What some of us have long called "prodigal" seems to have existed all along under the name https://en.wikipedia.org/wiki/Product_integral . Prodigal : Integral :: Geometric mean : Arithmetic mean. E.g., productIntegral[x!, {x, 0, t}] == E^Integrate[Log[x!], {x, 0, t}] == (2 π/E^t)^((1 + t)/2) Hyperfactorial[t] In[1176]:= Product[k^k, {k, n}] Out[1176]= Hyperfactorial[n] —rwg