Allouche JP., Shallit J. (1990) The ring of k-regular sequences. In: Choffrut C., Lengauer T. (eds) STACS 90. STACS 1990. Lecture Notes in Computer Science, vol 415. Springer, Berlin, Heidelberg
The automatic sequence is the central concept at the intersection of formal language theory and number theory. It was introduced by Cobham, and has been extensively studied by Christol, Kamae, Mendès France and Rauzy, and other writers. Since the range of an automatic sequence is finite, however, their descriptive power is severely limited.
In this paper, we generalize the concept of automatic sequence to the case where the sequence can take its values in a (possibly infinite) ring R; we call such sequences k-regular. (If R is finite, we obtain automatic sequences as a special case.) We argue that k-regular sequences provide a good framework for discussing many “naturally-occurring” sequences, and we support this contention by exhibiting many examples of k-regular sequences from numerical analysis, topology, number theory, combinatorics, analysis of algorithms, and the theory of fractals.
We investigate the closure properties of k-regular sequences. We prove that the set of k-regular sequences forms a ring under the operations of term-by-term addition and convolution. Hence the set of associated formal power series in R[[X]] also forms a ring.
We show how k-regular sequences are related to ℤ-rational formal series. We give a machine model for the k-regular sequences. We prove that all k-regular sequences can be computed quickly.
Let the pattern sequence ep (n) count the number of occurrences of the pattern P in the base-k expansion of n. Morton and Mourant showed that every sequence over ℤ has a unique expansion as sum of pattern sequences. We prove that this “Fourier” expansion maps k-regular sequences to k-regular sequences. In particular, the coefficients in the expansion of ep(an+b) form a k-automatic sequence.