Harmonic Number as Unsigned Stirling Number of First Kind over Factorial

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Theorem

$H_n = \dfrac { {n + 1} \brack 2} {n!}$

where:

$H_n$ denotes the $n$th harmonic number
$n!$ denotes the $n$th factorial
$\ds { {n + 1} \brack 2}$ denotes an unsigned Stirling number of the first kind.


Proof

The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

$H_n = \dfrac { {n + 1} \brack 2} {n!}$


$\map P 0$ is the case:

\(\ds H_0\) \(=\) \(\ds 0\)
\(\ds \) \(=\) \(\ds \dfrac {1 \brack 2} {0!}\) Unsigned Stirling Number of the First Kind of Number with Greater

Thus $\map P 0$ is seen to hold.


Basis for the Induction

$\map P 1$ is the case:

\(\ds H_1\) \(=\) \(\ds 1\)
\(\ds \) \(=\) \(\ds \dfrac {2 \brack 2} {1!}\) Unsigned Stirling Number of the First Kind of Number with Self


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 $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is the induction hypothesis:

$H_k = \dfrac { {k + 1} \brack 2} {k!}$


from which it is to be shown that:

$H_{k + 1} = \dfrac { {k + 2} \brack 2} {\paren {k + 1}!}$


Induction Step

This is the induction step:


\(\ds \dfrac { {k + 2} \brack 2} {\paren {k + 1}!}\) \(=\) \(\ds \dfrac {\paren {k + 1} { {k + 1} \brack 2} + { {k + 1} \brack 1} } {\paren {k + 1}!}\) Definition of Unsigned Stirling Numbers of the First Kind
\(\ds \) \(=\) \(\ds \dfrac {\paren {k + 1} { {k + 1} \brack 2} + k!} {\paren {k + 1}!}\) Unsigned Stirling Number of the First Kind of n+1 with 1
\(\ds \) \(=\) \(\ds \dfrac {\paren { {k + 1} \brack 2} } {k!} + \dfrac 1 {k + 1}\) simplifying
\(\ds \) \(=\) \(\ds H_k + \dfrac 1 {k + 1}\) Induction Hypothesis
\(\ds \) \(=\) \(\ds H_{k + 1}\) Definition of Harmonic Number

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \Z_{\ge 0}: H_n = \dfrac { {n + 1} \brack 2} {n!}$

$\blacksquare$


Sources

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