In mathematics, the n-th hyperharmonic number of order r, denoted by H n ( r ) {\displaystyle H_{n}^{(r)}} , is recursively defined by the relations:
H n ( 0 ) = 1 n , {\displaystyle H_{n}^{(0)}={\frac {1}{n}},}
and
H n ( r ) = ∑ k = 1 n H k ( r − 1 ) ( r > 0 ) . {\displaystyle H_{n}^{(r)}=\sum _{k=1}^{n}H_{k}^{(r-1)}\quad (r>0).}
In particular, H n = H n ( 1 ) {\displaystyle 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.
Identities involving hyperharmonic numbers By definition, the hyperharmonic numbers satisfy the recurrence relation
H n ( r ) = H n − 1 ( r ) + H n ( r − 1 ) . {\displaystyle 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 ) = ( n + r − 1 r − 1 ) ( H n + r − 1 − H r − 1 ) . {\displaystyle 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 = 1 n ! [ n + 1 2 ] . {\displaystyle H_{n}={\frac {1}{n!}}\left[{n+1 \atop 2}\right].}
reads as
H n ( r ) = 1 n ! [ n + r r + 1 ] r , {\displaystyle H_{n}^{(r)}={\frac {1}{n!}}\left[{n+r \atop r+1}\right]_{r},}
where [ n r ] r {\displaystyle \left[{n \atop r}\right]_{r}} 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.
H n ( r ) ∼ 1 ( r − 1 ) ! ( n r − 1 ln ( n ) ) , {\displaystyle 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
∑ n = 1 ∞ H n ( r ) n m < + ∞ {\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}^{(r)}}{n^{m}}}<+\infty }
when m>r.
… excerpt ends here. Continue reading the full article.
