Definition:Harmonic Numbers/General Definition/Complex Extension
Definition
Let $r \in \R_{>0}$.
For $z \in \C \setminus \Z_{< 0}$ the harmonic numbers order $r$ can be extended to the complex plane as:
- $\ds \harm r z = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + z}^r} }$
Notation
There appears to be no standard notation for the harmonic numbers.
The notation given here, and used on $\mathsf{Pr} \infty \mathsf{fWiki}$ throughout, is an adaptation for $\mathsf{Pr} \infty \mathsf{fWiki}$ of an idea by Donald E. Knuth, where he used $H_n^{\paren r}$.
Knuth's notation proves unwieldy when extended to the complex numbers, and so we have adopted the more conventional mapping notation $\harm r n$ and hence $\harm r z$.
Ramanujan used $\ds \map {\phi_r} n$.
Other notations that can also be found in the literature include $h_n$, $S_n$ and $\map \psi {n + 1} + \gamma$.
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
 | This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: The following, basically, write a page that does this and put a link to that here instead You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page. |
- Definition:Riemann Zeta Function: if $r > 1$, then $\ds \map \zeta r = \sum_{k \mathop = 1}^\infty \dfrac 1 {k^r} = \harm r \infty = \lim_{z \mathop \to \infty} \harm r z$
- 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