Editing Harmonic number (section)

==Generalizations==

===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.

===Multiplication formulas===
The [[multiplication theorem]] applies to harmonic numbers. Using [[polygamma]] functions, we obtain
<math display="block">\begin{align}
H_{2x} & =\frac{1}{2}\left(H_x+H_{x-\frac{1}{2}}\right)+\ln 2 \\
H_{3x} &= \frac{1}{3}\left(H_x+H_{x-\frac{1}{3}}+H_{x-\frac{2}{3}}\right)+\ln 3,
\end{align}</math>
or, more generally,
<math display="block">H_{nx}=\frac{1}{n}\left(H_x+H_{x-\frac{1}{n}}+H_{x-\frac{2}{n}}+\cdots +H_{x-\frac{n-1}{n}} \right) + \ln n.</math>

For generalized harmonic numbers, we have
<math display="block">\begin{align}
H_{2x,2} &= \frac{1}{2}\left(\zeta(2)+\frac{1}{2}\left(H_{x,2}+H_{x-\frac{1}{2},2}\right)\right) \\
H_{3x,2} &= \frac{1}{9}\left(6\zeta(2)+H_{x,2}+H_{x-\frac{1}{3},2}+H_{x-\frac{2}{3},2}\right),
\end{align}</math>
where <math>\zeta(n)</math> is the [[Riemann zeta function]].

===Hyperharmonic numbers===

The next generalization was discussed by [[John Horton Conway|J. H. Conway]] and [[Richard K. Guy|R. K. Guy]] in their 1995 book ''[[The Book of Numbers (maths)|The Book of Numbers]]''.<ref name=ConwayGuy/>{{rp|258}} Let
<math display="block"> H_n^{(0)} = \frac1n. </math>
Then the nth [[hyperharmonic number]] of order ''r'' (''r>0'') is defined recursively as
<math display="block"> H_n^{(r)} = \sum_{k=1}^n H_k^{(r-1)}. </math>
In particular, <math>H_n^{(1)}</math> is the ordinary harmonic number <math>H_n</math>.

=== Roman Harmonic numbers ===
The [[Roman Harmonic numbers]],<ref>{{Cite journal |last=Sesma |first=J. |date=2017 |title=The Roman harmonic numbers revisited |url=http://dx.doi.org/10.1016/j.jnt.2017.05.009 |journal=Journal of Number Theory |volume=180 |pages=544–565 |doi=10.1016/j.jnt.2017.05.009 |issn=0022-314X|arxiv=1702.03718 }}</ref> named after [[Steven Roman]], were introduced by [[Daniel E. Loeb|Daniel Loeb]] and [[Gian-Carlo Rota]] in the context of a generalization of [[umbral calculus]] with logarithms.<ref>{{Cite journal |last1=Loeb |first1=Daniel E |last2=Rota |first2=Gian-Carlo |date=1989 |title=Formal power series of logarithmic type |journal=Advances in Mathematics |volume=75 |issue=1 |pages=1–118 |doi=10.1016/0001-8708(89)90079-0 |issn=0001-8708|doi-access=free }}</ref>  There are many possible definitions, but one of them, for  <math>n,k \geq 0</math>, is<math display="block"> c_n^{(0)} = 1, </math>and<math display="block"> c_n^{(k+1)} = \sum_{i=1}^n\frac{c_i^{(k)}}{i}. </math>Of course,<math display="block"> c_n^{(1)} = H_n. </math>

If <math>n \neq 0</math>, they satisfy<math display="block"> c_n^{(k+1)} - \frac{c_n^{(k)}}{n} = c_{n-1}^{(k+1)}. </math>Closed form formulas are<math display="block"> c_n^{(k)} = n! (-1)^k s(-n,k), </math>where <math>s(-n,k)</math> is [[Stirling numbers of the first kind]] generalized to negative first argument, and<math display="block"> c_n^{(k)} = \sum_{j=1}^n \binom{n}{j} \frac{(-1)^{j-1}}{j^k}, </math>which was found by [[Donald Knuth]].

In fact, these numbers were defined in a more general manner using Roman numbers and [[Roman factorials]], that include negative values for <math>n</math>. This generalization was useful in their study to define [[Harmonic logarithms]].