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OrthogonalSeries

 SimplifyCoefficients
 simplify the coefficients of an orthogonal series

 Calling Sequence SimplifyCoefficients(S, funct, collect_expr, other_args)

Parameters

 S - orthogonal series funct - name chosen from: collect, expand, factor, normal, and simplify collect_expr - (optional) expression(s) to be collected other_args - (optional) complementary arguments if funct is collect

Description

 • The SimplifyCoefficients(S, funct) calling sequence applies the funct function, where funct is not collect, to the coefficients of the series $S$ (both particular and general coefficients).
 • The SimplifyCoefficients(S, collect, collect_expr, other_args) function applies the collect function to the coefficients of the series S (both particular and general coefficients). The user must specify the expression(s) to be collected by collect_expr. The user can specify a function to be applied to the collected expressions by other_args.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $R≔\mathrm{Create}\left(\left[\frac{1}{a},{a}^{2},\frac{1}{a+1}\right],\mathrm{GegenbauerC}\left(n,\frac{2}{3},x\right)\right)$
 ${R}{≔}\frac{{\mathrm{GegenbauerC}}{}\left({0}{,}\frac{{2}}{{3}}{,}{x}\right)}{{a}}{+}{{a}}^{{2}}{}{\mathrm{GegenbauerC}}{}\left({1}{,}\frac{{2}}{{3}}{,}{x}\right){+}\frac{{\mathrm{GegenbauerC}}{}\left({2}{,}\frac{{2}}{{3}}{,}{x}\right)}{{a}{+}{1}}$ (1)
 > $\mathrm{res}≔\mathrm{Derivate}\left(R,x,\mathrm{operator}=\mathrm{struct},\mathrm{root}=1\right)$
 ${\mathrm{res}}{≔}{-}\frac{{14}}{{45}}{}\frac{{{4}}^{{1}{/}{3}}{}\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{a}}^{{2}}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{JacobiP}}{}\left({0}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right)}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\left(\frac{{4}}{{15}}{}\frac{{{4}}^{{1}{/}{3}}{}\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{a}}^{{2}}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right)}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{-}\frac{{5}}{{9}}{}\frac{{{4}}^{{1}{/}{3}}{}\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right)}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}\left({a}{+}{1}\right)}\right){}{\mathrm{JacobiP}}{}\left({1}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right){+}\frac{{20}}{{39}}{}\frac{{{4}}^{{1}{/}{3}}{}\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{JacobiP}}{}\left({2}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right)}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}\left({a}{+}{1}\right)}$ (2)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{res},\mathrm{normal}\right)$
 ${-}\frac{{14}}{{45}}{}\frac{{{4}}^{{1}{/}{3}}{}\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{{a}}^{{2}}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{JacobiP}}{}\left({0}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right)}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}}{+}\frac{{1}}{{45}}{}\frac{{{4}}^{{1}{/}{3}}{}\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}\left({12}{}{{a}}^{{3}}{+}{12}{}{{a}}^{{2}}{-}{25}\right){}{\mathrm{JacobiP}}{}\left({1}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right)}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}\left({a}{+}{1}\right)}{+}\frac{{20}}{{39}}{}\frac{{{4}}^{{1}{/}{3}}{}\sqrt{{\mathrm{π}}}{}\sqrt{{3}}{}{\mathrm{Γ}}{}\left(\frac{{5}}{{6}}\right){}{\mathrm{JacobiP}}{}\left({2}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right)}{{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}\left({a}{+}{1}\right)}$ (3)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{res},\mathrm{simplify}\right)$
 ${-}\frac{{14}}{{15}}{}{{a}}^{{2}}{}{\mathrm{JacobiP}}{}\left({0}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right){+}\frac{{1}}{{15}}{}\frac{\left({12}{}{{a}}^{{3}}{+}{12}{}{{a}}^{{2}}{-}{25}\right){}{\mathrm{JacobiP}}{}\left({1}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right)}{{a}{+}{1}}{+}\frac{{20}}{{13}}{}\frac{{\mathrm{JacobiP}}{}\left({2}{,}\frac{{1}}{{6}}{,}\frac{{7}}{{6}}{,}{x}\right)}{{a}{+}{1}}$ (4)
 > $\mathrm{coef}≔\frac{u\left(n\right)-u\left(n+1\right)}{{n}^{2}}+\frac{u\left(n+2\right)-u\left(n\right)}{n\left(n+1\right)}$
 ${\mathrm{coef}}{≔}\frac{{u}{}\left({n}\right){-}{u}{}\left({n}{+}{1}\right)}{{{n}}^{{2}}}{+}\frac{{u}{}\left({n}{+}{2}\right){-}{u}{}\left({n}\right)}{{n}{}\left({n}{+}{1}\right)}$ (5)
 > $\mathrm{R1}≔\mathrm{Create}\left(\left\{\mathrm{coef},n=1..\mathrm{∞}\right\},\mathrm{GegenbauerC}\left(n,a,x\right)\right)$
 ${\mathrm{R1}}{≔}{\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{u}{}\left({n}\right){-}{u}{}\left({n}{+}{1}\right)}{{{n}}^{{2}}}{+}\frac{{u}{}\left({n}{+}{2}\right){-}{u}{}\left({n}\right)}{{n}{}\left({n}{+}{1}\right)}\right){}{\mathrm{GegenbauerC}}{}\left({n}{,}{a}{,}{x}\right)$ (6)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{R1},\mathrm{collect},u\right)$
 ${\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\left(\frac{{1}}{{{n}}^{{2}}}{-}\frac{{1}}{{n}{}\left({n}{+}{1}\right)}\right){}{u}{}\left({n}\right){-}\frac{{u}{}\left({n}{+}{1}\right)}{{{n}}^{{2}}}{+}\frac{{u}{}\left({n}{+}{2}\right)}{{n}{}\left({n}{+}{1}\right)}\right){}{\mathrm{GegenbauerC}}{}\left({n}{,}{a}{,}{x}\right)$ (7)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{R1},\mathrm{collect},u,\mathrm{normal}\right)$
 ${\sum }_{{n}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{u}{}\left({n}\right)}{{{n}}^{{2}}{}\left({n}{+}{1}\right)}{-}\frac{{u}{}\left({n}{+}{1}\right)}{{{n}}^{{2}}}{+}\frac{{u}{}\left({n}{+}{2}\right)}{{n}{}\left({n}{+}{1}\right)}\right){}{\mathrm{GegenbauerC}}{}\left({n}{,}{a}{,}{x}\right)$ (8)