Harmonic Number - Generating Functions

Generating Functions

A generating function for the harmonic numbers is

$\sum_{n=1}^\infty z^n H_n = \frac {-\ln(1-z)}{1-z},$

where is the natural logarithm. An exponential generating function is

$\sum_{n=1}^\infty \frac {z^n}{n!} H_n = -e^z \sum_{k=1}^\infty \frac{1}{k} \frac {(-z)^k}{k!} = e^z \mbox {Ein}(z)$

where is the entire exponential integral. Note that

$\mbox {Ein}(z) = \mbox{E}_1(z) + \gamma + \ln z = \Gamma (0,z) + \gamma + \ln z\,$

where is the incomplete gamma function.

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