Definition:Stirling Numbers of the First Kind/Unsigned/Definition 2

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Unsigned Stirling numbers of the first kind are defined as the polynomial coefficients $\displaystyle {n \brack k}$ which satisfy the equation:

$\displaystyle x^{\underline n} = \sum_k \left({-1}\right)^{n - k} {n \brack k} x^k$

where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.


The notation $\displaystyle {n \brack k}$ for the unsigned Stirling numbers of the first kind is that proposed by Jovan Karamata and publicised by Donald E. Knuth.

The notation $\map s {n, k}$ for the signed Stirling numbers of the first kind is similar to variants of that sometimes given for the unsigned.

Usage is inconsistent in the literature.

Also see

Source of Name

This entry was named for James Stirling.

Technical Note

The $\LaTeX$ code for \(\displaystyle {n \brack k}\) is \displaystyle {n \brack k} .

The braces around the n \brack k are important.

The \displaystyle is needed to create the symbol in its proper house display style.