Harmonic number

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The term harmonic number has multiple meanings. For other meanings, see harmonic number (disambiguation).

Image:HarmonicNumbers.svg

In mathematics, the n-th harmonic number is n times the inverse of the harmonic mean of the first n integers. More simply, it is

<math>H_n= \sum_{k=1}^n \frac{1}{k}</math>.

Harmonic numbers were studied in antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function and appear in various expressions for various special functions.

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Introduction

The generalized harmonic number of order <math>n</math> of m is given by

<math>H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}</math>.

Note that <math>n</math> may be equal to <math>\infty</math>, provided <math>m > 1</math>.

And If <math>m \le 1</math>, while <math>n=\infty</math>, the harmonic series does not converge and hence the harmonic number does not exist.

Other notations occasionally used include

<math>H_{n,m}= H_n^{(m)} = H_m(n)</math>

The special case of <math>m=1</math> is simply called a harmonic number and is frequently written without the superscript, as

<math>H_n= \sum_{k=1}^n \frac{1}{k}</math>.

In the limit of <math>n\rightarrow \infty</math>, the generalized harmonic number converges to the Riemann zeta function

<math>\lim_{n\rightarrow \infty} H_{n,m} = \zeta(m)</math>

The related sum <math>\sum_{k=1}^n k^m</math> occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of Stirling numbers.

For <math>m=1</math>, the asymptotic expansion is given by

<math>H_{n,1} = \gamma + \ln{n} + \frac{1}{2}n^{-1} - \frac{1}{12}n^{-2} + \frac{1}{120}n^{-4} + \mathcal{O}(n^{-6})</math>

where <math>\gamma</math> is the Euler-Mascheroni constant <math>0.5772156649\dots</math>

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Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function:

<math> \psi(n) = H_{n-1} - \gamma\, </math>

This relation is also frequently used to define the extension of the harmonic numbers to non-integer n. The harmonic numbers are also frequently used to define γ, in that

<math> \gamma = \lim_{n \rightarrow \infty}{\left(H_n - \ln(n)\right)} </math>

although

<math> \gamma = \lim_{n \rightarrow \infty}{\left(H_n - \ln\left(n+{1 \over 2}\right)\right)} </math>

converges more quickly.


An integral representation is given by Euler:

<math> H_n = \int_0^1 \frac{1 - x^n}{1 - x}\,dx </math>

This representation can be easily shown to satisfy the recurrence relation by the formula:

<math> \int_0^1 x^n\,dx = \frac{1}{n + 1}</math>

and then

<math> x^{n} + \frac{1 - x^n}{1 - x} = \frac{1 - x^{n+1}}{1 - x} </math>

inside the integral.

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Generalizations

Euler's integral formula for the harmonic numbers follows from the integral identity

<math>\int_a^1 \frac {1-x^s}{1-x} dx =

- \sum_{k=1}^\infty \frac {1}{k} {s \choose k} (a-1)^k</math>

which holds for general complex-valued s, for the suitably extended binomial coefficients. By choosing a=0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane. This integral relation is easily derived by manipulating the Newton series

<math>\sum_{k=0}^\infty {s \choose k} (-x)^k = (1-x)^s,</math>

which is just the Newton's generalized binomial theorem. The interpolating function is in fact just the digamma function:

<math>\psi(s+1)+\gamma = \int_0^1 \frac {1-x^s}{1-x} dx</math>

where <math>\psi(x)</math> is the digamma, and <math>\gamma</math> is the Euler-Mascheroni constant. The integration process may be repeated to obtain

<math>H_{s,2}=-\sum_{k=1}^\infty \frac {(-1)^k}{k} {s \choose k} H_k</math>
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Generating functions

A generating function for the generalized harmonic numbers is:

<math>\sum_{n=1}^\infty z^n H_{n,m} =

\frac {\mbox{Li}_m(z)}{1-z}</math>

where <math>\mbox{Li}_m(z)</math> is the polylogarithm, and <math>|z|<1</math>. As a special case, one has

<math>\sum_{n=1}^\infty z^n H_n =

\frac {-\ln(1-z)}{1-z}</math>

where <math>\ln(z)</math> is the logarithm. An exponential generating function is

<math>\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)</math>

where <math>\mbox{Ein}(z)</math> is the entire exponential integral. Note that

<math>\mbox {Ein}(z) = \mbox{E}_1(z) + \gamma + \ln z =

\Gamma (0,z) + \gamma + \ln z\,</math>

where <math>\Gamma (0,z)</math> is the incomplete gamma function.

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References

it:Numero armonico generalizzato nl:Harmonisch getal

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