I would like to be able to use the output coefficients from pchip to do various things, such as estimate the derivative at a point. This is easy enough to do with spline, since one knows that the coefficients correspond to a generic cubic polynomial in each subinterval. You can differentiate the cubic analytically and proceed, and it's clear what the coefficients mean.
However, I see absolutely nothing in the documentation indicating precisely what the "basis functions" are associated with the cubic Hermite interpolant. I suppose this is standard, but I wouldn't know it from the Matlab documentation. I know there are two references cited, but that's a bit much to go through in my opinion for such a simple piece of information, which I think should just be spelled out in a couple of lines in the help for pchip.