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Overview of the LinearFunctionalSystems Package

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

List of LinearFunctionalSystems Package Commands

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{LinearFunctionalSystems}\right):$
 > $M≔\mathrm{Matrix}\left(2,4,\left[-1,-1,n+1,0,-1,-1,0,n+1\right]\right):$
 > $\mathrm{MatrixTriangularization}\left(M,2,n,'\mathrm{trail}'\right)$
 $\left[\left[\begin{array}{cccc}{-1}& {-1}& {n}{+}{1}& {0}\\ {-1}& {-1}& {0}& {n}{+}{1}\end{array}\right]{,}{table}{}\left(\left[\right]\right){,}\left[\begin{array}{c}{0}\\ {0}\end{array}\right]{,}{\varnothing }{,}{\varnothing }\right]$ (1)
 > $\mathrm{MatrixTriangularization}\left(M,2,n,'\mathrm{lead}'\right)$
 $\left[\left[\begin{array}{cccc}{-1}& {-1}& {n}{+}{1}& {0}\\ {-}{n}{-}{2}& {n}{+}{2}& {0}& {0}\end{array}\right]{,}{table}{}\left(\left[\right]\right){,}\left[\begin{array}{c}{0}\\ {0}\end{array}\right]{,}{\varnothing }{,}{\varnothing }\right]$ (2)
 > $\mathrm{sys}≔\left[\left(x+3\right)\left(x+6\right)\left(x+1\right)\left(x+5\right)x\mathrm{y1}\left(x+1\right)-\left(x-1\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+1\right)\mathrm{y1}\left(x\right)-x\left({x}^{6}+11{x}^{5}+41{x}^{4}+65{x}^{3}+50{x}^{2}-36\right)\mathrm{y2}\left(x\right)+6\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+1\right)x\mathrm{y4}\left(x\right),\left(x+6\right)\left(x+2\right)\mathrm{y2}\left(x+1\right)-{x}^{2}\mathrm{y2}\left(x\right),\left(x+6\right)\left(x+1\right)\left(x+5\right)x\mathrm{y3}\left(x+1\right)+\left(x+6\right)\left(x+1\right)\left(x-1\right)\mathrm{y1}\left(x\right)-x\left({x}^{5}+7{x}^{4}+11{x}^{3}+4{x}^{2}-5x+6\right)\mathrm{y2}\left(x\right)-\mathrm{y3}\left(x\right)\left(x+6\right)\left(x+1\right)\left(x+5\right)x+\left(x+6\right)\left(x+1\right)x\cdot 3\left(x+3\right)\mathrm{y4}\left(x\right),\left(x+6\right)\mathrm{y4}\left(x+1\right)+{x}^{2}\mathrm{y2}\left(x\right)-\left(x+6\right)\mathrm{y4}\left(x\right)\right]:$
 > $\mathrm{vars}≔\left[\mathrm{y1}\left(x\right),\mathrm{y2}\left(x\right),\mathrm{y3}\left(x\right),\mathrm{y4}\left(x\right)\right]:$
 > $\mathrm{PolynomialSolution}\left(\mathrm{sys},\mathrm{vars}\right)$
 $\left[{0}{,}{0}{,}{{\mathrm{_c}}}_{{1}}{,}{0}\right]$ (3)
 > $\mathrm{RationalSolution}\left(\mathrm{sys},\mathrm{vars}\right)$
 $\left[\frac{{{\mathrm{_c}}}_{{1}}}{\left({x}{-}{1}\right){}\left({x}{+}{2}\right){}\left({x}{+}{4}\right){}\left({x}{+}{3}\right)}{,}{0}{,}\frac{{20}{}{{x}}^{{5}}{}{{\mathrm{_c}}}_{{2}}{+}{200}{}{{x}}^{{4}}{}{{\mathrm{_c}}}_{{2}}{+}{700}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}{+}{1000}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}{+}{5}{}{x}{}{{\mathrm{_c}}}_{{1}}{+}{480}{}{x}{}{{\mathrm{_c}}}_{{2}}{+}{4}{}{{\mathrm{_c}}}_{{1}}}{{20}{}{x}{}\left({x}{+}{1}\right){}\left({x}{+}{2}\right){}\left({x}{+}{4}\right){}\left({x}{+}{3}\right)}{,}{0}\right]$ (4)
