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

The sum of digits functions of the Zeckendorf and the base phi expansions