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

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Theorem

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

where:

$\harm r x$ denotes the general harmonic number of order $r$ evaluated at $x$
$\map \zeta r$ is the Riemann zeta function
$\map \zeta {r, x}$ is the Hurwitz zeta function
$r$ and $x$ are complex numbers with $\map \Re r > 1$ and $x \notin \Z_{<0}$


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
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \frac 1 {k^r} - \sum_{k \mathop = 1}^{\infty} \frac 1 {\paren {k + x}^r}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \frac 1 {k^r} - \sum_{k \mathop = 0}^{\infty} \frac 1 {\paren {k + \paren {x + 1} }^r}\) reindexing the sum
\(\ds \) \(=\) \(\ds \map \zeta r - \map \zeta {r, x + 1}\) Definition of Hurwitz Zeta Function and Definition of Riemann Zeta Function

$\blacksquare$


Sources

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