Difference Independence of the Euler Gamma Function.

In this paper, the authors established a sharp version of the difference analogue of the celebrated Hölder's theorem concerning the differential independence of the Euler gamma function Γ. More precisely, if P is a polynomial of n + 1 variables in ℂ[X, Y0, ⋯, Yn−1] such that P (s , Γ (s + a 0) , ⋯ , Γ (s + a n − 1)) ≡ 0 for some (a0, ⋯, an−1) ∈ ℂn and ai − aj ∉ ℤ for any 0 ≤ i ≤ j ≤ n − 1, then they have P ≡ 0. . Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by Hölder in 1886, and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006. [ABSTRACT FROM AUTHOR]