Hyperharmonic number

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.^[4]

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\ge1,m\ge 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^[5] 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.^[6] 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.^[7] These authors have also shown that $H_n^{(r)}$ is never integer when n is even or a prime power, or r is odd.

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.
1 2 Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130: 360–369. doi:10.1016/j.jnt.2009.08.005.
↑ Mező, István; Dil, Ayhan (2009). "Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence". Central European Journal of Mathematics. 7 (2): 310–321. doi:10.2478/s11533-009-0008-5.
↑ 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.
↑ Göral, Haydar; Doğa Can, Sertbaş (2017). "Almost all hyperharmonic numbers are not integers". Journal of Number Theory (171): 495–526. doi:10.1016/j.jnt.2016.07.023.

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[ConwayGuy-1] John H., Conway; Richard K., Guy (1995). The book of numbers. Copernicus.

[BGG-2] Benjamin, A. T.; Gaebler, D.; Gaebler, R. (2003). "A combinatorial approach to hyperharmonic numbers". Integers (3): 1–9.

[MezoDil-3] 1 2 Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130: 360–369. doi:10.1016/j.jnt.2009.08.005.

[MezoDil2009-4] Mező, István; Dil, Ayhan (2009). "Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence". Central European Journal of Mathematics. 7 (2): 310–321. doi:10.2478/s11533-009-0008-5.

[Mezo-5] 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.

[AB-6] Amrane, R. A.; Belbachir, H. (2010). "Non-integerness of class of hyperharmonic numbers". Annales Mathematicae et Informaticae (37): 7–11.

[GS-7] Göral, Haydar; Doğa Can, Sertbaş (2017). "Almost all hyperharmonic numbers are not integers". Journal of Number Theory (171): 495–526. doi:10.1016/j.jnt.2016.07.023.