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.

The long form, LinearFunctionalSystems:-command, is always available. The short form can be used after loading the package.

The following is a list of available commands.

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

$\mathrm{with}\left(\mathrm{LinearFunctionalSystems}\right)\:$

$M≔\mathrm{Matrix}\left(2\,4\,\left[-1\,-1\,n+1\,0\,-1\,-1\,0\,n+1\right]\right)\:$

$\mathrm{MatrixTriangularization}\left(M\,2\,n\,'\mathrm{trail}'\right)$

$\left[\left[\begin{array}{cccc}\mathrm{-1}& \mathrm{-1}& n+1& 0\\ \mathrm{-1}& \mathrm{-1}& 0& n+1\end{array}\right]\,table\left(\left[\right]\right)\,\left[\begin{array}{c}0\\ 0\end{array}\right]\,\varnothing \,\varnothing \right]$

$\mathrm{MatrixTriangularization}\left(M\,2\,n\,'\mathrm{lead}'\right)$

$\left[\left[\begin{array}{cccc}\mathrm{-1}& \mathrm{-1}& n+1& 0\\ -n-2& n+2& 0& 0\end{array}\right]\,table\left(\left[\right]\right)\,\left[\begin{array}{c}0\\ 0\end{array}\right]\,\varnothing \,\varnothing \right]$

$\mathrm{sys}≔\left[\left(x+3\right)\left(x+6\right)\left(x+1\right)\left(x+5\right)x\mathrm{y1}\left(x+1\right)-\left(x-1\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+1\right)\mathrm{y1}\left(x\right)-x\left(x^6+11x^5+41x^4+65x^3+50x^2-36\right)\mathrm{y2}\left(x\right)+6\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+1\right)x\mathrm{y4}\left(x\right)\,\left(x+6\right)\left(x+2\right)\mathrm{y2}\left(x+1\right)-x^2\mathrm{y2}\left(x\right)\,\left(x+6\right)\left(x+1\right)\left(x+5\right)x\mathrm{y3}\left(x+1\right)+\left(x+6\right)\left(x+1\right)\left(x-1\right)\mathrm{y1}\left(x\right)-x\left(x^5+7x^4+11x^3+4x^2-5x+6\right)\mathrm{y2}\left(x\right)-\mathrm{y3}\left(x\right)\left(x+6\right)\left(x+1\right)\left(x+5\right)x+\left(x+6\right)\left(x+1\right)x\cdot 3\left(x+3\right)\mathrm{y4}\left(x\right)\,\left(x+6\right)\mathrm{y4}\left(x+1\right)+x^2\mathrm{y2}\left(x\right)-\left(x+6\right)\mathrm{y4}\left(x\right)\right]\:$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\,\mathrm{y4}\left(x\right)\right]\:$

$\mathrm{PolynomialSolution}\left(\mathrm{sys}\,\mathrm{vars}\right)$

$\left[0\,0\,{\mathrm{\_c}}_1\,0\right]$

$\mathrm{RationalSolution}\left(\mathrm{sys}\,\mathrm{vars}\right)$

$\left[\frac{{\mathrm{\_c}}_1}{\left(x-1\right)\left(x+2\right)\left(x+4\right)\left(x+3\right)}\,0\,\frac{20x^5{\mathrm{\_c}}_2+200x^4{\mathrm{\_c}}_2+700x^3{\mathrm{\_c}}_2+1000x^2{\mathrm{\_c}}_2+5x{\mathrm{\_c}}_1+480x{\mathrm{\_c}}_2+4{\mathrm{\_c}}_1}{20x\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+3\right)}\,0\right]$

$\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{sys}\,\mathrm{vars}\right)$

