Polynomial Sequences of Binomial Type and Path Integrals

Polynomial sequences pn(x) of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express pn(x) as a path integral in the “phase space” N× [−π, π]. The Hamiltonian is h(φ) = ∑n=0 p ′ n(0)/n!e inφ and it produces a Schrödinger type equation for pn(x). This establishes a bridge between enumerative combinatorics and quantum field theory. It also provides an algorithm for parallel quantum computation.