Polynomial Sequences of Binomial Type and Path Integrals

Vladimir V. Kisil

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arXiv · math/9808040

Polynomial Sequences of Binomial Type and Path Integrals

Vladimir V. Kisil

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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 $h(ϕ)=\sum_{n=0}^\infty p_n'(0)/n! e^{inϕ}$ 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.

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BibTeX RIS

Vladimir V. Kisil. 2001-10-22. Polynomial Sequences of Binomial Type and Path Integrals. https://arxiv.org/abs/math/9808040

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