $2$ in the ring of integers. Hankel matrices with only $0$ and $1$ entries in e.g. finite fields gave recently access to counterexamples to the so-called $X$-adic Liouville conjecture. This justifies as well as motivates our consideration of further matrices with $0$ and $1$ entries.