The hyperharmonic number 
is obtained by repeated partial sums
of the reciprocals of the positive integers.
Set 
for 
and 
. For positive
integer order 
and 
,
 |
(1)
|
Thus 
is the ordinary harmonic number 
. For 
, the closed form is
 |
(2)
|
where the prefactor is a binomial coefficient (Benjamin et al. 2003). Repeated partial sums multiply
the ordinary generating function by 
at each step, giving
 |
(3)
|
for 
and 
.
The hyperharmonic number is implemented in the Wolfram Language as HyperHarmonicNumber[p, n].
