Definition:Harmonic Numbers/General Definition

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This page is about General Harmonic Numbers. For other uses, see Harmonic.


It has been suggested that this page be renamed.
In particular: Needs a "function" in its name after all. Ramanujan's function?
To discuss this page in more detail, feel free to use the talk page.


Definition

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

For $n \in \N_{> 0}$ the harmonic numbers order $r$ are defined as follows:

$\ds \map {H^{\paren r} } n = \sum_{k \mathop = 1}^n \frac 1 {k^r}$


Complex Extension

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:

$\ds \harm r z = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + z}^r} }$



This article is complete as far as it goes, but it could do with expansion.
In particular: I think it's worth adding a section here (usual transclusion rules apply of course) relating this to the harmonic numbers themselves, and perhaps to include a separate page for $\map {H^{\paren 1} } z$ for complex $z$ to be defined as a direct extension of $H_n$. Interesting exercise in elementary function theory.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
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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 $\map \psi {n + 1} + \gamma$.


Examples

General Harmonic Number of Order $1$ at $\dfrac 1 2$

$\harm 1 {\dfrac 1 2} = 2 - 2 \ln 2$


General Harmonic Number of Order $1$ at $-\dfrac 1 2$

$\harm 1 {-\dfrac 1 2} = -2 \ln 2$


General Harmonic Number of Order $1$ at $\dfrac 1 3$

$\harm 1 {\dfrac 1 3} = 3 - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$


General Harmonic Number of Order $1$ at $-\dfrac 1 3$

$\harm 1 {-\dfrac 1 3} = -\dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$


General Harmonic Number of Order $1$ at $\dfrac 2 3$

$\harm 1 {\dfrac 2 3} = \dfrac 3 2 - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$


General Harmonic Number of Order $1$ at $-\dfrac 2 3$

$\harm 1 {-\dfrac 2 3} = -\dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$


General Harmonic Number of Order $2$ at $\dfrac 1 2$

$\harm 2 {\dfrac 1 2} = 4 - 2 \map \zeta 2$


General Harmonic Number of Order $2$ at $-\dfrac 1 2$

$\harm 2 {-\dfrac 1 2} = -2 \map \zeta 2$


Also see

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


Technical Note

The $\LaTeX$ code for \(\harm {r} {z}\) is \harm {r} {z} .

When either of the arguments is a single character, it is usual to omit the braces:

\harm r z


Sources

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