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OrthogonalSeries

 Coefficients
 extract a coefficient from a series

 Calling Sequence Coefficients(S) Coefficients(S, k) Coefficients(S, [k1,.., kn])

Parameters

 S - orthogonal series k, k1, .., kn - integers

Description

 • The Coefficients(S, k) and Coefficients(S, [k1,.., kn]) calling sequences extract the coefficients of index k and $\left[\mathrm{k1},\left(\right)..\left(\right),\mathrm{kn}\right]$, respectively, from the series S. In these calling sequences, the dimension of the index must correspond to the dimension of S. Otherwise, an error is returned.
 • The Coefficients(S) calling sequence returns the general term coefficient of the series S.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $S≔\mathrm{Create}\left(\frac{n+1}{m+1},\mathrm{GegenbauerC}\left(n,1,x\right),\mathrm{LaguerreL}\left(m,1,y\right)\right)$
 ${S}{:=}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({n}{+}{1}\right){}{\mathrm{GegenbauerC}}{}\left({n}{,}{1}{,}{x}\right)}{{m}{+}{1}}\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{1}{,}{y}\right)$ (1)
 > $\mathrm{Coefficients}\left(S\right)$
 $\frac{{n}{+}{1}}{{m}{+}{1}}$ (2)
 > $\mathrm{Coefficients}\left(S,\left[2,3\right]\right)$
 $\frac{{3}}{{4}}$ (3)
 > $\mathrm{S1}≔\mathrm{Create}\left(\left[\left(1,2\right)=4,\left(3,4\right)=-7\right],\mathrm{GegenbauerC}\left(n,1,x\right),\mathrm{LaguerreL}\left(m,1,y\right)\right)$
 ${\mathrm{S1}}{:=}{4}{}{\mathrm{GegenbauerC}}{}\left({1}{,}{1}{,}{x}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{1}{,}{y}\right){-}{7}{}{\mathrm{GegenbauerC}}{}\left({3}{,}{1}{,}{x}\right){}{\mathrm{LaguerreL}}{}\left({4}{,}{1}{,}{y}\right)$ (4)
 > $\mathrm{Coefficients}\left(\mathrm{S1}\right);$$\mathrm{Coefficients}\left(\mathrm{S1},1\right);$$\mathrm{Coefficients}\left(\mathrm{S1},\left[1,2\right]\right);$$\mathrm{Coefficients}\left(\mathrm{S1},\left[2,2\right]\right)$
 ${0}$
 ${4}$
 ${0}$ (5)
 > $\mathrm{S1}≔\mathrm{Create}\left(\left\{\left[\left(1,2\right)=4,\left(3,4\right)=-7\right],a\left(n,m\right)\right\},\mathrm{GegenbauerC}\left(n,1,x\right),\mathrm{LaguerreL}\left(m,1,y\right)\right)$
 ${\mathrm{S1}}{:=}{4}{}{\mathrm{GegenbauerC}}{}\left({1}{,}{1}{,}{x}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{1}{,}{y}\right){-}{7}{}{\mathrm{GegenbauerC}}{}\left({3}{,}{1}{,}{x}\right){}{\mathrm{LaguerreL}}{}\left({4}{,}{1}{,}{y}\right){+}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}{}\left({n}{,}{m}\right){}{\mathrm{GegenbauerC}}{}\left({n}{,}{1}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{1}{,}{y}\right)$ (6)
 > $\mathrm{Coefficients}\left(\mathrm{S1}\right);$$\mathrm{Coefficients}\left(\mathrm{S1},\left[1,2\right]\right);$$\mathrm{Coefficients}\left(\mathrm{S1},\left[2,2\right]\right)$
 ${a}{}\left({n}{,}{m}\right)$
 ${a}{}\left({1}{,}{2}\right){+}{4}$
 ${a}{}\left({2}{,}{2}\right)$ (7)