Nth Derivative of General Harmonic Number Order One

Theorem

Let $n \in \Z_{>0}$.

The $n$th derivative of $\harm 1 x$ with respect to $x$ is:

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

where:

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

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

$x \in \C$ with $x \notin \Z_{<0}$

Proof

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

\(=\)

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

Definition of General Harmonic Numbers

\(\ds \)

\(=\)

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

Linear Combination of Convergent Series

\(\ds \)

\(=\)

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

Nth Derivative of Reciprocal of Mth Power: Corollary

\(\ds \)

\(=\)

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

add $0$

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

Definition of Riemann Zeta Function

$\blacksquare$