For each (commutative unitary) ring $R$, let $\mathfrak{R}(R)$ be the set of all linear recurrences over $R$, that is, the set of all sequences $(a(n))_{n \geq 0}$ in $R$ such that $$a(n+k) = r_1 a(n+k-1) + \cdots + r_k a(n) \quad \forall n \geq 0 ,$$ for some $r_1, \ldots, r_k \in R$.
It is known that $\mathfrak{R}(R)$ equipped with pointwise addition and pointwise multiplication of sequences is a ring (see [1]).
My question is: How much is known about the factorization of elements of $\mathfrak{R}(R)$ ?
Well, to ask that for a generic ring is probably too much. It seems reasonable to start from the case $R = D$ integral domain of characteristic $0$. In fact, the case $R = \mathbb{Z}$ is already enough interesting in my opinion. Thus, if you known only partial or special results, do not hesitate to let me know.
Let $K$ be the field of fractions of $D$ and let $L$ be the algebraic closure of $K$.
I try to summarize what I found in the literature.
Generalized power sums
It is well-known that any $(a(n))_{n \geq 0} \in \mathfrak{R}(D)$ satisfies $$(\star) \quad a(n) = \sum_{i=1}^m A_i(n) \alpha_i^n \quad \forall n \geq 0, $$ where $\alpha_1, \ldots, \alpha_m \in L$ are all the distinct roots of the polynomial $$x^k - r_1 x^{k-1} - \cdots - r_k$$ and $A_1, \ldots, A_n \in L[X]$ are such that $\deg(A_i) + 1$ is the multiplicity of the root $\alpha_i$. Functions of the form $(\star)$ are called generalized power sums. Actually, one can prove (in several ways) that the generalized power sums are exactly the elements of $\mathfrak{R}(L)$.
It seems that Bézivin developed [2] a theory of factorization for generalized power sums. Unfortunately, his paper is in French and I can't understand it.
Exponential polynomials
When $R$ is a subring of $\mathbb{C}$ it seems that the theory of exponential polynomials could be of help. Broadly speaking, an exponential polynomials is a formal expression of the form $$\sum_{i=1}^m A_i(Z) e^{\omega_i Z} ,$$ where $A_1, \ldots, A_m \in \mathbb{C}[Z]$ and $\omega_1, \ldots, \omega_k \in \mathbb{C}^*$ are pairwise distinct. The set of exponential polynomials is a ring $\mathbb{C}_Z$ in a natural way. The units of $\mathbb{C}_Z$ are $A e^{\omega Z}$, with $A,\omega \in \mathbb{C}^*$. The ring $\mathbb{C}_Z$ is not a UFD since any so called simple exponential polynomials $1 - Ae^{\omega Z}$ has the factor $1 - A^{1/n}e^{(\omega/n) Z}$, for any positive integer $n$. However, it holds [3] a kind of unique factorization theorem for $\mathbb{C}_Z$.
Theorem (Ritt). An exponential polynomial in $Z$ factors, uniquely up to units and associates, as a product of a polynomial in $Z$, a finite product of simple polynomials $1 - A_i e^{\omega_i Z}$ with the $\omega_i$ not rational multiples one of the other, and a finite product of exponential polynomials each irreducible in the ring of exponential polynomials.
With the notation of $(\star)$, for any $(a(n))_{n \geq 0}$ there exists an exponential polynomial $$\tilde{a}(Z) = \sum_{i=1}^m A_i(Z) e^{\log(\alpha_i) Z} ,$$ where $\log(\alpha_i)$ is some choice of a complex logarithm for $\alpha_i$, such that $a(n) = \tilde{a}(n)$ for each integer $n \geq 0$. Hence, one can hope to use Ritt's theorem to prove a factorization theorem in $\mathfrak{R}(R)$. Unfortunately, there is no global lift $\mathfrak{R}(R) \to \mathbb{C}_Z$ such that if $a(n) = b(n)c(n)$ in $\mathfrak{R}(R)$ then $\tilde{a}(Z) = \tilde{b}(n)\tilde{c}(n)$ in $\mathbb{C}_Z$. In fact, let $a(n) = 1^n$ and $b(n) = (-1)^n$, so that $\tilde{a}(Z) = e^{\log(1)Z}$ and $\tilde{b}(Z) = e^{\log(-1)Z}$. Now for any positive integer $k$ it holds $a(n) = (b(n))^{2k}$ in $\mathfrak{R}(R)$. However, for any choice of $\log(1)$ and $\log(-1)$ there exists a positive integer $k$ such that $\tilde{a}(Z) \neq (\tilde{b}(Z))^{2k}$ in $\mathbb{C}_Z$. Notably, when only some particular factorizations of $a(n)$ are considered, a lift may exists (see Sec. 5.1 of [4]).
Thank you in advance for any suggestion/reference.
[1] U. Cerruti and F. Vaccarino, R-algebras of linear recurrent sequences, J. Algebra 175 (1995), no. 1, 332--338.
[2] J.-P. Bézivin, Factorisation de suites récurrentes linéaires, Groupe d'étude d'Analyse ultramétrique, tome 7-8 (1979-1981), no. 33, 1--9.
[3] G. R. Everest and A. J. van der Poorten, Factorisation in the ring of exponential polynomials, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1293--1298.
[4] J.-P. Bézivin, A. Pethö, and A. J. van der Porten, A full characterization of divisibility sequences, Am. J. Math. 112 (1990), no. 6, 985--1001.