Stirling numbers

In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in the 18th century. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind.

Several different notations for Stirling numbers are in use. Stirling numbers of the first kind are written with a small s, and those of the second kind with a capital S. Stirling numbers of the second kind are never negative, but those of the first kind can be negative; hence, there are notations for the "unsigned Stirling numbers of the first kind", which are Stirling numbers without their signs. Common notations are:

for ordinary (signed) Stirling numbers of the first kind,

for unsigned Stirling numbers of the first kind, and

for Stirling numbers of the second kind.

Abramowitz and Stegun use an uppercase S and a blackletter S, respectively, for the first and second kinds of Stirling number. The notation of brackets and braces, in analogy to binomial coefficients, was introduced in 1935 by Jovan Karamata and promoted later by Donald Knuth. (The bracket notation conflicts with a common notation for Gaussian coefficients.) The mathematical motivation for this type of notation, as well as additional Stirling number formulae, may be found on the page for Stirling numbers and exponential generating functions.

The Stirling numbers of the first kind are the coefficients in the expansion

where ${\displaystyle (x)_{n}}$ (a Pochhammer symbol) denotes the falling factorial,

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