 > $\mathrm{sol}≔\mathrm{SeriesSolution}\left(\mathrm{sys},\mathrm{vars}\right)$
 ${\mathrm{sol}}{≔}\left[{x}{}\left({40320}{}{{\mathrm{_c}}}_{{1}}{-}\frac{{11621}{}{{\mathrm{_c}}}_{{5}}}{{7150}}\right){+}{\mathrm{O}}{}\left({{x}}^{{2}}\right){,}{362880}{}{x}{}{{\mathrm{_c}}}_{{2}}{-}{362880}{}{{\mathrm{_c}}}_{{2}}{+}{\mathrm{O}}{}\left({{x}}^{{2}}\right){,}{{\mathrm{_c}}}_{{3}}{+}{x}{}\left({362880}{}{{\mathrm{_c}}}_{{4}}{-}\frac{{92737}{}{{\mathrm{_c}}}_{{5}}}{{42900}}\right){+}{\mathrm{O}}{}\left({{x}}^{{2}}\right){,}{{\mathrm{_c}}}_{{5}}{+}{\mathrm{O}}{}\left({{x}}^{{2}}\right)\right]$ (5)
 > $\mathrm{ExtendSeries}\left(\mathrm{sol},5\right)$
 $\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){,}{362880}{}{x}{}{{\mathrm{_c}}}_{{2}}{-}{362880}{}{{\mathrm{_c}}}_{{2}}{-}{181440}{}{x}{}\left({x}{-}{1}\right){}{{\mathrm{_c}}}_{{2}}{+}{60480}{}{x}{}\left({x}{-}{1}\right){}\left({x}{-}{2}\right){}{{\mathrm{_c}}}_{{2}}{-}{15120}{}{x}{}\left({x}{-}{1}\right){}\left({x}{-}{2}\right){}\left({x}{-}{3}\right){}{{\mathrm{_c}}}_{{2}}{+}{3024}{}{x}{}\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]$ (6)
 > $B≔\mathrm{Matrix}\left(4,4,\left[\left[\frac{\left({x}^{2}+3x+1\right)\left(x-1\right)}{\left(x+2\right)\left(x+5\right)x},\frac{26{x}^{3}+29{x}^{2}+8x-1+{x}^{5}+9{x}^{4}}{\left(x+2\right)\left(x+5\right)\left(x+1\right)},-x-1,-\frac{\left(x-1\right)\left({x}^{3}+7{x}^{2}+14x+9\right)}{\left(x+2\right)\left(x+5\right)}\right],\left[\frac{x-1}{\left(x+1\right)\left(x+5\right)x},\frac{x-1}{{\left(x+1\right)}^{2}\left(x+5\right)},0,\frac{x-1}{\left(x+5\right)\left(x+1\right)}\right],\left[\frac{x-1}{x+5},\frac{\left(x-1\right)x}{\left(x+5\right)\left(x+1\right)},-x,-\frac{{x}^{3}+3{x}^{2}-5x-5}{x+5}\right],\left[-\frac{x-1}{\left(x+5\right)x},-\frac{x-1}{\left(x+5\right)\left(x+1\right)},1,\frac{\left(x-1\right)\left(x+4\right)}{x+5}\right]\right]\right):$
 > $\mathrm{PolynomialSolution}\left(B,x,'\mathrm{difference}'\right)$
 $\left[{-}{x}{}{{\mathrm{_c}}}_{{1}}{,}{0}{,}{-}{x}{}{{\mathrm{_c}}}_{{1}}{+}{2}{}{{\mathrm{_c}}}_{{1}}{,}{{\mathrm{_c}}}_{{1}}\right]$ (7)
 > $\mathrm{RationalSolution}\left(B,x,'\mathrm{difference}'\right)$
