In [my other thread](https://math.stackexchange.com/questions/89853) I discussed a matrix-decomposition; for one matrix (***U***) I found now a description of its entries, which may best be denoted as "recursive harmonic numbers". However, googling with this term leads to hits on a different thing. What I have is the following. 

Define a recursive type of harmonic-numbers of two arguments: 

$ \qquad \small h\_{0,c} =1 \text{ for all } c\gt 0 $  

$ \qquad \small h\_{r,1} =1 \text{ for all } r\gt 0 $  

$ \qquad \small h\_{r,c} = h\_{r,c-1} + h\_{r-1,c}/c \text{ for all } c\gt r $ 

Then we have  

$\qquad \small h\_{0,\*} = (1,1,1,1,1,\ldots) $  

$\qquad \small h\_{1,\*} = (1,
 1+{1 \over 2}, 
 1+{1 \over 2}+{1 \over 3},
 1+{1 \over 2}+{1 \over 3}+{1 \over 4},
 \ldots) $
$\qquad \small h\_{2,\*} = (1,
 1+ {1 \over 2}\left( 1+{1 \over 2}\right), 
 1+ {1 \over 2}\left(1+{1 \over 2}\right)
 +{1 \over 3}\left(1+{1 \over 2}+{1 \over 3},\right),
 \ldots) $  

where the recursion becomes visible by  

$\qquad \small h\_{2,\*} = (1,
 1+ {1 \over 2} h\_{1,2}, 
 1+ {1 \over 2} h\_{1,2} + {1 \over 3} h\_{1,3} 
 \ldots) $ 

Is someone aware of such a generalization/extension? I have a vague idea that I've seen the terms of $\small h\_{2,\*} $ in some online available article not too long ago (arXiv?) but can't remember exactly. Also with "generalized harmonic numbers" mostly is meant either the summation to fractional bounds of the index and/or higher exponents in the denominator of the harmonic numbers, so this is different...

---

I've [found the arXiv-article](http://arxiv.org/abs/math/9502217). However, the recursive harmonic numbers are only mentioned as intermediate terms (pg 6, eq (6)) for a generalization of the Stirling numbers first kind (which is in the context of my own discussion not surprising). If there are some more references dealing more specific with these, this were very good...