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On the zeros of T-universal functions

Let {zn} be a sequence of complex numbers with zn → ∞ (n → ∞). We consider entire functions φ which behave universal under translates (for short: T-universal functions), i. e. for every compact set K with connected complement and every function f, which is continuous on K and holomorphic in the interior of K, there exists a subsequence {nk} of natural numbers, such that φ(z+znk) → f(z) uniformly on K. In this note we prove the existence of entire T-universal functions that have zeros at certain prescribed points and whose zeros are all known. In addition we will show that such functions may have a regular rate of growth of its zeros.