“Polynomial solutions of P-recursive equations”的意思、由来-开放百科全书

In mathematics a P-recursive equation can be solved for polynomial solutions. Sergei A. Abramov in 1989 and Marko Petkovšek in 1992 described an algorithm which finds all polynomial solutions of those recurrence equations with polynomial coefficients.^[1]^[2] The algorithm computes a degree bound for the solution in a first step. In a second step an ansatz for a polynomial of this degree is used and the unknown coefficients are computed by a system of linear equations. This article describes this algorithm.

In 1995 Abramov, Bronstein and Petkovšek showed that the polynomial case can be solved more efficiently by considering power series solution of the recurrence equation in a specific power basis (i.e. not the ordinary basis

).^[3]

Other algorithms which compute rational or hypergeometric solutions of a linear recurrence equation with polynomial coefficients also use algorithms which compute polynomial solutions.

Degree bound

Let

be a field of characteristic zero and

a recurrence equation of order

with polynomial coefficients

, polynomial right-hand side

and unknown polynomial sequence

. Furthermore

denotes the degree of a polynomial

(with

for the zero polynomial) and

denotes the leading coefficient of the polynomial. Moreover let

for

where

denotes the falling factorial and

the set of nonegative integers. Then

. This is called a degree bound for the polynomial solution

. This bound was shown by Abramov and Petkovšek.^[1]^[2]^[3]^[4]

Algorithm

The algorithm consists of two steps. In a first step the degree bound is computed. In a second step an ansatz with a polynomial

of that degree with arbitrary coefficients in

is made and plugged into the recurrence equation. Then the different powers are compared and a system of linear equations for the coefficients of

is set up and solved. This is called the method undetermined coefficients.^[5] The algorithm returns the general polynomial solution of a recurrence equation.

Example

Applying the formula for the degree bound on the recurrence equation

over

yields

. Hence one can use an ansatz with a quadratic polynomial

with

. Plugging this ansatz into the original recurrence equation leads to

This is equivalent to the following system of linear equations

with the solution

. Therefore the only polynomial solution is

References

1. ^¹{{Cite journal|last=Abramov|first=Sergei A.|date=1989|title=Problems in computer algebra that are connected with a search for polynomial solutions of linear differential and difference equations|url=|journal=Moscow University Computational Mathematics and Cybernetics|volume=3|pages=|via=}}
2. ^¹{{Cite journal|last=Petkovšek|first=Marko|date=1992|title=Hypergeometric solutions of linear recurrences with polynomial coefficients|journal=Journal of Symbolic Computation|volume=14|issue=2–3|pages=243–264|doi=10.1016/0747-7171(92)90038-6|issn=0747-7171}}
3. ^¹{{Cite book|last=Abramov|first=Sergei A.|last2=Bronstein|first2=Manuel|last3=Petkovšek|first3=Marko|date=1995|title=On polynomial solutions of linear operator equations|url=http://dl.acm.org/citation.cfm?id=220346.220384|journal=ISSAC '95 Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation|publisher=ACM|volume=|pages=290–296|doi=10.1145/220346.220384|isbn=978-0897916998|via=|citeseerx=10.1.1.46.9373}}
4. ^Weixlbaumer, Christian (2001). [https://www.risc.jku.at/research/combinat/software/DiffTools/pub/weixlbaumer01.pdf Solutions of difference equations with polynomial coefficients]. Diploma Thesis, Johannes Kepler Universität Linz
5. ^{{Cite book|url=https://www.math.upenn.edu/~wilf/Downld.html|title=A=B|last=Petkovšek|first=Marko|last2=Wilf|first2=Herbert S.|last3=Zeilberger|first3=Doron|date=1996|publisher=A K Peters|others=|isbn=978-1568810638|location=|pages=|oclc=33898705}}