In mathematics a P-recursive equation is a linear equation of
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s where the coefficient sequences can be represented as
polynomials. P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients. These equations play an important role in different areas of mathematics, specifically in
combinatorics. The sequences which are solutions of these equations are called
holonomic, P-recursive or D-finite.
From the late 1980s on the first algorithms were developed to find solutions for these equations. Sergei A. Abramov,
Marko Petkovšek
Marko Petkovšek is a Slovenian mathematician, born: 1955, working mainly in symbolic computation.
He is a professor of discrete and computational mathematics at the University of Ljubljana. He completed his Ph.D. at Carnegie Mellon University
u ...
and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and d'Alembertian solutions.
Definition
Let
be a
field of characteristic zero
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
(for example
),