$\mathrm{sol}≔\left[x\left(40320{\mathrm{\_c}}_1-\frac{11621{\mathrm{\_c}}_5}{7150}\right)+\mathrm{O}\left(x^2\right)\,362880x{\mathrm{\_c}}_2-362880{\mathrm{\_c}}_2+\mathrm{O}\left(x^2\right)\,{\mathrm{\_c}}_3+x\left(362880{\mathrm{\_c}}_4-\frac{92737{\mathrm{\_c}}_5}{42900}\right)+\mathrm{O}\left(x^2\right)\,{\mathrm{\_c}}_5+\mathrm{O}\left(x^2\right)\right]$

$\mathrm{ExtendSeries}\left(\mathrm{sol}\,5\right)$

$\left[x\left(40320{\mathrm{\_c}}_1-\frac{11621{\mathrm{\_c}}_5}{7150}\right)+x\left(x-1\right)\left(-40320{\mathrm{\_c}}_1+\frac{448{\mathrm{\_c}}_5}{3575}\right)+x\left(x-1\right)\left(x-2\right)\left(20160{\mathrm{\_c}}_1-\frac{2697{\mathrm{\_c}}_5}{100100}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(-6720{\mathrm{\_c}}_1+\frac{3617{\mathrm{\_c}}_5}{800800}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(1680{\mathrm{\_c}}_1-\frac{2923{\mathrm{\_c}}_5}{4804800}\right)+\mathrm{O}\left(x^6\right)\,362880x{\mathrm{\_c}}_2-362880{\mathrm{\_c}}_2-181440x\left(x-1\right){\mathrm{\_c}}_2+60480x\left(x-1\right)\left(x-2\right){\mathrm{\_c}}_2-15120x\left(x-1\right)\left(x-2\right)\left(x-3\right){\mathrm{\_c}}_2+3024x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right){\mathrm{\_c}}_2+\mathrm{O}\left(x^6\right)\,{\mathrm{\_c}}_3+x\left(362880{\mathrm{\_c}}_4-\frac{92737{\mathrm{\_c}}_5}{42900}\right)+x\left(x-1\right)\left(-181440{\mathrm{\_c}}_4+\frac{6937{\mathrm{\_c}}_5}{85800}\right)+x\left(x-1\right)\left(x-2\right)\left(60480{\mathrm{\_c}}_4-\frac{27109{\mathrm{\_c}}_5}{1801800}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(-15120{\mathrm{\_c}}_4+\frac{512{\mathrm{\_c}}_5}{225225}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(3024{\mathrm{\_c}}_4-\frac{40511{\mathrm{\_c}}_5}{144144000}\right)+\mathrm{O}\left(x^6\right)\,{\mathrm{\_c}}_5+\mathrm{O}\left(x^6\right)\right]$

$B≔\mathrm{Matrix}\left(4\,4\,\left[\left[\frac{\left(x^2+3x+1\right)\left(x-1\right)}{\left(x+2\right)\left(x+5\right)x}\,\frac{26x^3+29x^2+8x-1+x^5+9x^4}{\left(x+2\right)\left(x+5\right)\left(x+1\right)}\,-x-1\,-\frac{\left(x-1\right)\left(x^3+7x^2+14x+9\right)}{\left(x+2\right)\left(x+5\right)}\right]\,\left[\frac{x-1}{\left(x+1\right)\left(x+5\right)x}\,\frac{x-1}{{\left(x+1\right)}^2\left(x+5\right)}\,0\,\frac{x-1}{\left(x+5\right)\left(x+1\right)}\right]\,\left[\frac{x-1}{x+5}\,\frac{\left(x-1\right)x}{\left(x+5\right)\left(x+1\right)}\,-x\,-\frac{x^3+3x^2-5x-5}{x+5}\right]\,\left[-\frac{x-1}{\left(x+5\right)x}\,-\frac{x-1}{\left(x+5\right)\left(x+1\right)}\,1\,\frac{\left(x-1\right)\left(x+4\right)}{x+5}\right]\right]\right)\:$

$\mathrm{PolynomialSolution}\left(B\,x\,'\mathrm{difference}'\right)$

$\left[-x{\mathrm{\_c}}_1\,0\,-x{\mathrm{\_c}}_1+2{\mathrm{\_c}}_1\,{\mathrm{\_c}}_1\right]$

