Editing Harmonic number (section)

=== Roman Harmonic numbers ===
The [[Roman Harmonic numbers]],<ref>{{Cite journal |last=Sesma |first=J. |date=2017 |title=The Roman harmonic numbers revisited |url=http://dx.doi.org/10.1016/j.jnt.2017.05.009 |journal=Journal of Number Theory |volume=180 |pages=544–565 |doi=10.1016/j.jnt.2017.05.009 |issn=0022-314X|arxiv=1702.03718 }}</ref> named after [[Steven Roman]], were introduced by [[Daniel E. Loeb|Daniel Loeb]] and [[Gian-Carlo Rota]] in the context of a generalization of [[umbral calculus]] with logarithms.<ref>{{Cite journal |last1=Loeb |first1=Daniel E |last2=Rota |first2=Gian-Carlo |date=1989 |title=Formal power series of logarithmic type |journal=Advances in Mathematics |volume=75 |issue=1 |pages=1–118 |doi=10.1016/0001-8708(89)90079-0 |issn=0001-8708|doi-access=free }}</ref>  There are many possible definitions, but one of them, for  <math>n,k \geq 0</math>, is<math display="block"> c_n^{(0)} = 1, </math>and<math display="block"> c_n^{(k+1)} = \sum_{i=1}^n\frac{c_i^{(k)}}{i}. </math>Of course,<math display="block"> c_n^{(1)} = H_n. </math>

If <math>n \neq 0</math>, they satisfy<math display="block"> c_n^{(k+1)} - \frac{c_n^{(k)}}{n} = c_{n-1}^{(k+1)}. </math>Closed form formulas are<math display="block"> c_n^{(k)} = n! (-1)^k s(-n,k), </math>where <math>s(-n,k)</math> is [[Stirling numbers of the first kind]] generalized to negative first argument, and<math display="block"> c_n^{(k)} = \sum_{j=1}^n \binom{n}{j} \frac{(-1)^{j-1}}{j^k}, </math>which was found by [[Donald Knuth]].

In fact, these numbers were defined in a more general manner using Roman numbers and [[Roman factorials]], that include negative values for <math>n</math>. This generalization was useful in their study to define [[Harmonic logarithms]].