Stirling numbers of the first kind
This article is devoted to specifics of Stirling numbers of the first kind.
Identities linking the two kinds appear in the article on Stirling numbers in general.)
The original definition of Stirling numbers of the first kind was algebraic: they are the coefficients s(n, k) in the expansion of the falling factorial
into powers of the variable x:
The unsigned Stirling numbers may also be defined algebraically, as the coefficients of the rising factorial:
and 8 permutations of the form
The signs of the (signed) Stirling numbers of the first kind are predictable and depend on the parity of n − k. In particular,
The unsigned Stirling numbers of the first kind can be calculated by the recurrence relation
for n > 0.
It follows immediately that the (signed) Stirling numbers of the first kind satisfy the recurrence
Table of values
These values are easy to generate using the recurrence relation in the previous section.
Note that although
Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials.
Many relations for the Stirling numbers shadow similar relations on the binomial coefficients.
Expansions for fixed k
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.
In this form, the simple identities given above take the form
One may produce alternative forms for the Stirling numbers of the first kind with a similar approach preceded by some algebraic manipulation: since
This yields identities like
These relations can be generalized to give
where w(n, m) is defined recursively in terms of the generalized harmonic numbers by
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.
For all positive integer m and n, one has
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
The following congruence identity may be proved via a generating function-based approach:
then we may expand congruences for these Stirling numbers defined as the coefficients
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:
This series generalizes Hasse's series for the Hurwitz zeta-function (we obtain Hasse's series by setting k=1).
The next estimate given in terms of the Euler gamma constant applies:
Since permutations are partitioned by number of cycles, one has
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 (Section 26.8) include the following sums:
Additionally, if we define the second-order Eulerian numbers by the triangular recurrence relation
Other software packages for guessing formulas for sequences (and polynomial sequence sums) 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.
The Stirling number s(n,n-p) can be found from the formula
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
It can be proved by using mathematical induction.
- Stirling polynomials
- Stirling numbers
- Stirling numbers of the second kind
- Random permutation statistics
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Stirling numbers of the first kind.