$\mathrm{RationalSolution}\left(B\,x\,'\mathrm{difference}'\right)$

$\left[-\frac{x^6{\mathrm{\_c}}_1+7x^5{\mathrm{\_c}}_1-109x^4{\mathrm{\_c}}_1+x^4{\mathrm{\_c}}_2+353x^3{\mathrm{\_c}}_1-3x^3{\mathrm{\_c}}_2+2268x^2{\mathrm{\_c}}_1-19x^2{\mathrm{\_c}}_2-840x{\mathrm{\_c}}_1+7x{\mathrm{\_c}}_2+720{\mathrm{\_c}}_1-6{\mathrm{\_c}}_2}{\left(x^2-1\right)\left(x+3\right)\left(x+4\right)x}\,-\frac{4\left(120{\mathrm{\_c}}_1-{\mathrm{\_c}}_2\right)}{\left(x+4\right)\left(x+2\right)x^2}\,-\frac{x^5{\mathrm{\_c}}_1+8x^4{\mathrm{\_c}}_1-105x^3{\mathrm{\_c}}_1+x^3{\mathrm{\_c}}_2-260x^2{\mathrm{\_c}}_1+2x^2{\mathrm{\_c}}_2+764x{\mathrm{\_c}}_1-7x{\mathrm{\_c}}_2+1272{\mathrm{\_c}}_1-11{\mathrm{\_c}}_2}{\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+3\right)}\,\frac{x^2{\mathrm{\_c}}_1+6x{\mathrm{\_c}}_1-112{\mathrm{\_c}}_1+{\mathrm{\_c}}_2}{\left(x+4\right)\left(x+2\right)}\right]$

$\mathrm{sol}≔\mathrm{SeriesSolution}\left(B\,x\,'\mathrm{difference}'\right)$

$\mathrm{sol}≔\left[x\left(-5040{\mathrm{\_c}}_1+\frac{1014645{\mathrm{\_c}}_5}{16}+\frac{935265{\mathrm{\_c}}_6}{4}-{\mathrm{\_c}}_4\right)+\mathrm{O}\left(x^2\right)\,\frac{1633536{\mathrm{\_c}}_6}{7}+\frac{408384{\mathrm{\_c}}_5}{7}+40320{\mathrm{\_c}}_2+x\left(-40320{\mathrm{\_c}}_2-\frac{744003{\mathrm{\_c}}_5}{14}-\frac{1488006{\mathrm{\_c}}_6}{7}\right)+\mathrm{O}\left(x^2\right)\,\frac{128835{\mathrm{\_c}}_5}{8}+\frac{450765{\mathrm{\_c}}_6}{2}+40320{\mathrm{\_c}}_3+2{\mathrm{\_c}}_4+x\left(-\frac{128835{\mathrm{\_c}}_5}{8}-\frac{450765{\mathrm{\_c}}_6}{2}-40320{\mathrm{\_c}}_3-{\mathrm{\_c}}_4\right)+\mathrm{O}\left(x^2\right)\,{\mathrm{\_c}}_4+x\left(\frac{5985{\mathrm{\_c}}_6}{2}+\frac{6615{\mathrm{\_c}}_5}{8}\right)+\mathrm{O}\left(x^2\right)\right]$

