OrthogonalSeries - Maple Help

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OrthogonalSeries

 PolynomialMultiply
 multiply a series by a polynomial

 Calling Sequence PolynomialMultiply(p, S)

Parameters

 S - orthogonal series p - polynomial in the variables of S

Description

 • The PolynomialMultiply(p, S) function returns a series equal to the product of the series S and the polynomial p.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $S≔\mathrm{Create}\left(\left\{\frac{1}{n+1},n=3..7\right\},\mathrm{LaguerreL}\left(n,a,z\right)\right)$
 ${S}{:=}{\sum }_{{n}{=}{3}}^{{7}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{LaguerreL}}{}\left({n}{,}{a}{,}{z}\right)}{{n}{+}{1}}$ (1)
 > $\mathrm{S1}≔\mathrm{PolynomialMultiply}\left(3+z,S\right)$
 ${\mathrm{S1}}{:=}\left({-}\frac{{3}}{{4}}{-}\frac{{1}}{{4}}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{,}{z}\right){+}\left(\frac{{17}}{{10}}{+}\frac{{1}}{{20}}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{,}{z}\right){+}\left(\frac{{5}}{{4}}{+}\frac{{1}}{{8}}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({7}{,}{a}{,}{z}\right){-}{\mathrm{LaguerreL}}{}\left({8}{,}{a}{,}{z}\right){+}{\sum }_{{n}{=}{4}}^{{6}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{3}}{{n}{+}{1}}{+}\frac{{a}{-}{1}}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{a}{,}{z}\right)$ (2)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{S1},\mathrm{normal}\right)$
 $\left({-}\frac{{3}}{{4}}{-}\frac{{1}}{{4}}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{a}{,}{z}\right){+}\left(\frac{{17}}{{10}}{+}\frac{{1}}{{20}}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({3}{,}{a}{,}{z}\right){+}\left(\frac{{5}}{{4}}{+}\frac{{1}}{{8}}{}{a}\right){}{\mathrm{LaguerreL}}{}\left({7}{,}{a}{,}{z}\right){-}{\mathrm{LaguerreL}}{}\left({8}{,}{a}{,}{z}\right){+}{\sum }_{{n}{=}{4}}^{{6}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({3}{}{n}{+}{5}{+}{a}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{a}{,}{z}\right)}{\left({n}{+}{1}\right){}\left({n}{+}{2}\right)}$ (3)
 > $R≔\mathrm{Create}\left(\left\{\left[\left(1,1\right)=8\right]\right\},\mathrm{Charlier}\left(n,2,x\right),\mathrm{Charlier}\left(m,3,y\right)\right)$
 ${R}{:=}{8}{}{\mathrm{Charlier}}{}\left({1}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({1}{,}{3}{,}{y}\right)$ (4)
 > $\mathrm{R1}≔\mathrm{PolynomialMultiply}\left({\left(x+y\right)}^{3},R\right)$
 ${\mathrm{R1}}{:=}{144}{}{\mathrm{Charlier}}{}\left({0}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({0}{,}{3}{,}{y}\right){-}{776}{}{\mathrm{Charlier}}{}\left({0}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({1}{,}{3}{,}{y}\right){+}{432}{}{\mathrm{Charlier}}{}\left({0}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({2}{,}{3}{,}{y}\right){-}{800}{}{\mathrm{Charlier}}{}\left({1}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({0}{,}{3}{,}{y}\right){+}{3792}{}{\mathrm{Charlier}}{}\left({1}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({1}{,}{3}{,}{y}\right){-}{3192}{}{\mathrm{Charlier}}{}\left({1}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({2}{,}{3}{,}{y}\right){+}{1080}{}{\mathrm{Charlier}}{}\left({1}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({3}{,}{3}{,}{y}\right){-}{216}{}{\mathrm{Charlier}}{}\left({1}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({4}{,}{3}{,}{y}\right){+}{384}{}{\mathrm{Charlier}}{}\left({2}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({0}{,}{3}{,}{y}\right){-}{2320}{}{\mathrm{Charlier}}{}\left({2}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({1}{,}{3}{,}{y}\right){+}{1152}{}{\mathrm{Charlier}}{}\left({2}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({2}{,}{3}{,}{y}\right){-}{96}{}{\mathrm{Charlier}}{}\left({3}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({0}{,}{3}{,}{y}\right){+}{768}{}{\mathrm{Charlier}}{}\left({3}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({1}{,}{3}{,}{y}\right){-}{288}{}{\mathrm{Charlier}}{}\left({3}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({2}{,}{3}{,}{y}\right){-}{64}{}{\mathrm{Charlier}}{}\left({4}{,}{2}{,}{x}\right){}{\mathrm{Charlier}}{}\left({1}{,}{3}{,}{y}\right)$ (5)