General Harmonic Number Reflection Formula

Theorem

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

where:

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

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

$r \in \Z_{>0}$ and $x \in \C$ and $x \notin \Z$

Proof

Lemma 1

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.

$\Box$

Lemma 2

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

$\Box$

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

\(=\)

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

Lemma $2$

\(\ds \leadsto \ \ \)

\(\ds \map {\dfrac {\d^n} {\d x^n} } {\harm 1 {x - 1} - \harm 1 {-x} }\)

\(=\)

\(\ds \map {\dfrac {\d^n} {\d x^n} } {-\pi \map \cot {\pi x} }\)

taking the $n$th derivative of each side

\(\ds \leadsto \ \ \)

\(\ds \paren {\paren {-1}^{n + 1} n! \paren {\map \zeta {n + 1} - \harm {n + 1} {x - 1} } } - \paren {\paren {-1}^{n + 1} n! \paren {\map \zeta {n + 1} - \harm {n + 1} {-x} } \paren {-1}^n}\)

\(=\)

\(\ds \map {\dfrac {\d^n} {\d x^n} } {-\pi \map \cot {\pi x} }\)

Nth Derivative of General Harmonic Number Order One and Chain Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \paren {\paren {-1}^r \paren {r - 1}! \paren {\map \zeta r - \harm r {x - 1} } } - \paren {\paren {-1}^r \paren {r - 1}! \paren {\map \zeta r - \harm r {-x} } \paren {-1}^{r - 1} }\)

\(=\)

\(\ds \map {\dfrac {\d^{r - 1} } {\d x^{r - 1} } } {-\pi \map \cot {\pi x} }\)

$n \to r - 1$

\(\ds \leadsto \ \ \)

\(\ds \paren {\paren {\map \zeta r - \harm r {x - 1} } } + \paren {-1}^r \paren {\paren {\map \zeta r - \harm r {-x} } }\)

\(=\)

\(\ds \dfrac {\paren {-1}^r} {\paren {r - 1}! } \map {\dfrac {\d^{r - 1} } {\d x^{r - 1} } } {-\pi \map \cot {\pi x} }\)

multiplying both sides by $\dfrac {\paren {-1}^r} {\paren {r - 1}! }$

\(\ds \leadsto \ \ \)

\(\ds -\map \zeta r \paren {1 + \paren {-1}^r} + \paren {\harm r {x - 1} + \paren {-1}^r \harm r {-x} }\)

\(=\)

\(\ds \dfrac {\paren {-1}^r} {\paren {r - 1}!} \map {\dfrac {\d^{r - 1} } {\d x^{r - 1} } } {\pi \map \cot {\pi x} }\)

multiply both sides by $-1$

\(\ds \leadsto \ \ \)

\(\ds \harm r {x - 1} + \paren {-1}^r \harm r {-x}\)

\(=\)

\(\ds \paren {1 + \paren {-1}^r} \map \zeta r + \dfrac {\paren {-1}^r} {\paren {r - 1}! } \map {\dfrac {\d^{r - 1} } {\d x^{r - 1} } } {\pi \map \cot {\pi x} }\)

adding $\paren {1 + \paren {-1}^r} \map \zeta r$ to both sides

Finally, from the Lemma $1$, we note that:

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

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

$\blacksquare$

Sources

• 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $7$. Sums of Powers, Bernoulli Numbers and the Gamma Function