On other two representations of the C-recursive integer sequences by terms in modular arithmetic

Abstract

If $s\in N^N$ is a sequence satisfying a recurrence rule of the form:$s(n+d)+{\alpha }_{1}s(n+d-1)+\dots +{\alpha }_{d}s\left(n\right)=0$ with coefficients ${\alpha }_i\in Z$, then there exist $b,n_0\in N$ such that for all $n\ge n_0$ the following representations work:$s\left(n\right)=\lfloor \frac{[b^{n(d-2)+\lceil n/2\rceil }+A(b,n\left)\right]\mathrm{mod}B(b,n)}{b^{(d-1)n}}\rfloor ,$$s\left(n\right)=\frac{1}{\left|{\alpha }_d\right|}\{[(b^{n(d-1)+\lceil n/2\rceil }-b^n\mathrm{sgn}({\alpha }_d\left)A\right(b,n\left)\right)\mathrm{mod}B(b,n)\left]\mathrm{mod}{b}^n\right\}.$ Here $A(b,n)$ and $B(b,n)$ are polynomials with integer coefficients in $b^n$. They can be obtained by writing:$b^{n^2}f\left(b^{-n}\right)=\frac{A(b,n)}{B(b,n)},$ where the rational function $f\left(z\right)$ is the generating function of the sequence $\left(s\right(n\left)\right)$. If $s\in Z^N$, then s can be represented as the difference between any of the representations above for the sequence $\left(s\right(n)+c^{n+1})$ which belongs to $N^N$ and the geometric progression $c^{n+1}$. Here $c\in N$ is a sufficiently large constant.