THE GROUP GENERATED BY THE GAMMA FUNCTIONS Γ(ax + 1), AND ITS SUBGROUP OF THE ELEMENTS CONVERGING TO CONSTANTS

Abstract

Let G be the multiplicative group generated by the gamma functions Γ(ax + 1) (a = 1, 2,...), and H be the subgroup of all elements of G that converge to non-zero constants as x →∞. The quotient group G/H is the group of equivalence classes of G, where ƒand g are equivalent ⇔ ƒ∼Cg (x →∞) for some C ≠ 0. We show that G/H ~ Q⁺. Similar consideration is possible for the case that the gamma functions Γ(ax + 1) with a ∈R⁺ are concerned, and we show that G/H ~ Z × R × R. Also, several concrete examples of the elements of H are constructed, e.g. it holds that / → (n →∞), where denotes a multinomial coefficient.