General Harmonic Numbers/Examples/Order 2/Half

Example of General Harmonic Number

$\harm 2 {\dfrac 1 2} = 4 - 2 \map \zeta 2$

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

Proof

\(\ds \harm r x\)

\(=\)

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

Recurrence Relation for General Harmonic Numbers

\(\ds \leadsto \ \ \)

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

\(=\)

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

setting $x := \dfrac 1 2$ and $r := 2$

\(\ds \)

\(=\)

\(\ds - 2 \map \zeta 2 + 4\)

General Harmonic Number of Order $2$ at $-\dfrac 1 2$

\(\ds \)

\(=\)

\(\ds 4 - 2 \map \zeta 2\)

rearranging

$\blacksquare$