Recurrence Relation for Polygamma Function/Proof 1

Theorem

$\map {\psi_n} {z + 1} = \map {\psi_n} z + \paren {-1}^n n! z^{-n - 1}$

where:

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

$z \in \C \setminus \Z_{\le 0}$.

Proof

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:

\(\ds \map \psi {z + 1}\)

\(=\)

\(\ds \map \psi z + z^{-1}\)

Recurrence Relation for Digamma Function

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\d^n} {\d z^n} \map \psi {z + 1}\)

\(=\)

\(\ds \dfrac {\d^n} {\d z^n} \map \psi z + \dfrac {\d^n} {\d z^n} {z^{-1} }\)

taking $n$th derivative

\(\ds \leadsto \ \ \)

\(\ds \map {\psi_n} {z + 1}\)

\(=\)

\(\ds \map {\psi_n} z + \paren {-1}^n n! z^{-n - 1}\)

Definition of Polygamma Function and Nth Derivative of Reciprocal of Mth Power: Corollary

$\blacksquare$