Вычислимые и μ-рекурсивные операторы

[Skordev D.; Skordev Dimiter; Скордев Димитър]Bulgarian. Russian, German summar

Transcendence of Binomial and Lucas' Formal Power Series

AbstractThe formal power series[formula]is transcendental over Q(X) whentis an integer ≥2. This is due to Stanley forteven, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series modulo prime numbersp, and to use thep-Lucas property: ifn=∑nipiis the basepexpansion of the integern, then[equation]The series reduced modulopis then proved algebraic over Fp(X), the field of rational functions over the Galois field Fp, but its degree is not a bounded function ofp. We generalize this method to characterize all formal power series that have thep-Lucas property for “many” prime numbersp, and that are furthermore algebraic over Q(X)

www.elsevier.com/locate/tcs Decimation-invariant sequences and their automaticity

AbstractThis paper deals with one-dimensional bidirectional sequences ā:Z→V, V a finite set, such that any p-decimation (|p|⩾2) of the sequence reproduces the sequence (modulo a certain shift). We develop a procedure for solving the underlying decimation-invariance (DI) equations and find that the number of solutions is always finite. Conditions for equivalency among solutions of differently parametrized DI-problems, and for possible periodicity and symmetry of solutions, are derived. It is shown that the set of all possible p-based decimations of a such a DI sequence (the so-called full kernel of the sequence) is finite. This implies finiteness of the kernel for the separate right and left parts of the sequence, and also |p|-automaticity of these parts. An algorithm is presented that constructs the kernel and associated |p|-automaton of a DI-sequence explicitly