In mathematics, the n-th hyperharmonic number of order r, denoted by , is recursively defined by the relations:
In particular, is the n-th harmonic number.
Identities involving hyperharmonic numbers
By definition, the hyperharmonic numbers satisfy the recurrence relation
In place of the recurrences, there is a more effective formula to calculate these numbers:
The hyperharmonic numbers have a strong relation to combinatorics of permutations. The generalization of the identity
where is an r-Stirling number of the first kind.
The above expression with binomial coefficients easily gives that for all fixed order r>=2 we have.
that is, the quotient of the left and right hand side tends to 1 as n tends to infinity.
An immediate consequence is that
Generating function and infinite series
The generating function of the hyperharmonic numbers is
The exponential generating function is much more harder to deduce. One has that for all r=1,2,...
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.
The next relation connects the hyperharmonic numbers to the Hurwitz zeta function:
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 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. Especially, these authors proved that is not integer for all r<26 and n=2,3,... Extension to high orders was made by Göral and Sertbaş. These authors have also shown that is never integer when n is even or a prime power, or r is odd.
Another result is the following. Let be the number of non-integer hyperharmonic numbers such that . Then, assuming the Cramér's conjecture,
Note that the number of integer lattice points in is , which shows that most of the hyperharmonic numbers cannot be integer. The conjecture, however, is still open.
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- Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130 (2): 360–369. doi:10.1016/j.jnt.2009.08.005. hdl:2437/90539.
- 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.
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- Göral, Haydar; Doğa Can, Sertbaş (2017). "Almost all hyperharmonic numbers are not integers". Journal of Number Theory. 171 (171): 495–526. doi:10.1016/j.jnt.2016.07.023.
- Alkan, Emre; Göral, Haydar; Doğa Can, Sertbaş (2018). "Hyperharmonic numbers can rarely be integers". Integers (18).