Polygamma Reflection Formula

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Theorem

Let $z \in \C \setminus \Z$.

Let $\psi_n$ denote the $n$th polygamma function.

Then:

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


Lemma

The expression:

$\map \psi z - \map \psi {1 - z}$

is defined on the domain $\C \setminus \Z$.

That is, on the set of complex numbers but specifically excluding the integers.

$\Box$


Proof 1

By definition:

$\map {\psi_n} z = \dfrac {\d^n} {\d z^n} \map \psi z$

where:

$\psi$ denotes the digamma function
$z \in \C \setminus \Z_{\le 0}$.


Then:

\(\ds \map \psi z - \map \psi {1 - z}\) \(=\) \(\ds -\pi \cot \pi z\) Digamma Reflection Formula
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d^n} {\d z^n} \map \psi z - \dfrac {\d^n} {\d z^n} \map \psi {1 - z}\) \(=\) \(\ds -\pi \dfrac {\d^n} {\d z^n} \cot \pi z\) taking $n$th derivative
\(\ds \leadsto \ \ \) \(\ds \map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z}\) \(=\) \(\ds -\pi \dfrac {\d^n} {\d z^n} \cot \pi z\) Definition of Polygamma Function


Finally, from the Lemma, we note that:

$\map \psi z - \map \psi {1 - z}$

is defined on the domain $\C \setminus \Z$.

$\blacksquare$


Proof 2

\(\ds \map \Gamma z \map \Gamma {1 - z}\) \(=\) \(\ds \dfrac \pi {\sin \pi z}\) Euler's Reflection Formula
\(\ds \leadsto \ \ \) \(\ds \map \ln {\map \Gamma z \map \Gamma {1 - z} }\) \(=\) \(\ds \map \ln {\dfrac \pi {\sin \pi z} }\) applying $\ln$ on both sides
\(\ds \leadsto \ \ \) \(\ds \map \ln {\map \Gamma z} + \map \ln {\map \Gamma {1 - z} }\) \(=\) \(\ds \map \ln \pi - \map \ln {\sin \pi z}\) Sum of Logarithms and Difference of Logarithms
\(\ds \leadsto \ \ \) \(\ds \dfrac \d {\d z} \map \ln {\map \Gamma z} + \dfrac \d {\d z} \map \ln {\map \Gamma {1 - z} }\) \(=\) \(\ds \dfrac \d {\d z} \map \ln \pi - \dfrac \d {\d z} \map \ln {\sin \pi z}\) differentiation with respect to $z$
\(\ds \leadsto \ \ \) \(\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} - \dfrac {\map {\Gamma'} {1 - z} } {\map \Gamma {1 - z} }\) \(=\) \(\ds 0 - \pi \cot \pi z\) Derivative of Natural Logarithm Function, Derivative of Sine Function, Chain Rule for Derivatives, Derivative of Constant
\(\ds \leadsto \ \ \) \(\ds \map \psi z - \map \psi {1 - z}\) \(=\) \(\ds -\pi \cot \pi z\) Definition of Digamma Function
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d^n} {\d z^n} \map \psi z - \dfrac {\d^n} {\d z^n} \map \psi {1 - z}\) \(=\) \(\ds -\pi \dfrac {\d^n} {\d z^n} \cot \pi z\) taking $n$th derivative
\(\ds \leadsto \ \ \) \(\ds \map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z}\) \(=\) \(\ds -\pi \dfrac {\d^n} {\d z^n} \cot \pi z\) Definition of Polygamma Function


Finally, from the Lemma, we note that:

$\map \psi z - \map \psi {1 - z}$

is defined on the domain $\C \setminus \Z$.

$\blacksquare$


Also see


Sources