A note on matrices over $\mathbb{Z}$ with entries stemming from binomial coefficients and from Catalan numbers once pure and once taken modulo $2$
Roswitha Hofer
TL;DR
This note investigates structural coincidences between the integer Pascal matrices $P_1$ and $P_2$ and their modulo $2$ reductions $M_1$ and $M_2$, alongside Hankel matrices $H_1$ and $H_2$ constructed from Catalan-number-based sequences. A central result is the multiplicativity $M_1(a)M_1(b)=M_1(a+b)$ derived via Lucas' theorem modulo $2$, together with determinant relations for submatrices that mirror classical Pascal-matrix properties and inform low-discrepancy sequence constructions modulo primes. The Hankel analysis shows that $H_1$ and $H_2$ have unimodular principal minors and admits an $LDU$ decomposition (notably for $H_2$), while the associated formal Laurent series $\mathcal L_1$ and $\mathcal L_2$ admit continued fraction expansions linked to the paperfolding sequence. Together, these results connect determinant properties, 0-1 Hankel structures, and low-discrepancy/normal-number constructions, and they identify open questions about extending to higher-dimensional $(t,s)$-sequences and related conjectures in the area.
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.