Polynomial Sequences of Binomial Type Path Integrals

Abstract

Polynomial sequences $ p_n (x) $ of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express $ \mathbb{N} \times [-\pi, \pi] $ as a path integral in the "phase space" $ h (\phi) = \sum_{n=0}^{\infty} {P'}_n {(0)} / n! e^{in\phi} $ and it produces a Schrodinger 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 computation.