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
- Weisstein, Eric W. "Polygamma Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolygammaFunction.html