Multiplication Theorem - Polygamma Function

Polygamma Function

The polygamma function is the logarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative:

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

for, and, for, one has the digamma function:

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

Read more about this topic: Multiplication Theorem

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