 $\left[{-}\frac{{{x}}^{{6}}{}{{\mathrm{_c}}}_{{1}}{+}{7}{}{{x}}^{{5}}{}{{\mathrm{_c}}}_{{1}}{-}{109}{}{{x}}^{{4}}{}{{\mathrm{_c}}}_{{1}}{+}{{x}}^{{4}}{}{{\mathrm{_c}}}_{{2}}{+}{353}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{1}}{-}{3}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}{+}{2268}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{1}}{-}{19}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}{-}{840}{}{x}{}{{\mathrm{_c}}}_{{1}}{+}{7}{}{x}{}{{\mathrm{_c}}}_{{2}}{+}{720}{}{{\mathrm{_c}}}_{{1}}{-}{6}{}{{\mathrm{_c}}}_{{2}}}{\left({{x}}^{{2}}{-}{1}\right){}\left({x}{+}{3}\right){}\left({x}{+}{4}\right){}{x}}{,}{-}\frac{{4}{}\left({120}{}{{\mathrm{_c}}}_{{1}}{-}{{\mathrm{_c}}}_{{2}}\right)}{\left({x}{+}{4}\right){}\left({x}{+}{2}\right){}{{x}}^{{2}}}{,}{-}\frac{{{x}}^{{5}}{}{{\mathrm{_c}}}_{{1}}{+}{8}{}{{x}}^{{4}}{}{{\mathrm{_c}}}_{{1}}{-}{105}{}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{1}}{+}{{x}}^{{3}}{}{{\mathrm{_c}}}_{{2}}{-}{260}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{1}}{+}{2}{}{{x}}^{{2}}{}{{\mathrm{_c}}}_{{2}}{+}{764}{}{x}{}{{\mathrm{_c}}}_{{1}}{-}{7}{}{x}{}{{\mathrm{_c}}}_{{2}}{+}{1272}{}{{\mathrm{_c}}}_{{1}}{-}{11}{}{{\mathrm{_c}}}_{{2}}}{\left({x}{+}{1}\right){}\left({x}{+}{2}\right){}\left({x}{+}{4}\right){}\left({x}{+}{3}\right)}{,}\frac{{{x}}^{{2}}{}{{\mathrm{_c}}}_{{1}}{+}{6}{}{x}{}{{\mathrm{_c}}}_{{1}}{-}{112}{}{{\mathrm{_c}}}_{{1}}{+}{{\mathrm{_c}}}_{{2}}}{\left({x}{+}{4}\right){}\left({x}{+}{2}\right)}\right]$ (8)
 > $\mathrm{sol}≔\mathrm{SeriesSolution}\left(B,x,'\mathrm{difference}'\right)$
 ${\mathrm{sol}}{≔}\left[{x}{}\left({-}{5040}{}{{\mathrm{_c}}}_{{1}}{+}\frac{{1014645}{}{{\mathrm{_c}}}_{{5}}}{{16}}{+}\frac{{935265}{}{{\mathrm{_c}}}_{{6}}}{{4}}{-}{{\mathrm{_c}}}_{{4}}\right){+}{\mathrm{O}}{}\left({{x}}^{{2}}\right){,}\frac{{1633536}{}{{\mathrm{_c}}}_{{6}}}{{7}}{+}\frac{{408384}{}{{\mathrm{_c}}}_{{5}}}{{7}}{+}{40320}{}{{\mathrm{_c}}}_{{2}}{+}{x}{}\left({-}{40320}{}{{\mathrm{_c}}}_{{2}}{-}\frac{{744003}{}{{\mathrm{_c}}}_{{5}}}{{14}}{-}\frac{{1488006}{}{{\mathrm{_c}}}_{{6}}}{{7}}\right){+}{\mathrm{O}}{}\left({{x}}^{{2}}\right){,}\frac{{128835}{}{{\mathrm{_c}}}_{{5}}}{{8}}{+}\frac{{450765}{}{{\mathrm{_c}}}_{{6}}}{{2}}{+}{40320}{}{{\mathrm{_c}}}_{{3}}{+}{2}{}{{\mathrm{_c}}}_{{4}}{+}{x}{}\left({-}\frac{{128835}{}{{\mathrm{_c}}}_{{5}}}{{8}}{-}\frac{{450765}{}{{\mathrm{_c}}}_{{6}}}{{2}}{-}{40320}{}{{\mathrm{_c}}}_{{3}}{-}{{\mathrm{_c}}}_{{4}}\right){+}{\mathrm{O}}{}\left({{x}}^{{2}}\right){,}{{\mathrm{_c}}}_{{4}}{+}{x}{}\left(\frac{{5985}{}{{\mathrm{_c}}}_{{6}}}{{2}}{+}\frac{{6615}{}{{\mathrm{_c}}}_{{5}}}{{8}}\right){+}{\mathrm{O}}{}\left({{x}}^{{2}}\right)\right]$ (9)
 > $\mathrm{ExtendSeries}\left(\mathrm{sol},5\right)$
 $\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]$ (10)

References

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