Polygamma Function - Multiplication Theorem

Multiplication Theorem

The multiplication theorem gives

k^{m+1} \psi^{(m)}(kz) = \sum_{n=0}^{k-1} \psi^{(m)}\left(z+\frac{n}{k}\right)\qquad m \ge 1

and

k \psi^{(0)}(kz) = k\log(k)) + \sum_{n=0}^{k-1} \psi^{(0)}\left(z+\frac{n}{k}\right)

for the digamma function.

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