Sum over k to n of Unsigned Stirling Number of the First Kind of k with m by n factorial over k factorial
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Theorem
Let $m, n \in \Z_{\ge 0}$.
- $\ds \sum_{k \mathop \le n} {k \brack m} \frac {n!} {k!} = {n + 1 \brack m + 1}$
where:
- $\ds {k \brack m}$ denotes an unsigned Stirling number of the first kind
- $ n!$ denotes a factorial.
Proof
The proof proceeds by induction on $n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
- $\ds \forall n \in \Z_{\ge 0}: \sum_{k \mathop \le n} {k \brack m} \frac {n!} {k!} = {n + 1 \brack m + 1}$
Basis for the Induction
$\map P 0$ is the case:
\(\ds \) | \(\) | \(\ds \sum_{k \mathop \le 0} {k \brack m} \frac {0!} {k!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {0 \brack m} \frac 1 {0!}\) | Definition of Unsigned Stirling Numbers of the First Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{0 m}\) | Unsigned Stirling Number of the First Kind of 0 | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{1 \paren {m + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {1 \brack m + 1}\) | Unsigned Stirling Number of the First Kind of 1 |
So $\map P 0$ 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:
- $\ds \sum_{k \mathop \le r} {k \brack m} \frac {r!} {k!} = {r + 1 \brack m + 1}$
from which it is to be shown that:
- $\ds \sum_{k \mathop \le r + 1} {k \brack m} \frac {\paren {r + 1}!} {k!} = {r + 2 \brack m + 1}$
Induction Step
This is the induction step:
\(\ds \) | \(\) | \(\ds \sum_{k \mathop \le r + 1} {k \brack m} \frac {\paren {r + 1}!} {k!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {r + 1 \brack m} \frac {\paren {r + 1}!} {\paren {r + 1}!} + \paren {r + 1} \paren {\sum_{k \mathop \le r} {k \brack m} \frac {r!} {k!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {r + 1 \brack m} + \paren {r + 1} {r + 1 \brack m + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {r + 2 \brack m + 1}\) | Definition of Unsigned Stirling Numbers of the First Kind |
So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall m, n \in \Z_{\ge 0}: \sum_{k \mathop \le n} {k \brack m} \frac {n!} {k!} = {n + 1 \brack m + 1}$
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(56)$