Abramov's algorithm

In mathematics, particularly in computer algebra, Abramov's algorithm computes all rational solutions of a linear recurrence equation with polynomial coefficients. The algorithm was published by Sergei A. Abramov in 1989.

Universal denominator
The main concept in Abramov's algorithm is a universal denominator. Let $\mathbb{K}$ be a field of characteristic zero. The dispersion $\operatorname{dis} (p,q)$ of two polynomials $p, q \in \mathbb{K}[n]$  is defined as$$\operatorname{dis} (p,q) =\max \{ k \in \N \, : \, \deg (\gcd (p(n), q(n+k) )) \geq 1 \} \cup \{ -1 \},$$where $\N$  denotes the set of non-negative integers. Therefore the dispersion is the maximum $k \in \N$ such that the polynomial $p$  and the $k$ -times shifted polynomial $$q$$ have a common factor. It is $-1$ if such a $k$  does not exist. The dispersion can be computed as the largest non-negative integer root of the resultant $\operatorname{res}_n (p(n), q(n+k) ) \in \mathbb{K}[k]$. Let $\sum_{k=0}^r p_k(n) \, y (n+k) = f(n)$ be a recurrence equation of order $r$  with polynomial coefficients $$p_k \in \mathbb{K} [n]$$, polynomial right-hand side $f \in \mathbb{K}[n]$  and rational sequence solution  $y (n) \in \mathbb{K}(n)$. It is possible to write $y (n) = p(n)/q(n)$ for two relatively prime polynomials $p, q \in \mathbb{K}[n]$. Let $D=\operatorname{dis} (p_r(n-r), p_0 (n) )$ and$$u(n) = \gcd ([p_0 (n+D)]^{\underline{D+1}}, [p_r (n-r)]^{\underline{D+1}})$$where $[p(n)]^{\underline{k}}=p(n)p(n-1)\cdots p(n-k+1)$  denotes the falling factorial of a function. Then $q(n)$ divides $u(n)$. So the polynomial $u(n)$ can be used as a denominator for all rational solutions $y(n)$  and hence it is called a universal denominator.

Algorithm
Let again $\sum_{k=0}^r p_k(n) \, y (n+k) = f(n)$ be a recurrence equation with polynomial coefficients and $u(n)$  a universal denominator. After substituting $y (n) = z(n)/u(n)$ for an unknown polynomial $z(n) \in \mathbb{K} [n]$  and setting $\ell(n) = \operatorname{lcm}(u(n), \dots, u(n+r))$  the recurrence equation is equivalent to$$\sum_{k=0}^r p_k (n) \frac{z(n+k)}{u(n+k)} \ell(n) = f(n) \ell(n).$$As the $u(n+k)$  cancel this is a linear recurrence equation with polynomial coefficients which can be solved for an unknown polynomial solution $z(n)$. There are algorithms to find polynomial solutions. The solutions for $z(n)$ can then be used again to compute the rational solutions $y(n) = z(n)/u(n)$.

algorithm rational_solutions is input: Linear recurrence equation $\sum_{k=0}^r p_k(n) \, y (n+k) = f(n), p_k, f \in \mathbb{K}[n], p_0, p_r \neq 0$. output: The general rational solution $y$ if there are any solutions, otherwise false. $D = \operatorname{disp} (p_r(n-r), p_0 (n) )$ $u(n) = \gcd ([p_0 (n+D)]^{\underline{D+1}}, [p_r (n-r)]^{\underline{D+1}})$ $\ell(n) = \operatorname{lcm}(u(n), \dots, u(n+r))$ Solve $\sum_{k=0}^r p_k (n) \frac{z(n+k)}{u(n+k)} \ell(n) = f(n) \ell(n)$ for general polynomial solution $z(n)$ if solution $z(n)$ exists then return general solution $y (n) = z(n)/u(n)$ else return false end if

Example
The homogeneous recurrence equation of order $1$ $$(n-1) \, y(n) + (-n-1) \, y(n+1) = 0$$over $\Q$ has a rational solution. It can be computed by considering the dispersion$$D = \operatorname{dis} (p_1 (n-1), p_0 (n) ) = \operatorname{disp} (-n,n-1) = 1.$$This yields the following universal denominator:$$ u(n) = \gcd ([p_0 (n+1)]^{\underline{2}}, [p_r (n-1)]^{\underline{2}}) = (n-1)n$$and$$ \ell(n) = \operatorname{lcm} (u(n), u(n+1) ) = (n-1)n(n+1).$$Multiplying the original recurrence equation with $ \ell(n)$ and substituting $ y(n) = z(n)/u(n)$  leads to$$ (n-1)(n+1)\, z(n) + (-n-1) (n-1) \, z(n+1) = 0.$$This equation has the polynomial solution $ z(n) = c$  for an arbitrary constant $ c \in \Q$. Using $ y(n) = z(n) / u(n)$ the general rational solution is$$ y(n) = \frac{c}{(n-1)n}$$for arbitrary $ c \in \Q$.