# Is it the 'd' or 'D' operator?

Philip J. Davis' article on the history of the gamma function (PDF) mentions how Leibniz proposed the iterated differential operator (p. 851 in the upper right corner, or p. 3 of the PDF, about half-way down the page).

Dr. Davis uses a lower-case 'd' in his article. However, most contemporary literature (er, Web sites), shows an upper-case 'D' for this. Examples:

I'm copy-editing Davis' paper (just for fun and to practice LaTeX), and so now I'm curious:

## What did Leibniz use - 'd' or 'D'?

Please note that I'm asking specifically about the iterated differential operator (often called a Riemann-Liouville fractional derivative, or RLFD), and not the 'd' in Leibniz's $$\frac{\text{d}y}{\text{d}x}$$ notation (although they are related).

• Lower case. See hsm.stackexchange.com/q/3323/6514, in particular the answer by Mauro Allegranza. – Torsten Schoeneberg Feb 23 '19 at 3:38
• If by $D$ you mean an operator which acts on things of type $\mathbb{R}\to \mathbb{R}$ (modern functions), then Leibniz had none of that, since he did not have modern functions. He had a small $d$, (the one you explicitly want to exclude), which acted on variable quantities, which modern mathematicians don't know how to make sense of. – Michael Bächtold Feb 28 '19 at 16:25
• The first hit that I got for your Google search is en.wikipedia.org/wiki/… and here we can see both a decorated $d$ applied to variable quantities and a decorated $D$ applied to modern functions. And as Michael noted above, Leibniz had only the former. (Although it seems to me that that bit on Wikipedia might better be rewritten to use only the latter.) – Toby Bartels May 2 '19 at 19:29