Talk:Digamma Function of One Sixth/Proof 2

From the Polygamma Reflection Formula, we have:

$\map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z} = -\pi \dfrac {\d^n} {\d z^n} \cot \pi z$

This proof asserts:

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

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

\(=\)

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

Polygamma Reflection Formula

Would it be clearer like this:

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

\(\ds \map {\psi_0} {\frac 1 6} - \map {\psi_0} {\frac 5 6}\)

\(=\)

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

Polygamma Reflection Formula $z = \frac 1 6$

This specific case (Digamma case: $\map {\psi_0} z$ ) shows up in the sixth line of Polygamma Reflection Formula

Suboptimal. This is confusing to the casual browser who needs to then make an extra excursion to make the mental link to how the polygamma function is defined and how the polygamma reflection formula is applied to the digamma function. --prime mover (talk) 06:10, 10 January 2024 (UTC)

In my sources, when the subscript is omitted, it is digamma and a 1 subscrpt is trigamma, 2 is quadgamma, etc. --Robkahn131 (talk) 00:49, 10 January 2024 (UTC)

It may be in your source but it's not on $\mathsf{Pr} \infty \mathsf{fWiki}$. --prime mover (talk) 06:10, 10 January 2024 (UTC)

I contend that what I have put in place is better.

If you wish to add further proofs that use the polygamma function, then you could always implement something like Polygamma Function of One Sixth and implement this as a special case, or whatever. --prime mover (talk) 07:13, 10 January 2024 (UTC)