The multiple harmonic number of depth 
is a finite nested sum with indices
, where 
is a nonnegative integer.
With weights 
and exponents 
, it is
 |
The weights are omitted from the notation when they all equal 1. At depth one with unit weight, 
is the generalized harmonic number 
. For positive
integer exponents, taking 
to infinity gives a multidimensional
polylogarithm for 
for all 
.
For unit weights and 
,
it gives a value of the multivariate zeta
function.
The multiple harmonic star sum 
uses weak inequalities between successive
indices and unit weights (Zhao 2016). At depth two, separating the equal-index terms
gives
 |
The strict sums are implemented in the Wolfram Language as MultipleHarmonicNumber[n, {a1, ..., ak}, {z1, ..., zk}]. The final list can be omitted for unit weights.
Multiple Harmonic Sums." 2004.