Stirling numbers of the first kind


In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles.

Definitions

The original definition of Stirling numbers of the first kind was algebraic: they are the coefficients s in the expansion of the falling factorial
into powers of the variable x:
For example,, leading to the values,, and.
Subsequently, it was discovered that the absolute values |s| of these numbers are equal to the number of permutations of certain kinds. These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted or. They may be defined directly to be the number of permutations of n elements with k disjoint cycles. For example, of the permutations of three elements, there is one permutation with three cycles, three permutations with two cycles and two permutations with one cycle. Thus,, and. These can be seen to agree with the previous calculation of for.
The unsigned Stirling numbers may also be defined algebraically, as the coefficients of the rising factorial:
The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources.

Further example

The image at right shows that : the symmetric group on 4 objects has 3 permutations of the form
and 8 permutations of the form

Signs

The signs of the Stirling numbers of the first kind are predictable and depend on the parity of. In particular,

Recurrence relation

The unsigned Stirling numbers of the first kind can be calculated by the recurrence relation
for, with the initial conditions
for n > 0.
It follows immediately that the Stirling numbers of the first kind satisfy the recurrence

Table of values

Below is a triangular array of unsigned values for the Stirling numbers of the first kind, similar in form to Pascal's triangle. These values are easy to generate using the recurrence relation in the previous section.
0123456789
01---------
101--------
2011-------
30231------
4061161-----
50245035101----
6012027422585151---
7072017641624735175211--
805040130681313267691960322281-
904032010958411812467284224494536546361

Properties

Simple identities

Note that although
and
Also
and
Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials. Many relations for the Stirling numbers shadow similar relations on the binomial coefficients. The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences. Generalizations of the Stirling numbers of both kinds to arbitrary complex-valued inputs may be extended through the relations of these triangles to the Stirling convolution polynomials.

Other relations

Expansions for fixed ''k''

Since the Stirling numbers are the coefficients of a polynomial with roots 0, 1,...,, one has by Vieta's formulas that
In other words, the Stirling numbers of the first kind are given by elementary symmetric polynomials evaluated at 0, 1,...,. In this form, the simple identities given above take the form
and so on.
One may produce alternative forms for the Stirling numbers of the first kind with a similar approach preceded by some algebraic manipulation: since
it follows from Newton's formulas that one can expand the Stirling numbers of the first kind in terms of generalized harmonic numbers. This yields identities like
where Hn is the harmonic number and Hn is the generalized harmonic number
These relations can be generalized to give
where w is defined recursively in terms of the generalized harmonic numbers by
For fixed these weighted harmonic number expansions are generated by the generating function
where the notation means extraction of the coefficient of from the following formal power series.
More generally, sums related to these weighted harmonic number expansions of the Stirling numbers of the first kind can be defined through generalized zeta series transforms of generating functions.
One can also "invert" the relations for these Stirling numbers given in terms of the -order harmonic numbers to write the integer-order generalized harmonic numbers in terms of weighted sums of terms involving the Stirling numbers of the first kind. For example, when the second-order and third-order harmonic numbers are given by
More generally, one can invert the Bell polynomial generating function for the Stirling numbers expanded in terms of the -order harmonic numbers to obtain that for integers

Factorial-related sums

For all positive integer m and n, one has
where is the rising factorial. This formula is a dual of Spivey's result for the Bell numbers.
Other related formulas involving the falling factorials, Stirling numbers of the first kind, and in some cases Stirling numbers of the second kind include the following:

Inversion relations and the Stirling transform

For any pair of sequences, and, related by a finite sum Stirling number formula given by
for all integers, we have a corresponding inversion formula for given by
These inversion relations between the two sequences translate into functional equations between the sequence exponential generating functions given by the Stirling transform as
and
The differential operators and are related by the following formulas for all integers :
Another pair of "inversion" relations involving the Stirling numbers relate the forward differences and the ordinary derivatives of a function,, which is analytic for all by the formulas

Congruences

The following congruence identity may be proved via a generating function-based approach:
More recent results providing Jacobi-type J-fractions that generate the single factorial function and generalized factorial-related products lead to other new congruence results for the Stirling numbers of the first kind.
For example, working modulo we can prove that
and working modulo we can similarly prove that
More generally, for fixed integers if we define the ordered roots
then we may expand congruences for these Stirling numbers defined as the coefficients
in the following form where the functions,, denote fixed
polynomials of degree in for each,, and :
Section 6.2 of the reference cited above provides more explicit expansions related to these congruences for the -order harmonic numbers and for the generalized factorial products,. In the previous examples, the notation denotes Iverson's convention.

