Define a transform on polynomials which is linear and acts on each monomial as $$\widehat{z^k} = \frac{(1+z)(2+z)\ldots(k+z)}{k!}.$$ Does anyone know whether this has a name (and therefore has been studied)?

(Also, not sure what to tag this with... please suggest/edit.)

inverse Beta transformbecause, although off by a factor of 1, your expression above can be rewritten as $$\frac{\Gamma(z+k+1)}{\Gamma(k+1)\Gamma(z+1)}$$ – Suvrit May 13 '12 at 23:47