Editing Harmonic number (section)

===Generalized harmonic numbers===
The ''n''th '''generalized harmonic number''' of order ''m'' is given by
<math display="block">H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}.</math>

(In some sources, this may also be denoted by <math display="inline">H_n^{(m)}</math> or <math display="inline">H_m(n).</math>)

The special case ''m'' = 0 gives <math>H_{n,0}= n.</math>  The special case ''m'' = 1 reduces to the usual harmonic number:
<math display="block">H_{n, 1} = H_n = \sum_{k=1}^n \frac{1}{k}.</math>

The limit of <math display="inline">H_{n, m}</math> as {{math|''n'' → ∞}} is finite if {{math|''m'' > 1}}, with the generalized harmonic number bounded by and converging to the [[Riemann zeta function]]
<math display="block">\lim_{n\rightarrow \infty} H_{n,m} = \zeta(m).</math>

The smallest natural number ''k'' such that ''k<sup>n</sup>'' does not divide the denominator of generalized harmonic number ''H''(''k'', ''n'') nor the denominator of alternating generalized harmonic number ''H′''(''k'', ''n'') is, for ''n''=1, 2, ... :
:77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, ... {{OEIS|id=A128670}}

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

Some integrals of generalized harmonic numbers are
<math display="block">\int_0^a H_{x,2} \, dx = a \frac {\pi^2}{6}-H_{a}</math>
and
<math display="block">\int_0^a H_{x,3} \, dx = a A - \frac {1}{2} H_{a,2},</math> where ''A'' is [[Apéry's constant]] ''ζ''(3),
and
<math display="block">\sum_{k=1}^n H_{k,m}=(n+1)H_{n,m}- H_{n,m-1} \text{ for } m \geq 0 .</math>

Every generalized harmonic number of order ''m'' can be written as a function of harmonic numbers of order <math>m-1</math> using
<math display="block">H_{n,m} = \sum_{k=1}^{n-1} \frac {H_{k,m-1}}{k(k+1)} + \frac {H_{n,m-1}}{n} </math> &nbsp; for example: <math>H_{4,3} = \frac {H_{1,2}}{1 \cdot 2} + \frac {H_{2,2}}{2 \cdot 3} + \frac {H_{3,2}}{3 \cdot 4} + \frac {H_{4,2}}{4} </math>

A [[generating function]] for the generalized harmonic numbers is
<math display="block">\sum_{n=1}^\infty z^n H_{n,m} = \frac {\operatorname{Li}_m(z)}{1-z},</math>
where <math>\operatorname{Li}_m(z)</math> is the [[polylogarithm]], and {{math|{{mabs|''z''}} < 1}}. The generating function given above for {{math|1=''m'' = 1}} is a special case of this formula.

A '''fractional argument for generalized harmonic numbers''' can be introduced as follows:

For every <math>p,q>0</math> integer, and <math>m>1</math> integer or not, we have from polygamma functions:
<math display="block">H_{q/p,m}=\zeta(m)-p^m\sum_{k=1}^\infty \frac{1}{(q+pk)^m}</math>
where <math>\zeta(m)</math> is the [[Riemann zeta function]]. The relevant recurrence relation is
<math display="block">H_{a,m}=H_{a-1,m}+\frac{1}{a^m}.</math>
Some special values are<math display="block">\begin{align}
H_{\frac{1}{4},2} &= 16-\tfrac{5}{6}\pi^2 -8G\\
H_{\frac{1}{2},2} &= 4-\frac{\pi^2}{3} \\
H_{\frac{3}{4},2} &= \frac{16}{9}-\frac{5}{6}\pi^2 + 8G \\
H_{\frac{1}{4},3} &= 64-\pi^3-27\zeta(3) \\
H_{\frac{1}{2},3} & =8-6\zeta(3) \\
H_{\frac{3}{4},3} &= \left(\frac{4}{3}\right)^3+\pi^3 -27\zeta(3)
\end{align}</math>where ''G'' is [[Catalan's constant]]. In the special case that <math>p = 1</math>, we get
<math display="block">H_{n,m}=\zeta(m, 1) - \zeta(m, n+1),</math>


where <math>\zeta(m, n)</math> is the [[Hurwitz zeta function]]. This relationship is used to calculate harmonic numbers numerically.