Properties of General Harmonic Numbers

Theorem

This page gathers together some of the properties of the general harmonic numbers:

Let $r \in \R_{>0}$.

For $z \in \C \setminus \Z_{< 0}$ the harmonic numbers order $r$ can be extended to the complex plane as:

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

Digamma Function in terms of General Harmonic Number

$\ds \map \psi {z + 1} = -\gamma + \harm 1 z$

Recurrence Relation for General Harmonic Numbers

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

General Harmonic Number Reflection Formula

$\harm r {x - 1} + \paren {-1}^r \harm r {-x} = \paren {1 + \paren {-1}^r} \map \zeta r + \dfrac {\paren {-1}^r} {\paren {r - 1}!} \map {\dfrac {\d^{r - 1} } {\d x^{r - 1} } } {\pi \map \cot {\pi x} }$

Nth Derivative of General Harmonic Number Order One

$\dfrac {\d^n} {\d x^n} \harm 1 x = \paren {-1}^{n + 1} n! \paren {\map \zeta {n + 1} - \harm {n + 1} x}$

General Harmonic Numbers in terms of Riemann Zeta and Hurwitz Zeta Functions

$\harm r x = \map \zeta r - \map \zeta {r, x + 1}$

Sum of General Harmonic Numbers in terms of Riemann Zeta Function

$\ds \harm r x - \dfrac 1 {n^r} \sum_{k \mathop = 0}^{n - 1} \harm r {x - \dfrac k n} = \paren {1 - n^{1 - r} } \map \zeta r$

General Harmonic Number Additive Formula

$\ds \harm 1 {n x} = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x - \dfrac k n } } + \ln n$