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Logarithmic (Log-Series, Series) Distribution

docs/tutorial:stats:discrete_logser

The logarithmic distribution with parameter $p$ has a probability mass function with terms proportional to the Taylor series expansion of $lo g (1 - p)$

where

Φ (z, s, a) = k = 0 \sum \infty \frac{z ^{k}}{( a + k ) ^{s}}

is the Lerch Transcendent. Also define $r = lo g (1 - p)$

Thus,

μ_{n}^{'} = M^{(n)} (t)_{t = 0} = \frac{Li _{1 - n} ( p e ^{t} )}{lo g ( 1 - p )}_{t = 0} = - \frac{Li _{1 - n} ( p )}{lo g ( 1 - p )} .

Implementation: scipy.stats.logser