Pascal Eigenspaces and Invariant Sequences of the First or Second Kind

An infinite real sequence $\{a_n\}$ is called an invariant sequence of the first (resp., second) kind if $a_n={\sum }_{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)(-1{)}^k{a}_k$ (resp., $a_n={\sum }_{k=n}^{\mathrm{\infty }}\left(\genfrac{}{}{0px}{}{k}{n}\right)(-1{)}^k{a}_k$). 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.