General Harmonic Numbers/Examples/Order 1/Two Thirds

Example of General Harmonic Number

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

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

Proof

\(\ds \harm 1 x\)

\(=\)

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

Recurrence Relation for General Harmonic Numbers

\(\ds \leadsto \ \ \)

\(\ds \harm 1 {\dfrac 2 3}\)

\(=\)

\(\ds \harm 1 {-\dfrac 1 3} + \dfrac 1 {\frac 2 3}\)

setting $x := \dfrac 2 3$ and $r = 1$

\(\ds \)

\(=\)

\(\ds \paren {-\dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3} } + \frac 3 2\)

Example: $\harm 1 {-1 / 3}$

\(\ds \)

\(=\)

\(\ds \frac 3 2 - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}\)

$\blacksquare$