Definition: Factorial Polynomials

Let $n\ge 0$ be a natural number $a_0,a_1,\ldots,a_n\in\mathbb C$ be complex numbers. A factorial polynomial of degree $n$ is a function $\phi:\mathbb C\to\mathbb C$ defined using the falling factorial powers by $$\phi(x)=a_nx^{\underline{n}}+a_{n-1}x^{\underline{n-1}}+\ldots+a_1x^{\underline{1}}+a_0$$ for some $a_n\neq 0.$

  1. Proposition: Factorial Polynomials have a Unique Representation
  2. Proposition: Factorial Polynomials vs. Polynomials

Definitions: 1
Proofs: 2 3 4
Propositions: 5 6
Solutions: 7
Theorems: 8

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  1. Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960