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In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, the Stirling numbers and the Bernoulli numbers.
Definition and examples
For nonnegative integers k, the Stirling polynomials Sk(x) are defined by the generating function equation
The first 10 Stirling polynomials are:
Special values include:
- , where denotes Stirling numbers of the second kind. Conversely, ;
- , where are Bernoulli numbers;
- , where are Stirling numbers of the first kind. They may be recovered by .
The sequence is of binomial type, since . Moreover, this basic recursion holds: .
Explicit representations involving Stirling numbers can be deduced with Lagrange's interpolation formula:
Here, are Laguerre polynomials.
These following relations hold as well:
where is the Stirling number of the second kind and
where is the Stirling number of the first kind.
By differentiating the generating function it readily follows that
Relations to other polynomials
The relation is given by .
- Erdeli, A., Magnus, W. and Oberhettinger, F and Tricomi, F. G. Higher Transcendental Functions. Volume III:. New York.