Together in hyperharmony

Harmonic numbers are the partial sums of the harmonic series:

\displaystyle  H_n=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{n}

As every freshman knows (?), they grow without bound: {H_n\rightarrow\infty} as {n\rightarrow\infty}. It is also well known that the sequence {H_n-\log n} has a finite limit, called the Euler-Mascheroni constant {\gamma\approx 0.577\dots}. On top of that, Bernoulli numbers show up in the asymptotic expansion for {H_n},

\displaystyle  H_n=\log n+\gamma -\sum_{k=1}^\infty \frac{B_k}{k n^{k}} =\log n+\gamma +\frac{1}{2n}-\frac{1}{12n^2}-\dots

where I take the position that {B_1=-1/2}. The larger {n} gets, the fewer terms are needed for given precision.

And apparently, {\log H_n \, e^{H_n}} have something to do with the Riemann hypothesis.

In their Book of Numbers, J. H. Conway and R. K. Guy briefly entertained the idea of defining “hyperharmonic numbers” by repeating the process of partial summation.

\displaystyle   H_n^{(2)}  = H_1+H_2+\dots + H_n \\ \\ H_n^{(3)}   = H_1^{(2)}+H_2^{(2)}+\dots + H_n^{(2)}

and so on. It is natural to guess that this will produce sequences of increasing complexity. But Conway and Guy immediately destroy this illusion with the formula

\displaystyle H_n^{(r)} = \binom{n+r-1}{r-1}(H_{n+r-1}-H_{r-1})

The base case {r=2} is a neat arithmetical trick:

\displaystyle   H_1+\dots+H_n  = \frac{n}{1}+\frac{n-1}{2}+\frac{n-2}{3}+\dots +\frac{1}{n} \\ \\   = \frac{n+1}{1}-1 +\frac{n+1}{2}-1 + \frac{n+1}{3} -1 + \dots +\frac{n+1}{n} -1 \\ \\  = (n+1)H_n-n = (n+1)H_{n+1} -(n+1) = \binom{n+1}{1} (H_{n+1} -1)

For {n\ge 2}, the harmonic numbers {H_n} are never integers, although {H_{30}=3.994987\dots} makes a solid effort. The reason is simple: the highest power of {2} not exceeding {n} appears in some {1/2^k} and in no other term. Nothing to cancel it with.

The hyperharmonic numbers are conjectured to be non-integer as well, and this has been verified for many values of {r} but not in full generality.

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