Generalized Lucas congruences and linear p-schemes

Generalized Lucas congruences and linear p-schemes
Joel A. Henningsen, Armin Straub — Advances in Applied Mathematics — Volume 141, 2022, Pages 1-20, #102409


We observe that a sequence satisfies Lucas congruences modulo \(p\) if and only if its values modulo \(p\) can be described by a linear \(p\)-scheme, as introduced by Rowland and Zeilberger, with a single state. This simple observation suggests natural generalizations of the notion of Lucas congruences. To illustrate this point, we prove explicit generalized Lucas congruences for integer sequences that can be represented as the constant terms of \(P(x,y)^n Q(x,y)\) where \(P\) and \(Q\) are certain Laurent polynomials.


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    author = {Joel A. Henningsen and Armin Straub},
    title = {Generalized Lucas congruences and linear $p$-schemes},
    journal = {Advances in Applied Mathematics},
    year = {2022},
    volume = {141},
    pages = {1--20, #102409},
    doi = {10.1016/j.aam.2022.102409},