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Proposition: Factorial Polynomials have a Unique Representation

A factorial polynomial $\phi(x)$ of degree $n$ has a unique representation $$\phi(x)=a_nx^{\underline{n}}+a_{n-1}x^{\underline{n-1}}+\ldots+a_1x^{\underline{1}}+a_0,\quad a_n\neq 0,$$ that is, if $\phi(x)$ has some other representation $$\begin{align}\phi(x)&=b_{n+k}x^{\underline{n+k}}+b_{n+k-1}x^{\underline{n+k-1}}+\ldots+b_{n+1}x^{\underline{n+1}}+\nonumber\\
&\quad+b_nx^{\underline{n}}+b_{n-1}x^{\underline{n-1}}+\ldots+b_1x^{\underline{1}}+b_0\nonumber\end{align},$$ then $$b_{n+k}=b_{n+k-1}=\cdots=b_{n+1}=0$$ and $$b_r=a_r,\quad r=0,1,\ldots,n.$$

| | | | | created: 2020-03-30 22:11:48 | modified: 2020-03-30 22:16:28 | by: bookofproofs | references: [8404]

1.Proof: (related to "Factorial Polynomials have a Unique Representation")

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Bibliography (further reading)

[8404] Miller, Kenneth S.: “An Introduction to the Calculus of Finite Differences And Difference Equations”, Dover Publications, Inc, 1960