Stirling Number - Notation

Notation

Several different notations for the 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 large S. The 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 the Stirling numbers without their signs, Common notations are:

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

for the unsigned Stirling numbers of the first kind, and

for the 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 the binomial coefficients, was introduced in 1935 by Jovan Karamata and promoted later by Donald Knuth. (The bracket notation conflicts with a common notation for the 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.