\(p\)-linear schemes for sequences modulo \(p^{r}\) (Q6585094)

In this paper the author studies infinite sequences of integers that have special properties when they are reduced modulo a prime \(p\) or a prime power \(p^r\). The most familiar sequences are those sequences \(a_k\) that have the so-called Lucas property modulo almost all primes \(p\), \N\[\Na_{k_m p^m+\cdots+k_1 p+k_0}\equiv a_{k_m}\cdots a_{k_1}a_{k_0}\pmod p \N\]\Nfor all \(k_i\) (\(0\le k_i\le p-1\)). Some of the examples are the exponential sequences like \(2^k\), the central binomial coefficients \(\binom{2 k}{k}\), the Apéry numbers \(\sum_{m=0}^k\binom{k}{m}^2\binom{k+m}{m}^2\) and Franel-type numbers \(\sum_{m=0}^k\binom{k}{m}^\ell\). They are generalizations of finite \(p\)-automata. In this paper the author constructs such \(p\)-linear schemes and give upper bounds for the number of states which, for fixed \(r\), do not depend on \(p\).