# Pascal Eigenspaces and Invariant Sequences of the First or Second Kind

## TL;DR

This paper investigates invariant sequences of the first and second kinds, exploring their properties and relationships through Pascal matrices and eigenspaces, providing a comprehensive review and new insights.

## Contribution

It introduces a unified framework for analyzing invariant sequences of both kinds using Pascal matrix eigenspaces and explores their interconnections.

## Key findings

- Characterization of invariant sequences via Pascal eigenspaces
- Relationships between sequences of the first and second kinds
- New insights into the structure of Pascal-type matrices

## Abstract

An infinite real sequence $\{a_n\}$ is called an invariant sequence of the first (resp., second) kind if $a_n=\sum_{k=0}^n {n \choose k} (-1)^k a_k$ (resp., $a_n=\sum_{k=n}^{\infty} {k \choose n} (-1)^k a_k$). We review and investigate invariant sequences of the first and second kinds, and study their relationships using similarities of Pascal-type matrices and their eigenspaces.