Solving Linear Recurrence Equations with Polynomial Coefficients

Abstract

Summation is closely related to solving linear recurrence equations, since an indefinite sum satisfies a first-order linear recurrence with constant coefficients, and a definite proper-hypergeometric sum satisfies a linear recurrence with polynomial coefficients. Conversely, d'Alembertian solutions of linear recurrences can be expressed as nested indefinite sums with hypergeometric summands. We sketch the simplest algorithms for finding polynomial, rational, hypergeometric, d'Alembertian, and Liouvillian solutions of linear recurrences with polynomial coefficients, and refer to the relevant literature for state-of-the-art algorithms for these tasks. We outline an algorithm for finding the minimal annihilator of a given P-recursive sequence, prove the salient closure properties of d'Alembertian sequences, and present an alternative proof of a recent result of Reutenauer's that Liouvillian sequences are precisely the interlacings of d'Alembertian ones.

Summation Theory II: Characterizations of RΠΣ⁎-extensions and algorithmic aspects

“…Note further that Picard-Vessiot extensions and basic RΠΣ * -extensions cover the common class of basic RΠΣ * -extensions where the Π-monomials are not nested, i.e., where their shift-quotients are from F. This subclass enables one to model nested sum expressions over hypergeometric products and more generally over simple products (see Section 7). In particular, it allows one to represent the so-called d'Alembertian solutions [8,10] of a linear recurrence, but also the so-called Liouvillian solutions [22] by exploiting ideas from [42,40]. We notice that for this class the proof of Theorem 3.3 simplifies significantly (see Footnote 1).…”

Convolutions of Liouvillian sequences

“…It is easy to see that both A(K) and L(K) are closed under multisection (cf. [9] and [6]). With convolution, the situation is much more varied already for hypergeometric operands, as demonstrated by the following three examples.…”

Computer Algebra and Polynomials

“…Next, the resulting cones need to be triangulated to make them simplicial. There are sophisticated algorithms available for both tasks [32,33,44,45]. It is also possible to compute a decomposition of P into simplicial cones directly, without computing vertices or triangulating, using the Polyhedral Omega algorithm [23].…”

“…We conclude the introduction by remarking that all the presented algorithms play an important role in concrete problem solving like in the fields of combinatorics [12,47], numerics [23], number theory [36,40] or particle physics [14,15]. In particular, if one solves linear recurrence relations in terms of d'Alembertian solutions [9,43,44,48], a subclass of Liouvillian solutions [26,45], one obtains highly nested indefinite nested sums. It is then a necessary task to simplify these sums by fast parameterized telescoping algorithms.…”