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'' $\Space{N}{} \times {[-π,π]}$. The Hamiltonian is $h(\varphi )={\sum }_{n=0}^{\infty }{p}_n^{\prime }(0)/n!e^{in\varphi }$ and it produces a Schrödinger 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, Schrödinger equation, propagator, wave function, cumulants, quantum computation.

16 pages; LaTeX; no pictures; Complete revision on 19.10.2001

Cited by in corpus (3)

• A Treatise on Quantum Clifford Algebras
• Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle
• Tokens: An Algebraic Construction Common in Combinatorics, Analysis, and Physics