General Harmonic Numbers/Examples/Order 1/Minus Two Thirds

Example of General Harmonic Number

$\harm 1 {-\dfrac 2 3} = -\dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$

where $\harm 1 {-\dfrac 2 3}$ denotes the general harmonic number of order $1$ evaluated at $-\dfrac 2 3$.

$\ds \harm 1 z$

$=$

$\ds \map \psi {z + 1} + \gamma$

$\ds \leadsto \ \ $

$\ds \harm 1 {-\dfrac 2 3}$

$=$

$\ds \map \psi {-\dfrac 2 3 + 1} + \gamma$

setting $z := -\dfrac 2 3$

$\ds $

$=$

$\ds \map \psi {\dfrac 1 3} + \gamma$

$\ds $

$=$

$\ds \paren {-\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3} } + \gamma$

$\ds $

$=$

$\ds -\dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$

$\blacksquare$