In my post on hypertetrahedral polytopic roots, I gave a few formulae for functions such as the triangular root.
Here I give some very similar formulae for some falling and rising roots.
The notations for rising and falling factorials or powers are, respectively:
If then , and so (inventing some notation):
It is easy to see that these are related: this is because
We also have that
From these together with my earlier post, we therefore have, for example:
Similarly the other formulae given in that earlier post give rise easily to a few further formulae.