Equivalence of Definitions of Stirling Numbers of the Second Kind
Theorem
The following definitions of the concept of Stirling Numbers of the Second Kind are equivalent:
Definition 1
Stirling numbers of the second kind are defined recursively by:
- $\ds {n \brace k} := \begin{cases}
\delta_{n k} & : k = 0 \text{ or } n = 0 \\ & \\ \ds {n - 1 \brace k - 1} + k {n - 1 \brace k} & : \text{otherwise} \\ \end{cases}$
Definition 2
Stirling numbers of the second kind are defined as the coefficients $\ds {n \brace k}$ which satisfy the equation:
- $\ds x^n = \sum_k {n \brace k} x^{\underline k}$
where $x^{\underline k}$ denotes the $k$th falling factorial of $x$.
in the sense that the coefficients of the falling factorial powers in the summand are uniquely defined by the given recurrence relation.
Proof
The proof proceeds by induction.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
- the coefficients of the falling factorial powers in the expression $\ds x^n = \sum_k {n \brace k} x^{\underline k}$ are uniquely defined by $\ds {n \brace k} = {n - 1 \brace k - 1} + k {n - 1 \brace k}$
where $\ds {n \brace k} = \delta_{n k}$ where $k = 0$ or $n = 0$.
First the case where $n = 0$ is attended to.
We have:
\(\ds x^0\) | \(=\) | \(\ds 1\) | Definition of Integer Power | |||||||||||
\(\ds \) | \(=\) | \(\ds x^{\underline 0}\) | Number to Power of Zero Falling is One | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k \delta_{0 k} x^{\underline k}\) | Definition of Kronecker Delta |
Thus, in the expression:
- $\ds x^0 = \sum_k {0 \brace k} x^{\underline k}$
we have:
- $\ds {0 \brace 0} = 1$
and for all $k \in \Z_{>0}$:
- $\ds {0 \brace k} = 0$
That is:
- $\ds {0 \brace k} = \delta_{0 k}$
Hence the result holds for $n = 0$.
Basis for the Induction
$\map P 1$ is the case:
We have:
\(\ds x^1\) | \(=\) | \(\ds x\) | Definition of Integer Power | |||||||||||
\(\ds \) | \(=\) | \(\ds x^{\underline 1}\) | Number to Power of One Falling is Itself | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k \delta_{1 k} x^{\underline k}\) | Definition of Kronecker Delta |
Then:
\(\ds {1 \brace k}\) | \(=\) | \(\ds {0 \brace k - 1} + k {0 \brace k}\) | by hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{0 \paren {k - 1} } + k \delta_{0 k}\) | by hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{0 \paren {k - 1} }\) | as $k \delta_{0 k} = 0$ for all $k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{1 k}\) |
Thus, in the expression:
- $\ds x^1 = \sum_k {1 \brace k} x^{\underline k}$
we have:
- $\ds {1 \brace 1} = 1$
and for all $k \in \Z$ where $k \ne 1$:
- $\ds {1 \brace k} = 0$
That is:
- $\ds {1 \brace k} = \delta_{1 k}$
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P r$ is true, where $r \ge 1$, then it logically follows that $\map P {r + 1}$ is true.
So this is the induction hypothesis:
- The coefficients in the expression $\ds x^r = \sum_k {r \brace k} x^{\underline k}$ are uniquely defined by $\ds {r \brace k} = {r - 1 \brace k - 1} + k {r - 1 \brace k}$
from which it is to be shown that:
- The coefficients in the expression $\ds x^{r + 1} = \sum_k {r + 1 \brace k} x^{\underline k}$ are uniquely defined by $\ds {r + 1 \brace k} = {r \brace k - 1} + k {r \brace k}$
Induction Step
This is the induction step:
Anticipating the expected result, we use $\ds {r + 1 \brace k}$ to denote the coefficients of the $k$th falling factorial power in the expansion of $x^{r + 1}$.
Thus:
\(\ds x^{r + 1}\) | \(=\) | \(\ds x x^r\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \paren {\sum_k {r \brace k} x^{\underline k} }\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k {r \brace k} x x^{\underline k}\) | $x$ is constant in this context | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k {r \brace k} \paren {x^{\underline {k + 1} } + k x^{\underline k} }\) | Product of Number by its Falling Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k {r \brace k} x^{\underline {k + 1} } + \sum_k {r \brace k} k x^{\underline k}\) | Sum of Summations equals Summation of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k {r \brace k - 1} x^{\underline k} + \sum_k {r \brace k} k x^{\underline k}\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k \paren { {r \brace k - 1} x^{\underline k} + {r \brace k} k x^{\underline k} }\) | Sum of Summations equals Summation of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_k \paren { {r \brace k - 1} + k {r \brace k} } x^{\underline k}\) |
Thus the coefficients of the falling factorial powers are defined by the recurrence relation:
- $\ds {r + 1 \brace k} = {r \brace k - 1} + k {r \brace k}$
as required.
So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- for all $n \in \Z_{\ge 0}$, the coefficients of the falling factorial powers in the expression $\ds x^n = \sum_k {n \brace k} x^{\underline k}$ are uniquely defined by:
- $\ds {n \brace k} = {n - 1 \brace k - 1} + k {n - 1 \brace k}$
- where $\ds {n \brace k} = \delta_{n k}$ where $k = 0$ or $n = 0$.
$\blacksquare$
Also see
- Equivalence of Definitions of Unsigned Stirling Numbers of the First Kind
- Equivalence of Definitions of Signed Stirling Numbers of the First Kind
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $35$