Recurrence Relation for General Harmonic Numbers
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Theorem
- $\harm r x = \harm r {x - 1} + \dfrac 1 {x^r}$
where:
- $\harm r x$ denotes the general harmonic number of order $r$ evaluated at $x$.
Proof
\(\ds \harm r x\) | \(=\) | \(\ds \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} }\) | Definition of General Harmonic Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} } - \dfrac 1 {x^r} + \dfrac 1 {x^r}\) | add $0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {\paren {k - 1} + x }^r} } + \dfrac 1 {x^r}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + \paren {x - 1} }^r} } + \dfrac 1 {x^r}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \harm r {x - 1} + \dfrac 1 {x^r}\) |
$\blacksquare$