General Harmonic Number Reflection Formula/Lemma 2

General Harmonic Number Reflection Formula: Lemma 2

$\ds \harm 1 {x - 1} - \harm 1 {-x} = -\pi \map \cot {\pi x}$

where:

$\harm 1 x$ denotes the general harmonic number of order $1$ evaluated at $x$

$\map \zeta r$ is the Riemann zeta function

$x \in \C$ and $x \notin \Z$

Proof

\(\ds \harm 1 {x - 1} - \harm 1 {-x}\)

\(=\)

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

Definition of General Harmonic Numbers

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

Definition of Riemann Zeta Function

\(\ds \)

\(=\)

\(\ds -\paren {\frac 1 {x^1} + \sum_{k \mathop = 1}^\infty \frac {2 x} {\paren {x^2 - k^2}^1} }\)

\(\ds \)

\(=\)

\(\ds -\pi \map \cot {\pi x}\)

Mittag-Leffler Expansion for Cotangent Function

$\blacksquare$