# Stirling numbers of the first kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. Stirling numbers of the first kind_sentence_0

In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). Stirling numbers of the first kind_sentence_1

(The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. Stirling numbers of the first kind_sentence_2

Identities linking the two kinds appear in the article on Stirling numbers in general.) Stirling numbers of the first kind_sentence_4

## Definitions Stirling numbers of the first kind_section_0

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 Stirling numbers of the first kind_sentence_5

into powers of the variable x: Stirling numbers of the first kind_sentence_6

The unsigned Stirling numbers may also be defined algebraically, as the coefficients of the rising factorial: Stirling numbers of the first kind_sentence_7

### Further example Stirling numbers of the first kind_section_1

and 8 permutations of the form Stirling numbers of the first kind_sentence_8

### Signs Stirling numbers of the first kind_section_2

The signs of the (signed) Stirling numbers of the first kind are predictable and depend on the parity of n − k. In particular, Stirling numbers of the first kind_sentence_9

## Recurrence relation Stirling numbers of the first kind_section_3

The unsigned Stirling numbers of the first kind can be calculated by the recurrence relation Stirling numbers of the first kind_sentence_10

for n > 0. Stirling numbers of the first kind_sentence_11

It follows immediately that the (signed) Stirling numbers of the first kind satisfy the recurrence Stirling numbers of the first kind_sentence_12

## Table of values Stirling numbers of the first kind_section_4

Below is a triangular array of unsigned values for the Stirling numbers of the first kind, similar in form to Pascal's triangle. Stirling numbers of the first kind_sentence_13

These values are easy to generate using the recurrence relation in the previous section. Stirling numbers of the first kind_sentence_14

## Properties Stirling numbers of the first kind_section_5

### Simple identities Stirling numbers of the first kind_section_6

Note that although Stirling numbers of the first kind_sentence_15

and Stirling numbers of the first kind_sentence_16

Also Stirling numbers of the first kind_sentence_17

and Stirling numbers of the first kind_sentence_18

Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials. Stirling numbers of the first kind_sentence_19

Many relations for the Stirling numbers shadow similar relations on the binomial coefficients. Stirling numbers of the first kind_sentence_20

The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences. Stirling numbers of the first kind_sentence_21

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. Stirling numbers of the first kind_sentence_22

### Other relations Stirling numbers of the first kind_section_7

#### Expansions for fixed k Stirling numbers of the first kind_section_8

Since the Stirling numbers are the coefficients of a polynomial with roots 0, 1, ..., n − 1, one has by Vieta's formulas that Stirling numbers of the first kind_sentence_23

In other words, the Stirling numbers of the first kind are given by elementary symmetric polynomials evaluated at 0, 1, ..., n − 1. Stirling numbers of the first kind_sentence_24

In this form, the simple identities given above take the form Stirling numbers of the first kind_sentence_25

One may produce alternative forms for the Stirling numbers of the first kind with a similar approach preceded by some algebraic manipulation: since Stirling numbers of the first kind_sentence_26

it follows from Newton's formulas that one can expand the Stirling numbers of the first kind in terms of generalized harmonic numbers. Stirling numbers of the first kind_sentence_27

This yields identities like Stirling numbers of the first kind_sentence_28

These relations can be generalized to give Stirling numbers of the first kind_sentence_29

where w(n, m) is defined recursively in terms of the generalized harmonic numbers by Stirling numbers of the first kind_sentence_30

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. Stirling numbers of the first kind_sentence_31

#### Factorial-related sums Stirling numbers of the first kind_section_9

For all positive integer m and n, one has Stirling numbers of the first kind_sentence_32

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: Stirling numbers of the first kind_sentence_33

#### Inversion relations and the Stirling transform Stirling numbers of the first kind_section_10

These inversion relations between the two sequences translate into functional equations between the sequence exponential generating functions given by the Stirling (generating function) transform as Stirling numbers of the first kind_sentence_34

and Stirling numbers of the first kind_sentence_35

#### Congruences Stirling numbers of the first kind_section_11

The following congruence identity may be proved via a generating function-based approach: Stirling numbers of the first kind_sentence_36

then we may expand congruences for these Stirling numbers defined as the coefficients Stirling numbers of the first kind_sentence_37

### Generating functions Stirling numbers of the first kind_section_12

A variety of identities may be derived by manipulating the generating function: Stirling numbers of the first kind_sentence_38

Using the equality Stirling numbers of the first kind_sentence_39

it follows that Stirling numbers of the first kind_sentence_40

(This identity is valid for formal power series, and the sum converges in the complex plane for |z| < 1.) Stirling numbers of the first kind_sentence_41

Other identities arise by exchanging the order of summation, taking derivatives, making substitutions for z or u, etc. For example, we may derive: Stirling numbers of the first kind_sentence_42

and Stirling numbers of the first kind_sentence_43

or Stirling numbers of the first kind_sentence_44

and Stirling numbers of the first kind_sentence_45

This series generalizes Hasse's series for the Hurwitz zeta-function (we obtain Hasse's series by setting k=1). Stirling numbers of the first kind_sentence_46

### Asymptotics Stirling numbers of the first kind_section_13

The next estimate given in terms of the Euler gamma constant applies: Stirling numbers of the first kind_sentence_47

### Finite sums Stirling numbers of the first kind_section_14

Since permutations are partitioned by number of cycles, one has Stirling numbers of the first kind_sentence_48

The identity Stirling numbers of the first kind_sentence_49

can be proved by the techniques on the page Stirling numbers and exponential generating functions. Stirling numbers of the first kind_sentence_50

The table in section 6.1 of Concrete Mathematics provides a plethora of generalized forms of finite sums involving the Stirling numbers. Stirling numbers of the first kind_sentence_51

Several particular finite sums relevant to this article include Stirling numbers of the first kind_sentence_52

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: Stirling numbers of the first kind_sentence_53

Additionally, if we define the second-order Eulerian numbers by the triangular recurrence relation Stirling numbers of the first kind_sentence_54

Software tools for working with finite sums involving Stirling numbers and Eulerian numbers are provided by the utilities in Mathematica. Stirling numbers of the first kind_sentence_55

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. Stirling numbers of the first kind_sentence_56

Furthermore, infinite series involving the finite sums with the Stirling numbers often lead to the special functions. Stirling numbers of the first kind_sentence_57

For example Stirling numbers of the first kind_sentence_58

or Stirling numbers of the first kind_sentence_59

and Stirling numbers of the first kind_sentence_60

or even Stirling numbers of the first kind_sentence_61

where γm are the Stieltjes constants and δm,0 represents the Kronecker delta function. Stirling numbers of the first kind_sentence_62

### Explicit formula Stirling numbers of the first kind_section_15

The Stirling number s(n,n-p) can be found from the formula Stirling numbers of the first kind_sentence_63

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. Stirling numbers of the first kind_sentence_64

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. Stirling numbers of the first kind_sentence_65

### Relations to natural logarithm function Stirling numbers of the first kind_section_16

The nth derivative of the μth power of the natural logarithm involves the signed Stirling numbers of the first kind: Stirling numbers of the first kind_sentence_66

It can be proved by using mathematical induction. Stirling numbers of the first kind_sentence_67

## Generalizations Stirling numbers of the first kind_section_17

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 Stirling numbers of the first kind_sentence_68