Polynomial Sequences of Binomial Type and Path Integrals
Vladimir V. Kisil
Abstract
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)=Σn=0∞ pn'(0)/n! einϕ 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 computations. Keywords: Feynman path integral, umbral calculus, polynomial sequence of binomial type, token, Schrödinger equation, propagator, wave function, cumulants, quantum computation.
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