Polynomial Sequences of Binomial Type and Path Integrals

Abstract

Polynomial sequences $p_n(x)$ of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express $p_n(x)$ as a \emph{path integral} in the ``phase space'' $\Space{N}{} \times {[-\pi,\pi]}$. The Hamiltonian is $h(\phi)=\sum_{n=0}^\infty p_n'(0)/n! e^{in\phi}$ and it produces a Schr\"odinger type equation for $p_n(x)$. This establishes a bridge between enumerative combinatorics and quantum field theory. It also provides an algorithm for parallel quantum computations. Keywords: Feynman path integral, umbral calculus, polynomial sequence of binomial type, token, Schr\"odinger equation, propagator, wave function, cumulants, quantum computation.