Study of the rank- and size-frequency functions in the case of power law growth of sources and items and proof of Heaps’ law

Supposing that the number of sources and the number of items in sources grow in time according to power laws, we present explicit formulae for the size- and rank-frequency functions in such systems. Size-frequency functions can decrease or increase while rank-frequency functions only decrease. The latter can be convex, concave, S-shaped (first convex, then concave) or reverse S-shaped (first concave, then convex). We also prove that, in such systems, Heaps’ law on the relation between the number of sources and items is valid.