Overview of the LinearFunctionalSystems Package

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  - LinearFunctionalSystems
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	Calling Sequence
	LinearFunctionalSystems[command](arguments) command(arguments)

Description

•	The LinearFunctionalSystems package is useful for solving the following types of problems.

* Find polynomial solutions of a linear functional system of equations with polynomial coefficients.

* Find rational solutions of a linear functional system of equations with polynomial coefficients.

* Find formal power series solutions of a linear functional system of equations with polynomial coefficients.

* Find the universal denominator of the rational solutions of a linear functional system of equations with polynomial coefficients

* Transform a matrix recurrence system into an equivalent system with nonsingular leading or trailing matrix.

•

For a given linear functional system of equations, the main functionality of this package is to transform the given system into an equivalent system with a nonsingular leading or trailing matrix. The construction of this equivalent system can be solved by using the EG-elimination algorithm by S.A. Abramov.

•	Each command in the LinearFunctionalSystems package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

As the underlying implementation of the LinearFunctionalSystems package is a module, it is also possible to use the form LinearFunctionalSystems:-command to access a command from the package. For more information, see Module Members.

List of LinearFunctionalSystems Package Commands

•	The following is a list of available commands.

AreSameSolution	CanonicalSystem	ExtendRegularSolution	ExtendSeries
HomogeneousSystem	IsSolution	LogarithmicSolution	MatrixTriangularization
PolynomialSolution	Properties	Rank	RationalSolution
RegularSolution	SeriesSolution	UniversalDenominator

To display the help page for a particular LinearFunctionalSystems command, see Getting Help with a Command in a Package.

Examples

[Matrix([[-1, -1, n+1, 0], [-1, -1, 0, n+1]]), TABLE([]), Vector[column]([[0], [0]]), {}, {}]

(1)

[Matrix([[-1, -1, n+1, 0], [-n-2, n+2, 0, 0]]), TABLE([]), Vector[column]([[0], [0]]), {}, {}]

(2)

sys := [(x+3)*(x+6)*(x+1)*(x+5)*x*y1(x+1)-(x-1)*(x+2)*(x+3)*(x+6)*(x+1)*y1(x)-x*(x^6+11*x^5+41*x^4+65*x^3+50*x^2-36)*y2(x)+(6*(x+2))*(x+3)*(x+6)*(x+1)*x*y4(x), (x+6)*(x+2)*y2(x+1)-x^2*y2(x), (x+6)*(x+1)*(x+5)*x*y3(x+1)+(x+6)*(x+1)*(x-1)*y1(x)-x*(x^5+7*x^4+11*x^3+4*x^2-5*x+6)*y2(x)-y3(x)*(x+6)*(x+1)*(x+5)*x+3*(x+6)*(x+1)*x*(x+3)*y4(x), (x+6)*y4(x+1)+x^2*y2(x)-(x+6)*y4(x)]

(3)

[_c[1]/((x-1)*(x+2)*(x+4)*(x+3)), 0, (1/20)*(20*x^5*_c[2]+200*x^4*_c[2]+700*x^3*_c[2]+1000*x^2*_c[2]+480*x*_c[2]+5*x*_c[1]+4*_c[1])/(x*(x+1)*(x+2)*(x+4)*(x+3)), 0]

(4)

sol := [x*(40320*_c[1]-(11621/7150)*_c[5])+O(x^2), -362880*_c[2]+362880*x*_c[2]+O(x^2), _c[3]+x*(362880*_c[4]-(92737/42900)*_c[5])+O(x^2), _c[5]+O(x^2)]

(5)

(6)

B := Matrix(4, 4, [[(x^2+3*x+1)*(x-1)/((x+2)*(x+5)*x), (26*x^3+29*x^2+8*x-1+x^5+9*x^4)/((x+2)*(x+5)*(x+1)), -x-1, -(x-1)*(x^3+7*x^2+14*x+9)/((x+2)*(x+5))], [(x-1)/((x+1)*(x+5)*x), (x-1)/((x+1)^2*(x+5)), 0, (x-1)/((x+1)*(x+5))], [(x-1)/(x+5), (x-1)*x/((x+1)*(x+5)), -x, -(x^3+3*x^2-5*x-5)/(x+5)], [-(x-1)/(x*(x+5)), -(x-1)/((x+1)*(x+5)), 1, (x-1)*(x+4)/(x+5)]])

(7)

(8)

sol := [x*(-5040*_c[1]+(1014645/16)*_c[5]+(935265/4)*_c[6]-_c[4])+O(x^2), (1633536/7)*_c[6]+(408384/7)*_c[5]+40320*_c[2]+x*(-40320*_c[2]-(744003/14)*_c[5]-(1488006/7)*_c[6])+O(x^2), (128835/8)*_c[5]+(450765/2)*_c[6]+40320*_c[3]+2*_c[4]+x*(-(128835/8)*_c[5]-(450765/2)*_c[6]-40320*_c[3]-_c[4])+O(x^2), _c[4]+x*((5985/2)*_c[6]+(6615/8)*_c[5])+O(x^2)]