The signed Stirling numbers of the first kind 
, for nonnegative integers 
and 
,
are defined such that the number of permutations
of 
elements which contain exactly 
permutation cycles is
the nonnegative number
 |
(1)
|
This means that 
for 
and 
. A related set of numbers is the associated
Stirling numbers of the first kind. The unsigned counts 
and the associated counts are 
and 
, respectively. The general count 
counts permutations
whose 
permutation cycles all have length at least the
positive integer 
.
The signed Stirling numbers of the first kind are variously denoted 
(Riordan 1980, Roman 1984), 
(Fort 1948, Abramowitz and Stegun 1972), 
(Jordan 1950). Abramowitz and Stegun (1972, p. 822)
summarize the various notational conventions, which can be a bit confusing (especially
since an unsigned version 
is also in common use). The signed Stirling
number of the first kind 
is returned by StirlingS1[n,
m] in the Wolfram Language,
where they are denoted 
.
The triangle of signed Stirling numbers of the first kind is
 |
(2)
|
(OEIS A008275). Special values include (with 
in the last four formulas)
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  | ![1/2(-1)^(n-1)(n-1)![H_(n-1)^2-H_(n-1)^((2))]](/images/equations/StirlingNumberoftheFirstKind/Inline33.svg) |
(6)
|
 |  |  |
(7)
|
where 
is the Kronecker delta, 
is a harmonic number,
is a harmonic number of order 
, and 
is a binomial coefficient.
The generating function for the Stirling numbers of the first kind is
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
where 
is a falling factorial and 
is the rising factorial,
![sum_(k=m)^infty(s(k,m))/(k!)x^k=([ln(x+1)]^m)/(m!)](/images/equations/StirlingNumberoftheFirstKind/NumberedEquation3.svg) |
(12)
|
for 
(Abramowitz and Stegun 1972, p. 824) and
 |  |  |
(13)
|
 |  |  |
(14)
|
where the last expression is interpreted by its polynomial continuation at 
.
The Stirling numbers of the first kind satisfy the recurrence relation
 |
(15)
|
for 
and the sum identities
 |
(16)
|
for 
and
 |
(17)
|
for 
,
where 
is a binomial coefficient.
The Stirling numbers of the first kind 
are connected with the Stirling
numbers of the second kind 
. For example, the matrices 
and 
are inverses of
each other, where 
denotes the matrix with 
th entry 
for 
, ..., 
(G. Helms, pers. comm., Apr. 28, 2006).
Other formulas include
 |  |  |
(18)
|
 |  |  |
(19)
|
(Roman 1984, p. 67), as well as
 |
(20)
|
 |
(21)
|
 |
(22)
|
 |
(23)
|
A nonnegative (unsigned) version of the Stirling numbers gives the number of permutations of 
objects having 
permutation cycles (with
cycles in opposite directions counted as distinct) and is obtained by taking the
absolute value of the signed version. The nonnegative
Stirling numbers of the first kind are variously denoted
![S_1(n,m)=[n; m]=|s(n,m)|](/images/equations/StirlingNumberoftheFirstKind/NumberedEquation11.svg) |
(24)
|
(Graham et al. 1994). Diagrams illustrating 
, 
, 
, and 
(Dickau) are shown above.
The unsigned Stirling numbers of the first kind satisfy
 |
(25)
|
For a nonnegative integer 
and a positive real number 
,
they satisfy the exact identity
 |  |  |
(26)
|
 |  |  |
(27)
|
The displayed coefficient polynomials also give the beginning of an asymptotic series
for fixed noninteger 
as 
. The case 
gives a ratio of gamma functions 
(Gosper 1996).
." Grunert Archiv 65, 171-175,
1880.Butzer, P. L. and Hauss, M. "Stirling Functions of the
First and Second Kinds; Some New Applications." Israel Mathematical Conference
Proceedings: Approximation, Interpolation, and Summability, in Honor of Amnon Jakimovski
on his Sixty-Fifth Birthday (Ed. S. Baron and D. Leviatan). Ramat Gan,
Israel: IMCP, pp. 89-108, 1991.Carlitz, L. "On Some Polynomials
of Tricomi." Boll. Un. M. Ital. 13, 58-64, 1958.Carlitz,
L. "Note on Nörlund's [sic] Polynomial 
." Proc. Amer. Math. Soc. 11, 452-455,
1960.Comtet, L.