Recurrence Relation for Polygamma Function

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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 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:

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

$\blacksquare$


Proof 2

\(\ds \map \Gamma {z + 1}\) \(=\) \(\ds z \map \Gamma z\) Gamma Difference Equation
\(\ds \leadsto \ \ \) \(\ds \map \ln {\map \Gamma {z + 1} }\) \(=\) \(\ds \map \ln {z \map \Gamma z}\) applying $\ln$ on both sides
\(\ds \) \(=\) \(\ds \ln z + \map \ln {\map \Gamma z}\) Sum of Logarithms
\(\ds \leadsto \ \ \) \(\ds \dfrac \d {\d z} \map \ln {\map \Gamma {z + 1} }\) \(=\) \(\ds \dfrac \d {\d z} \ln z + \dfrac \d {\d z} \map \ln {\map \Gamma z}\) differentiation with respect to $z$
\(\ds \leadsto \ \ \) \(\ds \dfrac {\map {\Gamma'} {z + 1} } {\map \Gamma {z + 1} }\) \(=\) \(\ds z^{-1} + \dfrac {\map {\Gamma'} z} {\map \Gamma z}\) Derivative of Natural Logarithm Function, Chain Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \map \psi {z + 1}\) \(=\) \(\ds \map \psi z + z^{-1}\) Definition of 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

$\blacksquare$


Also presented as

The Recurrence Relation for Polygamma Function can also be presented as:

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


Also see


Sources

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