Let β be a fixed element of 𝔽q((X-1)) with polynomial part of degree ≥ 1, then any formal power series can be represented in base β, using the transformation Tβ : f ↦ {βf} of the unit disk [Formula: see text]. Any formal power series in [Formula: see text] is expanded in this way into dβ(f) = (ai(X))i≥1, where [Formula: see text]. The main aim of this paper is to characterize the formal power series β(|β| > 1), such that dβ(1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case).

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The aim of this work is to prove new results on a class of digital functions with special emphasis on shifted primes as arguments. Our method lies on the estimate of exponential sums of the form n x Λ(n) exp(2iπf (n + cn) + βn) where f a digital function, c = (cn) is an almostperiodic sequence in Z and β is a real parameter, which extend the works of and to the case of the shifted prime numbers satisfying a digital constraint.

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