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, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence, , defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, , which also satisfy a characteristic ordinary generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex-valued inputs. We consider the "convolution polynomial" variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the articles given in the references.

Definition and examples[edit]

For nonnegative integers k, the Stirling polynomials, Sk(x), are a Sheffer sequence for [1] defined by the exponential generating function

The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials) [2] each with exponential generating function

given by the relation .

The first 10 Stirling polynomials are:


Yet another variant of the Stirling polynomials is considered in [3] (see also the subsection on Stirling convolution polynomials below). In particular, the article by I. Gessel and R. P. Stanley defines the modified Stirling polynomial sequences, and where are the unsigned Stirling numbers of the first kind, in terms of the two Stirling number triangles for non-negative integers . For fixed , both and are polynomials of the input each of degree and with leading coefficient given by the double factorial term .


Special values include:

  • , where denotes Stirling numbers of the second kind. Conversely, ;
  • , where Bk are Bernoulli numbers under the convention B1 = −1/2;
  • ;
  • ;
  • ;
  • If and then we have that where are the Bernoulli numbers [4]
  • Similarly, we have that for integers , , 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

Stirling convolution polynomials[edit]

Definition and examples[edit]

Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article [5] and in the Concrete Mathematics reference. We first define these polynomials through the Stirling numbers of the first kind as

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

These Stirling "convolution" polynomials may be used to define the Stirling numbers, and , for integers and arbitrary complex values of . The next table provides several special cases of these Stirling polynomials for the first few .


Generating functions[edit]

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

More generally, if is a power series that satisfies , we have that

We also have the related series identity [6]

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

Properties and relations[edit]

For integers and , these polynomials satisfy the two Stirling convolution formulas given by


When , we also have that the polynomials, , are defined through their relations to the Stirling numbers

and their relations to the Bernoulli numbers given by

See also[edit]


  1. ^ See section 4.8.8 of The Umbral Calculus (1984) reference linked below.
  2. ^ See Norlund polynomials on MathWorld.
  3. ^ Gessel and Stanley (1978). "Stirling polynomials". J. of Combin. Theory Ser. A. 53: 24–33. 
  4. ^ Section 4.4.8 of The Umbral Calculus.
  5. ^ Knuth, D. E. (1992). "Convolution Polynomials" (PDF). Mathematica J. 2: 67–78.  The article contains definitions and properties of special convolution polynomial families defined by special generating functions of the form for . Special cases of these convolution polynomial sequences include the binomial power series, , so-termed tree polynomials, the Bell numbers, , and the Laguerre polynomials. For , the polynomials are said to be of binomial type, and moreover, satisfy the generating function relation for all , where is implicitly defined by a functional equation of the form . The article also discusses asymptotic approximations and methods applied to polynomial sequences of this type.
  6. ^ Section 7.4 of Concrete Mathematics.
  • Erdeli, A., Magnus, W. and Oberhettinger, F and Tricomi, F. G. Higher Transcendental Functions. Volume III:. New York. 
  • Graham, Knuth and Patashnik (1994). Concrete Mathematics: A Foundation for Computer Science. 
  • S. Roman (1984). The Umbral Calculus. 

External links[edit]