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| Difference Independence of the Euler Gamma Function* |
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Citation： |
Qiongyan WANG,Xiao YAO.Difference Independence of the Euler Gamma Function*[J].Chinese Annals of Mathematics B,2023,44(4):481~488 |
| Page view： 1452
Net amount： 901 |
Authors： |
Qiongyan WANG; Xiao YAO |
Foundation： |
This work was supported by the National Natural Science Foundation of China (No. 11901311). |
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| Abstract： |
In this paper, the authors established a sharp version of the difference ana logue of the celebrated H¨older’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 + a0), · · · , Γ(s + an-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 H¨older in 1886, and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006. |
Keywords： |
Algebraic difference independence, Euler gamma function, Algebraic difference equations |
Classification： |
11M06, 39A05 |
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