We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite words are letter-to-letter projections of fixed points of these morphisms. We characterize the first differences of both functions a) with generalized Beatty sequences, or unions of generalized Beatty sequences, and b) with morphic sequences.

中文翻译：

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Zeckendorf和基础phi展开的数字函数之和

我们考虑基本phi和自然数的Zeckendorf展开的数字总和函数。对于这两个数字和函数，我们在无穷大的字母上给出了态射，使得这些被视为无穷词的函数是这些态射的固定点的字母到字母的投影。我们用广义的Beatty序列或广义Beatty序列的并集来表征这两个函数的第一个差异，而b）则是形态词。