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What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?

[Background:] Looking at the powerseries for the gamma-function $ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $ then we can arrive at a decomposition $ \Gamma(1+x) = r(x) + g(x) $ ...