Recurrence Relation for General Harmonic Numbers

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 $x$.

$\ds \harm r x$

$=$

$\ds \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} }$

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 $

$=$

$\ds \harm r {x - 1} + \dfrac 1 {x^r}$

$\blacksquare$