Stirling polynomials

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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[edit]

For nonnegative integers k, the Stirling polynomials Sk(x) are defined by the generating function equation

The first 10 Stirling polynomials are:

k
0
1
2
3
4
5
6
7
8
9

Properties[edit]

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[edit]

Closely related to Stirling polynomials are Nørlund polynomials (or generalized Bernoulli polynomials) with generating function

The relation is given by .

See also[edit]

References[edit]

  • Erdeli, A., Magnus, W. and Oberhettinger, F and Tricomi, F. G. Higher Transcendental Functions. Volume III:. New York. 

External links[edit]