Digamma Function of One Sixth/Proof 2

Theorem

$\map \psi {\dfrac 1 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2$

Proof

\(\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}\)

\(=\)

\(\ds -\paren {n - 1} \gamma - n \ln n\)

Digamma Additive Formula: Corollary

\(\ds \leadsto \ \ \)

\(\ds \sum_{k \mathop = 1}^{6 - 1} \map \psi {\frac k 6}\)

\(=\)

\(\ds -\paren {6 - 1} \gamma - 6 \ln 6\)

\(\text {(1)}: \quad\)

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 1 6} + \map \psi {\frac 2 6} + \map \psi {\frac 3 6} + \map \psi {\frac 4 6} + \map \psi {\frac 5 6}\)

\(=\)

\(\ds -5 \gamma - 6 \ln 6\)

\(\text {(2)}: \quad\)

\(\ds \map \psi {\frac 1 6} - \map \psi {\frac 5 6}\)

\(=\)

\(\ds -\pi \map \cot {\frac \pi 6}\)

Digamma Reflection Formula

\(\ds \leadsto \ \ \)

\(\ds 2 \map \psi {\frac 1 6} + \map \psi {\frac 2 6} + \map \psi {\frac 3 6} + \map \psi {\frac 4 6}\)

\(=\)

\(\ds -5 \gamma - 6 \ln 6 - \pi \map \cot {\frac \pi 6}\)

adding lines $1$ and $2$

\(\ds \leadsto \ \ \)

\(\ds 2 \map \psi {\frac 1 6}\)

\(=\)

\(\ds -5 \gamma - 6 \ln 6 - \pi \map \cot {\frac \pi 6} - \map \psi {\frac 1 3} - \map \psi {\frac 1 2} - \map \psi {\frac 2 3}\)

rearranging

\(\ds \)

\(=\)

\(\ds -5 \gamma - 6 \ln 2 - 6 \ln 3 - \pi \times \sqrt 3\)

Sum of Logarithms, Cotangent of $30 \degrees$,

\(\ds \)

\(\)

\(\, \ds - \, \)

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

Digamma Function of One Third,

\(\ds \)

\(\)

\(\, \ds - \, \)

\(\ds \paren {-\gamma - 2 \ln 2}\)

Digamma Function of One Half,

\(\ds \)

\(\)

\(\, \ds - \, \)

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

and Digamma Function of Two Thirds

\(\ds \)

\(=\)

\(\ds -2 \gamma - 4 \ln 2 - 3 \ln 3 - \pi \sqrt 3\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds \map \psi {\frac 1 6}\)

\(=\)

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

dividing by $2$

$\blacksquare$