Harmonic Number

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

:

H_n= \sum_{k=1}^n rac{1}{k}

.

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 Function s.

INTRODUCTION

The generalized harmonic number of order

n

of ''m'' is given by

:

H_{n,m}=\sum_{k=1}^n rac{1}{k^m}

.

Note that

n

may be equal to

\infty

, provided

m > 1

.

And If

m \le 1

, while

n=\infty

, the harmonic series does not Converge and hence the harmonic number does not exist.

Other notations occasionally used include

:

H_{n,m}= H_n^{(m)} = H_m(n)

The special case of

m=1

is simply called a harmonic number and is frequently written without the superscript, as

:

H_n= \sum_{k=1}^n rac{1}{k}

.

In the limit of

n
ightarrow \infty

, the generalized harmonic number converges to the Riemann Zeta Function

:

\lim_{n
ightarrow \infty} H_{n,m} = \zeta(m)

The related sum

\sum_{k=1}^n k^m

occurs in the study of Bernoulli Number s; the harmonic numbers also appear in the study of Stirling Number s.

For

m=1

, the asymptotic expansion is given by

:

H_{n,1} = \gamma + \ln{n} + rac{1}{2}n^{-1} - rac{1}{12}n^{-2} + rac{1}{120}n^{-4} + \mathcal{O}(n^{-6})

where

\gamma

is the Euler-Mascheroni Constant

0.5772156649\dots

APPLICATIONS

The harmonic numbers appear in several calculation formulas, such as the Digamma Function :

:

\psi(n) = H_{n-1} - \gamma\,

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

:

\gamma = \lim_{n 
ightarrow \infty}{\left(H_n - \ln(n)
ight)}

although
:

\gamma = \lim_{n 
ightarrow \infty}{\left(H_n - \ln\left(n+{1 \over 2}
ight)
ight)}

converges more quickly.

An integral representation is given by Euler :

:

H_n = \int_0^1 rac{1 - x^n}{1 - x}\,dx

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

:

\int_0^1 x^n\,dx = rac{1}{n + 1}

and then

:

x^{n} + rac{1 - x^n}{1 - x} = rac{1 - x^{n+1}}{1 - x}

inside the integral.

GENERALIZATIONS

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

:

\int_a^1 rac {1-x^s}{1-x} dx = 
- \sum_{k=1}^\infty rac {1}{k} {s \choose k} (a-1)^k

which holds for general Complex-valued ''s'', for the suitably extended Binomial Coefficient s. 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

:

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

which is just the Newton's generalized Binomial Theorem . The interpolating function is in fact just the Digamma Function :

:

\psi(s+1)+\gamma = \int_0^1 rac {1-x^s}{1-x} dx

where

\psi(x)

is the digamma, and

\gamma

is the Euler-Mascheroni constant. The integration process may be repeated to obtain

:

H_{s,2}=-\sum_{k=1}^\infty rac {(-1)^k}{k} {s \choose k} H_k

GENERATING FUNCTIONS

A Generating Function for the generalized harmonic numbers is:

:

\sum_{n=1}^\infty z^n H_{n,m} = 
rac {\mbox{Li}_m(z)}{1-z}

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