Binomial matrices

Every s x s matrix A yields a composition map acting on polynomials on ℝ^s. Specifically, for every polynomial p we define the mapping C_A by the formula (C_AP)(x) := p(Ax), x ∈ ℝ^s. For each nonnegative integer n, homogeneous polynomials of degree n form an invariant subspace for C_A. We let A⁽ⁿ⁾ be the matrix representation of C_A relative to the monomial basis and call A⁽ⁿ⁾ a binomial matrix. This paper studies the asymptotic behavior of A⁽ⁿ⁾ as n → ∞. The special case of 2 × 2 matrices A with the property that A² = I corresponds to discrete Taylor series and motivated our original interest in binomial matrices.