Summation over Lower Index of Unsigned Stirling Numbers of the First Kind with Alternating Signs
Jump to navigation
Jump to search
Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
- $\ds \sum_k \paren {-1}^k {n \brack k} = \delta_{n 0} - \delta_{n 1}$
where:
- $\ds {n \brack k}$ denotes an unsigned Stirling number of the first kind
- $\delta_{n 0}$ denotes the Kronecker delta.
Proof
The proof proceeds by induction on $n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
- $\ds \sum_k \paren {-1}^k {n \brack k} = \delta_{n 0} - \delta_{n 1}$
$\map P 0$ is the case:
\(\ds \sum_k \paren {-1}^k {0 \brack k}\) | \(=\) | \(\ds \sum_k \delta_{0 k}\) | Unsigned Stirling Number of the First Kind of 0 | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | all terms vanish but for $k = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{0 0} - \delta_{0 1}\) | Definition of Kronecker Delta |
Thus $\map P 0$ is seen to hold.
$\map P 1$ is the case:
\(\ds \sum_k \paren {-1}^k {1 \brack k}\) | \(=\) | \(\ds \sum_k \paren {-1}^k \delta_{1 k}\) | Unsigned Stirling Number of the First Kind of 1 | |||||||||||
\(\ds \) | \(=\) | \(\ds -1\) | all terms vanish but for $k = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{1 0} - \delta_{1 1}\) | Definition of Kronecker Delta |
Thus $\map P 1$ is seen to hold.
Basis for the Induction
$\map P 2$ is the case:
\(\ds \sum_k \paren {-1}^k {2 \brack k}\) | \(=\) | \(\ds - {2 \brack 1} + {2 \brack 2}\) | Definition of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds - {2 \brack 1} + 1\) | Unsigned Stirling Number of the First Kind of Number with Self | |||||||||||
\(\ds \) | \(=\) | \(\ds -\binom 2 2 + 1\) | Unsigned Stirling Number of the First Kind of n with n-1 | |||||||||||
\(\ds \) | \(=\) | \(\ds -1 + 1\) | Binomial Coefficient with Self | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | Binomial Coefficient with Self | |||||||||||
\(\ds \) | \(=\) | \(\ds \delta_{2 0} - \delta_{2 1}\) | Definition of Kronecker Delta |
Thus $\map P 2$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P m$ is true, where $m \ge 2$, then it logically follows that $\map P {m + 1}$ is true.
So this is the induction hypothesis:
- $\ds \sum_k \paren {-1}^k {m \brack k} = \delta_{m 0} - \delta_{m 1} = 0$
from which it is to be shown that:
- $\ds \sum_k \paren {-1}^k {m + 1 \brack k} = \delta_{\paren {m + 1} 0} - \delta_{\paren {m + 1} 1} = 0$
Induction Step
This is the induction step:
\(\ds \sum_k \paren {-1}^k {m + 1 \brack k}\) | \(=\) | \(\ds \sum_k \paren {-1}^k \paren {m {m \brack k} + {m \brack k - 1} }\) | Definition of Unsigned Stirling Numbers of the First Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds m \sum_k \paren {-1}^k {m \brack k} + \sum_k \paren {-1}^k {m \brack k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m \sum_k \paren {-1}^k {m \brack k} - \sum_k \paren {-1}^k {m \brack k}\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {m - 1} \sum_k {m \brack k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {m + 1} \times 0\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
So $\map P m \implies \map P {m + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \in \Z_{\ge 0}: \sum_k \paren {-1}^k {n \brack k} = \delta_{n 0} - \delta_{n 1}$
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $39$