Definition:Complex Extension of General Harmonic Numbers/Also defined as

Complex Extension of General Harmonic Numbers: Also defined as

When Srinivasa Ramanujan introduced the complex extension of the general harmonic numbers, he defined and denoted them:

$\ds \map {\phi_r} n = \sum_{k \mathop = 1}^\infty \paren {k^r - \paren {k + x}^r}$

for $r \in \R_{<0}$.

This is seen to be the same as:

$\ds \map {\phi_{-r} } n = \sum_{k \mathop = 1}^\infty \paren {\dfrac 1 {k^r} - \dfrac 1 {\paren {k + x}^r} }$

for $r \in \R_{>0}$.

By writing:

$\map {\phi_{-r} } n =: \map {H^{\paren r} } n$

we recover the form used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

• Results about the general harmonic numbers can be found here.

Sources

• 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $7$. Sums of Powers, Bernoulli Numbers, and the Gamma Function