OrthogonalSeries - Maple Programming Help

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OrthogonalSeries

 ApplyOperator
 apply a differential or difference operator to a series

 Calling Sequence ApplyOperator(L,S)

Parameters

 L - differential or difference operator S - orthogonal series

Description

 • The ApplyOperator function applies the operator $L$ to the series $S$ using the elementary operations for series: differentiation, derivative representation, and multiplication by a polynomial.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $\mathrm{S1}≔\mathrm{Create}\left({2}^{n},\mathrm{LaguerreL}\left(n,1,x\right)\right)$
 ${\mathrm{S1}}{≔}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{{2}}^{{n}}{}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right)$ (1)
 > $\mathrm{R1}≔\mathrm{ApplyOperator}\left(\left[{x}^{2}{\mathrm{dx}}^{2}-7x\mathrm{dx}+3,\left[\mathrm{dx},x\right]\right],\mathrm{S1}\right)$
 ${\mathrm{R1}}{≔}{139}{}{\mathrm{LaguerreL}}{}\left({0}{,}{3}{,}{x}\right){+}{332}{}{\mathrm{LaguerreL}}{}\left({1}{,}{3}{,}{x}\right){+}{\sum }_{{n}{=}{2}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({-}{8}{}{{2}}^{{n}}{}{n}{+}{15}{}{{2}}^{{n}{+}{1}}{}{n}{+}{3}{}{{2}}^{{n}{+}{2}}{}{n}{-}{19}{}{{2}}^{{n}{+}{3}}{}{n}{+}{3}{}{{2}}^{{n}}{+}{22}{}{{2}}^{{n}{+}{1}}{-}{33}{}{{2}}^{{n}{+}{2}}{-}{12}{}{{2}}^{{n}{+}{3}}{+}{{2}}^{{n}{+}{4}}{}{{n}}^{{2}}{+}{{2}}^{{n}}{}{{n}}^{{2}}{-}{4}{}{{2}}^{{n}{+}{1}}{}{{n}}^{{2}}{+}{6}{}{{2}}^{{n}{+}{2}}{}{{n}}^{{2}}{-}{4}{}{{2}}^{{n}{+}{3}}{}{{n}}^{{2}}{+}{9}{}{{2}}^{{n}{+}{4}}{}{n}{+}{20}{}{{2}}^{{n}{+}{4}}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{3}{,}{x}\right)$ (2)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{R1},\mathrm{simplify}\right)$
 ${139}{}{\mathrm{LaguerreL}}{}\left({0}{,}{3}{,}{x}\right){+}{332}{}{\mathrm{LaguerreL}}{}\left({1}{,}{3}{,}{x}\right){+}{\sum }_{{n}{=}{2}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({{n}}^{{2}}{+}{26}{}{n}{+}{139}\right){}{{2}}^{{n}}{}{\mathrm{LaguerreL}}{}\left({n}{,}{3}{,}{x}\right)$ (3)
 > $\mathrm{S3}≔\mathrm{Create}\left(a\left(n,m\right),\mathrm{LaguerreL}\left(n,2,x\right),\mathrm{LaguerreL}\left(m,3,y\right)\right)$
 ${\mathrm{S3}}{≔}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}{}\left({n}{,}{m}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{2}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{3}{,}{y}\right)$ (4)
 > $R≔\mathrm{ApplyOperator}\left(\left[x\mathrm{dx}+y\mathrm{dy},\left[\mathrm{dx},x\right],\left[\mathrm{dy},y\right]\right],\mathrm{S3}\right)$
 ${R}{≔}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({n}{}{a}{}\left({n}{,}{m}\right){+}\left({-}{2}{}{n}{-}{4}\right){}{a}{}\left({n}{+}{1}{,}{m}\right){+}\left({n}{+}{4}\right){}{a}{}\left({n}{+}{2}{,}{m}\right){+}{m}{}{a}{}\left({n}{,}{m}\right){+}\left({-}{2}{}{m}{-}{5}\right){}{a}{}\left({n}{,}{m}{+}{1}\right){+}\left({m}{+}{5}\right){}{a}{}\left({n}{,}{m}{+}{2}\right)\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{2}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{3}{,}{y}\right)$ (5)
 > $\mathrm{SimplifyCoefficients}\left(R,\mathrm{collect},a\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\left({n}{+}{m}\right){}{a}{}\left({n}{,}{m}\right){+}\left({-}{2}{}{n}{-}{4}\right){}{a}{}\left({n}{+}{1}{,}{m}\right){+}\left({n}{+}{4}\right){}{a}{}\left({n}{+}{2}{,}{m}\right){+}\left({-}{2}{}{m}{-}{5}\right){}{a}{}\left({n}{,}{m}{+}{1}\right){+}\left({m}{+}{5}\right){}{a}{}\left({n}{,}{m}{+}{2}\right)\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{2}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{3}{,}{y}\right)$ (6)
 > $\mathrm{S5}≔\mathrm{Create}\left(\left\{\left[1=7\right],\frac{1}{n+1}\right\},\mathrm{LaguerreL}\left(n,1,x\right)\right)$
 ${\mathrm{S5}}{≔}{7}{}{\mathrm{LaguerreL}}{}\left({1}{,}{1}{,}{x}\right){+}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right)}{{n}{+}{1}}$ (7)
 > $\mathrm{R2}≔\mathrm{ApplyOperator}\left(\left[\left(1+x\right)d+a,\left[d,x\right]\right],\mathrm{S5}\right)$
 ${\mathrm{R2}}{≔}\left({-}\frac{{13}}{{2}}{}{a}{-}{29}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{2}{,}{x}\right){+}\left({7}{}{a}{+}{7}\right){}{\mathrm{LaguerreL}}{}\left({1}{,}{2}{,}{x}\right){+}{\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{a}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}{-}\frac{{1}}{{n}{+}{2}}{-}\frac{{1}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{2}{,}{x}\right)$ (8)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{R2},\mathrm{simplify}\right)$
 $\left({-}\frac{{13}}{{2}}{}{a}{-}{29}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{2}{,}{x}\right){+}\left({7}{}{a}{+}{7}\right){}{\mathrm{LaguerreL}}{}\left({1}{,}{2}{,}{x}\right){+}{\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({a}{-}{n}{-}{2}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{2}{,}{x}\right)}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}$ (9)