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.

Read more about this topic: Harmonic Number

Famous quotes containing the word functions:

“Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
—Henry David Thoreau (1817–1862)