General Harmonic Number Reflection Formula/Lemma 1

General Harmonic Number Reflection Formula: Lemma 1

Let $\harm r x$ denote the general harmonic number of order $r$ evaluated at $x$

The expression:

$\harm r {x - 1} + \paren {-1}^r \harm r {-x}$

is defined on the domain $\C \setminus \Z$.

That is, on the set of complex numbers but specifically excluding the integers.

Proof

From the definition of the general harmonic numbers:

$\harm r {-x}$ is defined for $-x \in \C \setminus \Z_{<0}$

Therefore:

$x \in \C \setminus \Z_{>0}$

and:

$\harm r {x - 1}$ is defined for $\paren {x - 1} \in \C \setminus \Z_{<0}$.

Therefore:

$x \in \C \setminus \Z_{\le 0}$

Therefore, $\harm r {x - 1} + \paren {-1}^r \harm r {-x}$ is defined for $z \in \C \setminus \Z$.

$\blacksquare$