hyperharmonic number

By definition, the hyperharmonic numbers satisfy the recurrence relation

:$H\_n^\{(r)\}\; =\; H\_\{n-1\}^\{(r)\}\; +\; H\_n^\{(r-1)\}.$

In place of the recurrences, there is a more effective formula to calculate these numbers:

: $H\_\{n\}^\{(r)\}=\backslash binom\{n+r-1\}\{r-1\}(H\_\{n+r-1\}-H\_\{r-1\}).$

The hyperharmonic numbers have a strong relation to combinatorics of permutations. The generalization of the identity

: $H\_n\; =\; \backslash frac\{1\}\{n!\}\backslash left[\{n+1\; \backslash atop\; 2\}\backslash right].$

reads as

: $H\_n^\{(r)\}\; =\; \backslash frac\{1\}\{n!\}\backslash left[\{n+r\; \backslash atop\; r+1\}\backslash right]\_r,$

where $\backslash left[\{n\; \backslash atop\; r\}\backslash right]\_r$ is an r-Stirling number of the first kind.{{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 }}

The above expression with binomial coefficients easily gives that for all fixed order r>=2 we have.{{cite journal | last1 = Mező | first1 = István| last2 = Dil | first2 = Ayhan | title = Hyperharmonic series involving Hurwitz zeta function | journal = Journal of Number Theory| year = 2010 | volume = 130 | issue = 2| pages = 360–369 | doi=10.1016/j.jnt.2009.08.005| hdl = 2437/90539 | hdl-access = free }}

: $H\_n^\{(r)\}\backslash sim\backslash frac\{1\}\{(r-1)!\}\backslash left(n^\{r-1\}\backslash ln(n)\backslash right),$

that is, the quotient of the left and right hand side tends to 1 as n tends to infinity.

An immediate consequence is that

:

when m>r.

The generating function of the hyperharmonic numbers is

: $\backslash sum\_\{n=0\}^\backslash infty\; H\_n^\{(r)\}z^n=-\backslash frac\{\backslash ln(1-z)\}\{(1-z)^r\}.$

The exponential generating function is much more harder to deduce. One has that for all r=1,2,...

: $\backslash sum\_\{n=0\}^\backslash infty\; H\_n^\{(r)\}\backslash frac\{t^n\}\{n!\}=e^t\backslash left(\backslash sum\_\{n=1\}^\{r-1\}H\_n^\{(r-n)\}\backslash frac\{t^n\}\{n!\}+\backslash frac\{(r-1)!\}\{(r!)^2\}t^r\backslash ,\; \_2\; F\_2\backslash left(1,1;r+1,r+1;-t\backslash right)\backslash right),$

where ₂F₂ 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.{{Cite journal

| last1 = Mező | first1 = István

| last2 = Dil | first2 = Ayhan

| title = Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

| year = 2009

| volume = 7

| issue = 2

| pages = 310–321

| journal = Central European Journal of Mathematics

| doi=10.2478/s11533-009-0008-5

| doi-access = free

}}

The next relation connects the hyperharmonic numbers to the Hurwitz zeta function:

: $\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{H\_n^\{(r)\}\}\{n^m\}=\backslash sum\_\{n=1\}^\backslash infty\; H\_n^\{(r-1)\}\backslash zeta(m,n)\backslash quad(r\backslash ge1,m\backslash ge\; r+1).$

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{{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 }} 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.{{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 }} Especially, these authors proved that $H\_n^\{(4)\}$ is not integer for all r<26 and n=2,3,... Extension to high orders was made by Göral and Sertbaş.{{cite journal | last1 = Göral | first1 = Haydar | last2 = Doğa Can | first2 = Sertbaş | title = Almost all hyperharmonic numbers are not integers | journal = Journal of Number Theory| volume = 171 | year = 2017 | issue = 171 | pages = 495–526 | doi=10.1016/j.jnt.2016.07.023| doi-access = free }} These authors have also shown that $H\_n^\{(r)\}$ is never integer when n is even or a prime power, or r is odd.

Another result is the following.{{cite journal | last1 = Alkan | first1 = Emre | last2 = Göral | first2 = Haydar | last3 = Doğa Can | first3 = Sertbaş | title = Hyperharmonic numbers can rarely be integers | journal = Integers | year = 2018 | issue = 18 }} Let $S(x)$ be the number of non-integer hyperharmonic numbers such that $(n,x)\backslash in[0,x]\backslash times[0,x]$. Then, assuming the Cramér's conjecture,

:$$

S(x)=x^2+O(x\log^3x).

Note that the number of integer lattice points in $[0,x]\backslash times[0,x]$ is $x^2+O(x^2)$, which shows that most of the hyperharmonic numbers cannot be integer.

The problem was finally settled by D. C. Sertbaş who found that there are infinitely many hyperharmonic integers, albeit they are quite huge. The smallest hyperharmonic number which is an integer found so far is{{cite journal | last1 = Doğa Can | first1 = Sertbaş | title = Hyperharmonic integers exist | journal = Comptes Rendus Mathématique | year = 2020 | issue = 358 }}

:$$

H_{33}^{(64(2^{2659}-1)+32)}.

• [http://mathworld.wolfram.com/HarmonicNumber.html Wolfram MathWorld]

{{reflist|refs=

{{Cite book

| last1 = John H. | first1 = Conway

| last2 = Richard K. | first2 = Guy

| title = The book of numbers

| year = 1995

| publisher = Copernicus

| url = https://books.google.com/books?id=0--3rcO7dMYC&pg=PP1

| isbn = 9780387979939

}}

}}

{{DEFAULTSORT:Hyperharmonic number}}

Category:Number theory