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| In [[mathematics]], the ''n''-th '''hyperharmonic number''' of order ''r'', denoted by <math>H_n^{(r)}</math>, is recursively defined by the relations:
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| : <math> H_n^{(0)} = \frac{1}{n} ,</math>
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| and
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| : <math> H_n^{(r)} = \sum_{k=1}^n H_k^{(r-1)}\quad(r>0). </math> | |
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| In particular, <math>H_n=H_n^{(1)}</math> is the ''n''-th [[harmonic number]].
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| The hyperharmonic numbers were 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}}
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| ==Identities involving hyperharmonic numbers==
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| By definition, the hyperharmonic numbers satisfy the [[recurrence relation]]
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| :<math> H_n^{(r)} = H_{n-1}^{(r)} + H_n^{(r-1)}. </math> | |
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| In place of the recurrences, there is a more effective formula to calculate these numbers:
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| : <math> H_{n}^{(r)}=\binom{n+r-1}{r-1}(H_{n+r-1}-H_{r-1}). </math>
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| The hyperharmonic numbers have a strong relation to combinatorics of permutations. The generalization of the identity
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| : <math> H_n = \frac{1}{n!}\left[{n+1 \atop 2}\right]. </math>
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| reads as
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| : <math> H_n^{(r)} = \frac{1}{n!}\left[{n+r \atop r+1}\right]_r, </math>
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| where <math>\left[{n \atop r}\right]_r</math> is an ''r''-Stirling number of the first kind.<ref name='BGG'>{{cite journal | last1 = Benjamin | first1 = A. T. | last2 = Gaebler | first2 = D. | last3 = Gaebler | first3 = R. | title = A combinatorial approach to hyperharmonic numbers | journal = Integers | year = 2003 | issue = 3 | pages = 1–9 }}</ref>
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| ==Asymptotics== | |
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| The above expression with binomial coefficients easily gives that for all fixed order ''r>=2'' we have.<ref name='MezoDil'>{{cite journal | last1 = Mező | first1 = István| last2 = Dil | first2 = Ayhan | title = Hyperharmonic series involving Hurwitz zeta function | journal = Journal of Number Theory| year = 2010 | issue = 130 | pages = 360–369 }}</ref>
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| : <math> H_n^{(r)}\sim\frac{1}{(r-1)!}\left(n^{r-1}\ln(n)\right),</math> | |
| that is, the quotient of the left and right hand side tends to 1 as ''n'' tends to infinity.
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| An immediate consequence is that
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| : <math> \sum_{n=1}^\infty\frac{H_n^{(r)}}{n^m}<+\infty </math> | |
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| when ''m>r''.
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| ==Generating function and infinite series==
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| The [[generating function]] of the hyperharmonic numbers is
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| : <math> \sum_{n=0}^\infty H_n^{(r)}z^n=-\frac{\ln(1-z)}{(1-z)^r}. </math>
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| The [[Exponential generating function#Exponential_generating_function|exponential generating function]] is much more harder to deduce. One has that for all ''r=1,2,...''
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| : <math>\sum_{n=0}^\infty H_n^{(r)}\frac{t^n}{n!}=e^t\left(\sum_{n=1}^{r-1}H_n^{(r-n)}\frac{t^n}{n!}+\frac{(r-1)!}{(r!)^2}t^r\, _2 F_2\left(1,1;r+1,r+1;-t\right)\right), | |
| </math>
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| where ''<sub>2</sub>F<sub>2</sub>'' is a [[hypergeometric function]]. The ''r=1'' case for the harmonic numbers is a classical result, the general one was proved in 2009 by I. Mező and A. Dil.<ref name = 'MezoDil'>{{Cite journal
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| | last1 = Mező | first1 = István
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| | last2 = Dil | first2 = Ayhan
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| | title = Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence
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| | year = 2009
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| | volume = 7
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| | issue = 2
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| | pages = 310–321
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| | journal = Central European Journal of Mathematics
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| }}</ref>
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| The next relation connects the hyperharmonic numbers to the [[Hurwitz zeta function]]:<ref name= MezoDil/>
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| : <math>\sum_{n=1}^\infty\frac{H_n^{(r)}}{n^m}=\sum_{n=1}^\infty H_n^{(r-1)}\zeta(m,n)\quad(r\ge1,m\ge r+1). </math>
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| ==An open conjecture==
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| It is known, that the harmonic numbers are never integers except the case ''n=1''. The same question can be posed with respect to the hyperharmonic numbers: are there integer hyperharmonic numbers? István Mező proved<ref name='Mezo'>{{cite journal | last1 = Mező | first1 = István | title = About the non-integer property of the hyperharmonic numbers | journal = Annales Universitatis Scientarium Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica | year = 2007 | issue = 50 | pages = 13–20 }}</ref> that if ''r=2'' or ''r=3'', these numbers are never integers except the trivial case when ''n=1''. He conjectured that this is always the case, namely, the hyperharmonic numbers of order ''r'' are never integers except when ''n=1''. This conjecture was justified for a class of parameters by R. Amrane and H. Belbachir.<ref name='AB'>{{cite journal | last1 = Amrane | first1 = R. A. | last2 = Belbachir | first2 = H. | title = Non-integerness of class of hyperharmonic numbers | journal = Annales Mathematicae et Informaticae | year = 2010 | issue = 37 | pages = 7–11 }}</ref> Especially, these authors proved that <math>H_n^{(4)}</math> is not integer for all n=2,3,...
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| ==External links==
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| * [http://mathworld.wolfram.com/HarmonicNumber.html Wolfram MathWorld]
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| ==Notes==
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| {{reflist|refs=
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| <ref name=ConwayGuy>
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| {{Cite book
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| | last1 = John H. | first1 = Conway
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| | last2 = Richard K. | first2 = Guy
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| | title = The book of numbers
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| | year = 1995
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| | publisher = Copernicus
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| }}</ref>
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| }}
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| {{DEFAULTSORT:Hyperharmonic number}}
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| [[Category:Number theory]]
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