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Expansions of falling and rising factorials

Stirling numbers express coefficients in expansions of falling and rising factorials (also known as the Pochhammer symbol) as polynomials.

That is, the falling factorial, defined as (x) n = x (x − 1) ⋯ (x − n + 1) , {\displaystyle \ (x)_{n}=x(x-1)\ \cdots (x-n+1)\,} is a polynomial in x of degree n whose expansion is

with (signed) Stirling numbers of the first kind as coefficients.

Note that (x) 0 ≡ 1 , {\displaystyle \ (x)_{0}\equiv 1\,} by convention, because it is an empty product. The notations x n _ {\displaystyle \ x^{\underline {n}}\ } for the falling factorial and x n ¯ {\displaystyle \ x^{\overline {n}}\ } for the rising factorial are also often used. (Confusingly, the Pochhammer symbol that many use for falling factorials is used in special functions for rising factorials.)

Similarly, the rising factorial, defined as x (n) = x (x + 1) ⋯ (x + n − 1) , {\displaystyle \ x^{(n)}\ =\ x(x+1)\ \cdots (x+n-1)\,} is a polynomial in x of degree n whose expansion is

with unsigned Stirling numbers of the first kind as coefficients.One of these expansions can be derived from the other by observing that x (n) = (− 1) n (− x) n. {\displaystyle \ x^{(n)}=(-1)^{n}(-x)_{n}~.}

Stirling numbers of the second kind express the reverse relations:

and