Generating functions

A variety of identities may be derived by manipulating the generating function:
Using the equality
it follows that
Other identities arise by exchanging the order of summation, taking derivatives, making substitutions for z or u, etc. For example, we may derive:
and
or
and
where and are the Riemann zeta function and the Hurwitz zeta function respectively, and even evaluate this integral
where is the Gamma function. There also exist more complicated expressions for the zeta-functions involving the Stirling numbers. One, for example, has
This series generalizes Hasse's series for the Hurwitz zeta-function.

Asymptotics

The next estimate given in terms of the Euler gamma constant applies:
For fixed we have the following estimate as :
We can also apply the saddle point asymptotic methods from Temme's article to obtain other estimates for the Stirling numbers of the first kind. These estimates are more involved and complicated to state. Nonetheless, we provide an example.
In particular, we define the log gamma function,, whose higher-order derivatives are given in terms of polygamma functions as
where we consider the saddle point of the function to be the solution of when. Then for and the constants
we have the following asymptotic estimate as :

Finite sums

Since permutations are partitioned by number of cycles, one has
The identity
can be proved by the techniques on the page
Stirling numbers and exponential generating functions.
The table in section 6.1 of Concrete Mathematics provides a plethora of generalized forms of finite sums involving the Stirling numbers. Several particular finite sums relevant to this article include
Other finite sum identities involving the signed Stirling numbers of the first kind found, for example, in the NIST Handbook of Mathematical Functions include the following sums:
Additionally, if we define the second-order Eulerian numbers by the triangular recurrence relation
we arrive at the following identity related to the form of the Stirling convolution polynomials which can be employed to generalize both Stirling number triangles to arbitrary real, or complex-valued, values of the input :
Particular expansions of the previous identity lead to the following identities expanding the Stirling numbers of the first kind for the first few small values of :
Software tools for working with finite sums involving Stirling numbers and Eulerian numbers are provided by the utilities in Mathematica. Other software packages for guessing formulas for sequences involving Stirling numbers and other special triangles is available for both Mathematica and Sage and , respectively.
Furthermore, infinite series involving the finite sums with the Stirling numbers often lead to the special functions. For example
or
and
or even
where γm are the Stieltjes constants and δm,0 represents the Kronecker delta function.

Explicit formula

The Stirling number s can be found from the formula
where The sum is a sum over all partitions of p.
Another exact nested sum expansion for these Stirling numbers is computed by elementary symmetric polynomials corresponding to the coefficients in of a product of the form. In particular, we see that
Newton's identities combined with the above expansions may be used to give an alternate proof of the weighted expansions involving the generalized harmonic numbers already noted above.
Another explicit formula for reciprocal powers of n is given by the following identity for integers :
Notice that this last identity immediately implies relations between the polylogarithm functions, the Stirling number exponential generating functions given above, and the Stirling-number-based power series for the generalized functions.

Relations to natural logarithm function

The nth derivative of the μth power of the natural logarithm involves the signed Stirling numbers of the first kind:
where is the falling factorial, and is the signed Stirling number.
It can be proved by using mathematical induction.

Generalizations

There are many notions of generalized Stirling numbers that may be defined in a number of differing combinatorial contexts. In so much as the Stirling numbers of the first kind correspond to the coefficients of the distinct polynomial expansions of the single factorial function,, we may extend this notion to define triangular recurrence relations for more general classes of products.
In particular, for any fixed arithmetic function and symbolic parameters, related generalized factorial products of the form
may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of in the expansions of and then by the next corresponding triangular recurrence relation:
These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the f-harmonic numbers,.
One special case of these bracketed coefficients corresponding to allows us to expand the multiple factorial, or multifactorial functions as polynomials in .
The Stirling numbers of both kinds, the binomial coefficients, and the first and second-order Eulerian numbers are all defined by special cases of a triangular super-recurrence of the form
for integers and where whenever or. In this sense, the form of the Stirling numbers of the first kind may also be generalized by this parameterized super-recurrence for fixed scalars .