Stirling numbers of the first kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations.

[1] The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices.

These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted

Alfréd Rényi observed that the unsigned Stirling number of the first kind

[2] It follows immediately that the signed Stirling numbers of the first kind satisfy the recurrence We prove the recurrence relation using the definition of Stirling numbers in terms of rising factorials.

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.

Using the Kronecker delta one has, and Also and Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials.

The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences.

For example, a permutation of n elements with n − 3 cycles must have one of the following forms: The three types may be enumerated as follows: Sum the three contributions to obtain Note that all the combinatorial proofs above use either binomials or multinomials of

Since the Stirling numbers are the coefficients of a polynomial with roots 0, 1, ..., n − 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, ..., n − 1.

from the following formal power series (see the non-exponential Bell polynomials and section 3 of [7]).

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.

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 Additionally, if we define the second-order Eulerian numbers by the triangular recurrence relation [10] 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

[11] The following congruence identity may be proved via a generating function-based approach:[12] 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.

: Section 6.2 of the reference cited above provides more explicit expansions related to these congruences for the

A variety of identities may be derived by manipulating the generating function (see change of basis): Using the equality it follows that and This identity is valid for formal power series, and the sum converges in the complex plane for |z| < 1.

Other identities arise by exchanging the order of summation, taking derivatives, making substitutions for z or u, etc.

There also exist more complicated expressions for the zeta-functions involving the Stirling numbers.

[15][16] The next estimate given in terms of the Euler gamma constant applies:[17] For fixed

we have the following estimate : It is well-known that we don't know any one-sum formula for Stirling numbers of the first kind.

The nth derivative of the μth power of the natural logarithm involves the signed Stirling numbers of the first kind:

Infinite series involving the finite sums with the Stirling numbers often lead to the special functions.

where γm are the Stieltjes constants and δm,0 represents the Kronecker delta function.

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.

, 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 this sense, the form of the Stirling numbers of the first kind may also be generalized by this parameterized super-recurrence for fixed scalars