SumTools[IndefiniteSum] - Maple Programming Help

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SumTools[IndefiniteSum]

 Polynomial
 compute closed forms of indefinite sums of polynomials

 Calling Sequence Polynomial(f, k)

Parameters

 f - polynomial in k k - name

Description

 • The Polynomial(f, k) command computes a closed form of the indefinite sum of f with respect to k.
 • Polynomials are summed using a formula based on Bernoulli polynomials.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}[\mathrm{IndefiniteSum}]\right):$
 > $p≔68{y}^{2}-17{x}^{2}{y}^{2}-98x{y}^{3}-36{x}^{4}y+40{x}^{3}{y}^{2}+22{x}^{2}{y}^{3}$
 ${p}{≔}{-}{36}{}{{x}}^{{4}}{}{y}{+}{40}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{22}{}{{x}}^{{2}}{}{{y}}^{{3}}{-}{17}{}{{x}}^{{2}}{}{{y}}^{{2}}{-}{98}{}{x}{}{{y}}^{{3}}{+}{68}{}{{y}}^{{2}}$ (1)
 > $\mathrm{dpx}≔\mathrm{Polynomial}\left(p,x\right)$
 ${\mathrm{dpx}}{≔}\frac{{391}}{{6}}{}{{y}}^{{2}}{}{x}{-}{60}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}\frac{{158}}{{3}}{}{x}{}{{y}}^{{3}}{+}\frac{{22}}{{3}}{}{{x}}^{{3}}{}{{y}}^{{3}}{-}\frac{{77}}{{3}}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}\frac{{37}}{{2}}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{10}{}{{y}}^{{2}}{}{{x}}^{{4}}{-}\frac{{36}}{{5}}{}{y}{}{{x}}^{{5}}{+}{18}{}{{x}}^{{4}}{}{y}{-}{12}{}{y}{}{{x}}^{{3}}{+}\frac{{6}}{{5}}{}{y}{}{x}$ (2)
 > $\mathrm{expand}\left(p-\left(\genfrac{}{}{0}{}{\mathrm{dpx}}{\phantom{x=x+1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{dpx}}}{x=x+1}-\mathrm{dpx}\right)\right)$
 ${0}$ (3)
 > $\mathrm{dpy}≔\mathrm{Polynomial}\left(p,y\right)$
 ${\mathrm{dpy}}{≔}{-}{18}{}{{y}}^{{2}}{}{{x}}^{{4}}{+}{18}{}{{x}}^{{4}}{}{y}{+}\frac{{40}}{{3}}{}{{x}}^{{3}}{}{{y}}^{{3}}{-}\frac{{50}}{{3}}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}\frac{{68}}{{3}}{}{{y}}^{{3}}{-}{20}{}{{x}}^{{3}}{}{{y}}^{{2}}{+}{14}{}{{x}}^{{2}}{}{{y}}^{{2}}{-}{34}{}{{y}}^{{2}}{+}\frac{{20}}{{3}}{}{y}{}{{x}}^{{3}}{-}\frac{{17}}{{6}}{}{{x}}^{{2}}{}{y}{+}\frac{{34}}{{3}}{}{y}{+}\frac{{11}}{{2}}{}{{x}}^{{2}}{}{{y}}^{{4}}{-}\frac{{49}}{{2}}{}{x}{}{{y}}^{{4}}{+}{49}{}{x}{}{{y}}^{{3}}{-}\frac{{49}}{{2}}{}{{y}}^{{2}}{}{x}$ (4)
 > $\mathrm{expand}\left(p-\left(\genfrac{}{}{0}{}{\mathrm{dpy}}{\phantom{y=y+1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{\mathrm{dpy}}}{y=y+1}-\mathrm{dpy}\right)\right)$
 ${0}$ (5)