Stirling polynomials

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Definition and examples Stirling polynomials_section_0

The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials) each with exponential generating function Stirling polynomials_sentence_0

The first 10 Stirling polynomials are given in the following table: Stirling polynomials_sentence_1

Properties Stirling polynomials_section_1

Stirling polynomials_unordered_list_0

  • Special values:Stirling polynomials_item_0_0

Stirling polynomials_unordered_list_1

Stirling polynomials_unordered_list_2

  • The following relations hold as well:Stirling polynomials_item_2_2

Stirling polynomials_unordered_list_3

  • By differentiating the generating function it readily follows thatStirling polynomials_item_3_3

Stirling convolution polynomials Stirling polynomials_section_2

Definition and examples Stirling polynomials_section_3

Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article and in the Concrete Mathematics reference. Stirling polynomials_sentence_2

We first define these polynomials through the Stirling numbers of the first kind as Stirling polynomials_sentence_3

It follows that these polynomials satisfy the next recurrence relation given by Stirling polynomials_sentence_4

Generating functions Stirling polynomials_section_4

This variant of the Stirling polynomial sequence has particularly nice ordinary generating functions of the following forms: Stirling polynomials_sentence_5

We also have the related series identity Stirling polynomials_sentence_6

and the Stirling (Sheffer) polynomial related generating functions given by Stirling polynomials_sentence_7

Properties and relations Stirling polynomials_section_5

and Stirling polynomials_sentence_8

and their relations to the Bernoulli numbers given by Stirling polynomials_sentence_9

See also Stirling polynomials_section_6

Stirling polynomials_unordered_list_4

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: polynomials.