$\mathrm{ExtendSeries}\left(\mathrm{sol}\,5\right)$

$\left[x\left(-5040{\mathrm{\_c}}_1+\frac{1014645{\mathrm{\_c}}_5}{16}+\frac{935265{\mathrm{\_c}}_6}{4}-{\mathrm{\_c}}_4\right)+x\left(x-1\right)\left(5040{\mathrm{\_c}}_1-\frac{931485{\mathrm{\_c}}_5}{16}-\frac{852105{\mathrm{\_c}}_6}{4}\right)+x\left(x-1\right)\left(x-2\right)\left(-2520{\mathrm{\_c}}_1+\frac{845985{\mathrm{\_c}}_5}{32}+\frac{767865{\mathrm{\_c}}_6}{8}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(840{\mathrm{\_c}}_1-\frac{248755{\mathrm{\_c}}_5}{32}-\frac{223555{\mathrm{\_c}}_6}{8}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(-210{\mathrm{\_c}}_1+\frac{207641{\mathrm{\_c}}_5}{128}+\frac{184121{\mathrm{\_c}}_6}{32}\right)+\mathrm{O}\left(x^6\right)\,\frac{1633536{\mathrm{\_c}}_6}{7}+\frac{408384{\mathrm{\_c}}_5}{7}+40320{\mathrm{\_c}}_2+x\left(-40320{\mathrm{\_c}}_2-\frac{744003{\mathrm{\_c}}_5}{14}-\frac{1488006{\mathrm{\_c}}_6}{7}\right)+x\left(x-1\right)\left(\frac{671238{\mathrm{\_c}}_6}{7}+\frac{335619{\mathrm{\_c}}_5}{14}+20160{\mathrm{\_c}}_2\right)+x\left(x-1\right)\left(x-2\right)\left(-6720{\mathrm{\_c}}_2-\frac{49584{\mathrm{\_c}}_5}{7}-\frac{198336{\mathrm{\_c}}_6}{7}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(1680{\mathrm{\_c}}_2+\frac{21327{\mathrm{\_c}}_5}{14}+\frac{42654{\mathrm{\_c}}_6}{7}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(-336{\mathrm{\_c}}_2-\frac{1740{\mathrm{\_c}}_5}{7}-\frac{6960{\mathrm{\_c}}_6}{7}\right)+\mathrm{O}\left(x^6\right)\,\frac{128835{\mathrm{\_c}}_5}{8}+\frac{450765{\mathrm{\_c}}_6}{2}+40320{\mathrm{\_c}}_3+2{\mathrm{\_c}}_4+x\left(-\frac{128835{\mathrm{\_c}}_5}{8}-\frac{450765{\mathrm{\_c}}_6}{2}-40320{\mathrm{\_c}}_3-{\mathrm{\_c}}_4\right)+x\left(x-1\right)\left(\frac{67725{\mathrm{\_c}}_5}{8}+\frac{228375{\mathrm{\_c}}_6}{2}+20160{\mathrm{\_c}}_3\right)+x\left(x-1\right)\left(x-2\right)\left(-6720{\mathrm{\_c}}_3-\frac{25005{\mathrm{\_c}}_5}{8}-\frac{78135{\mathrm{\_c}}_6}{2}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(1680{\mathrm{\_c}}_3+\frac{7275{\mathrm{\_c}}_5}{8}+\frac{20295{\mathrm{\_c}}_6}{2}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(-336{\mathrm{\_c}}_3-\frac{861{\mathrm{\_c}}_5}{4}-2100{\mathrm{\_c}}_6\right)+\mathrm{O}\left(x^6\right)\,{\mathrm{\_c}}_4+x\left(\frac{5985{\mathrm{\_c}}_6}{2}+\frac{6615{\mathrm{\_c}}_5}{8}\right)+x\left(x-1\right)\left(-\frac{5985{\mathrm{\_c}}_6}{2}-\frac{6615{\mathrm{\_c}}_5}{8}\right)+x\left(x-1\right)\left(x-2\right)\left(\frac{855{\mathrm{\_c}}_5}{2}+1500{\mathrm{\_c}}_6\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(-\frac{1005{\mathrm{\_c}}_6}{2}-\frac{1215{\mathrm{\_c}}_5}{8}\right)+x\left(x-1\right)\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(\frac{321{\mathrm{\_c}}_5}{8}+\frac{237{\mathrm{\_c}}_6}{2}\right)+\mathrm{O}\left(x^6\right)\right]$

Abramov, S.A. "EG-Eliminations." Journal of Difference Equations and Applications, Vol. 5. (1999): 393-433.