Hyperharmonic number

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In mathematics, the n-th hyperharmonic number of order r, denoted by $H_{n}^{{(r)}}$ , is recursively defined by the relations:

$H_{n}^{{(0)}}={\frac {1}{n}},$

and

$H_{n}^{{(r)}}=\sum _{{k=1}}^{n}H_{k}^{{(r-1)}}\quad (r>0).$

In particular, $H_{n}=H_{n}^{{(1)}}$ is the n-th harmonic number.

The hyperharmonic numbers were discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers.^[1]^:258

Identities involving hyperharmonic numbers

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:

$H_{{n}}^{{(r)}}={\binom {n+r-1}{r-1}}(H_{{n+r-1}}-H_{{r-1}}).$

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

$H_{n}={\frac {1}{n!}}\left[{n+1 \atop 2}\right].$

reads as

$H_{n}^{{(r)}}={\frac {1}{n!}}\left[{n+r \atop r+1}\right]_{r},$

where $\left[{n \atop r}\right]_{r}$ is an r-Stirling number of the first kind.^[2]

Asymptotics

The above expression with binomial coefficients easily gives that for all fixed order r>=2 we have.^[3]

$H_{n}^{{(r)}}\sim {\frac {1}{(r-1)!}}\left(n^{{r-1}}\ln(n)\right),$

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

An immediate consequence is that

$\sum _{{n=1}}^{\infty }{\frac {H_{n}^{{(r)}}}{n^{m}}}<+\infty$

when m>r.

Generating function and infinite series

The generating function of the hyperharmonic numbers is

$\sum _{{n=0}}^{\infty }H_{n}^{{(r)}}z^{n}=-{\frac {\ln(1-z)}{(1-z)^{r}}}.$

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

$\sum _{{n=0}}^{\infty }H_{n}^{{(r)}}{\frac {t^{n}}{n!}}=e^{t}\left(\sum _{{n=1}}^{{r-1}}H_{n}^{{(r-n)}}{\frac {t^{n}}{n!}}+{\frac {(r-1)!}{(r!)^{2}}}t^{r}\,_{2}F_{2}\left(1,1;r+1,r+1;-t\right)\right),$

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.^[3]

The next relation connects the hyperharmonic numbers to the Hurwitz zeta function:^[3]

$\sum _{{n=1}}^{\infty }{\frac {H_{n}^{{(r)}}}{n^{m}}}=\sum _{{n=1}}^{\infty }H_{n}^{{(r-1)}}\zeta (m,n)\quad (r\geq 1,m\geq r+1).$

An open conjecture

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^[4] 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.^[5] Especially, these authors proved that $H_{n}^{{(4)}}$ is not integer for all n=2,3,...

External links

Wolfram MathWorld

Notes

↑ John H., Conway; Richard K., Guy (1995). The book of numbers. Copernicus.
↑ Benjamin, A. T.; Gaebler, D.; Gaebler, R. (2003). "A combinatorial approach to hyperharmonic numbers". Integers (3): 1–9.
↑ 3.0 3.1 3.2 Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory (130): 360–369.
↑ Mező, István (2007). "About the non-integer property of the hyperharmonic numbers". Annales Universitatis Scientarium Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica (50): 13–20.
↑ Amrane, R. A.; Belbachir, H. (2010). "Non-integerness of class of hyperharmonic numbers". Annales Mathematicae et Informaticae (37): 7–11.

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