Equivalence of Definitions of Stirling Numbers of the Second Kind

Definition 2
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 $P \left({n}\right)$ be the proposition:
 * the coefficients of the falling factorial powers in the expression $\displaystyle x^n = \sum_k {n \brace k} x^{\underline k}$ are uniquely defined by $\displaystyle {n \brace k} =  {n - 1 \brace k - 1} + k {n - 1 \brace k}$

where $\displaystyle {n \brace k} = \delta_{n k}$ where $k = 0$ or $n = 0$.

First the case where $n = 0$ is attended to.

We have:

Thus, in the expression:
 * $\displaystyle x^0 = \sum_k {0 \brace k} x^{\underline k}$

we have:
 * $\displaystyle {0 \brace 0} = 1$

and for all $k \in \Z_{>0}$:
 * $\displaystyle {0 \brace k} = 0$

That is:
 * $\displaystyle {0 \brace k} = \delta_{0 k}$

Hence the result holds for $n = 0$.

Basis for the Induction
$\map P 1$ is the case:

We have:

Then:

Thus, in the expression:
 * $\displaystyle x^1 = \sum_k {1 \brace k} x^{\underline k}$

we have:
 * $\displaystyle {1 \brace 1} = 1$

and for all $k \in \Z$ where $k \ne 1$:
 * $\displaystyle {1 \brace k} = 0$

That is:
 * $\displaystyle {1 \brace k} = \delta_{1 k}$

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 $\displaystyle x^r = \sum_k {r \brace k} x^{\underline k}$ are uniquely defined by $\displaystyle {r \brace k} = {r - 1 \brace k - 1} + k {r - 1 \brace k}$

from which it is to be shown that:
 * The coefficients in the expression $\displaystyle x^{r + 1} = \sum_k {r + 1 \brace k} x^{\underline k}$ are uniquely defined by $\displaystyle {r + 1 \brace k} = {r \brace k - 1} + k {r \brace k}$

Induction Step
This is the induction step:

Anticipating the expected result, we use $\displaystyle {r + 1 \brace k}$ to denote the coefficients of the $k$th falling factorial power in the expansion of $x^{r + 1}$.

Thus:

Thus the coefficients of the falling factorial powers are defined by the recurrence relation:
 * $\displaystyle {r + 1 \brace k} = {r \brace k - 1} + k {r \brace k}$

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 $\displaystyle x^n = \sum_k {n \brace k} x^{\underline k}$ are uniquely defined by:
 * $\displaystyle {n \brace k} = {n - 1 \brace k - 1} + k {n - 1 \brace  k}$
 * where $\displaystyle {n \brace k} = \delta_{n k}$ where $k = 0$ or $n = 0$.

Also see

 * Equivalence of Definitions of Unsigned Stirling Numbers of the First Kind
 * Equivalence of Definitions of Signed Stirling Numbers of the First Kind