Hyperharmonic number

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In mathematics, the n-th hyperharmonic number of order r, denoted by , is recursively defined by the relations:

and

[citation needed]

In particular, is the n-th harmonic number.

The hyperharmonic numbers were discussed by J. H. Conway and R. K. Guy in their 1995 book .:258

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

reads as

where is an r-Stirling number of the first kind.

Asymptotics[]

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

when m>r.

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.

External links[]

Notes[]

  1. John H., Conway; Richard K., Guy (1995). The book of numbers. Copernicus. ISBN 9780387979939.
  2. Benjamin, A. T.; Gaebler, D.; Gaebler, R. (2003). "A combinatorial approach to hyperharmonic numbers". Integers (3): 1–9.
  3. ^ 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.
  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.
  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.
  6. Amrane, R. A.; Belbachir, H. (2010). "Non-integerness of class of hyperharmonic numbers". Annales Mathematicae et Informaticae (37): 7–11.
  7. 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.
  8. Alkan, Emre; Göral, Haydar; Doğa Can, Sertbaş (2018). "Hyperharmonic numbers can rarely be integers". Integers (18).