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.

$\mathrm{with}\left(\mathrm{OrthogonalSeries}\right)\:$

$R≔\mathrm{Create}\left(\left[\frac{1}{a}\,a^2\,\frac{1}{a+1}\right]\,\mathrm{GegenbauerC}\left(n\,\frac{2}{3}\,x\right)\right)$

$R≔\frac{\mathrm{GegenbauerC}\left(0\,\frac{2}{3}\,x\right)}{a}+a^2\mathrm{GegenbauerC}\left(1\,\frac{2}{3}\,x\right)+\frac{\mathrm{GegenbauerC}\left(2\,\frac{2}{3}\,x\right)}{a+1}$

$\mathrm{res}≔\mathrm{Derivate}\left(R\,x\,\mathrm{operator}=\mathrm{struct}\,\mathrm{root}=1\right)$

$\mathrm{res}≔-\frac{144^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{\pi }}\sqrt{3}{a}^2\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{JacobiP}\left(0\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{45{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\left(\frac{44^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{\pi }}\sqrt{3}{a}^2\mathrm{\Gamma }\left(\frac{5}{6}\right)}{15{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}-\frac{54^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{\pi }}\sqrt{3}\mathrm{\Gamma }\left(\frac{5}{6}\right)}{9{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2\left(a+1\right)}\right)\mathrm{JacobiP}\left(1\,\frac{1}{6}\,\frac{7}{6}\,x\right)+\frac{204^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{\pi }}\sqrt{3}\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{JacobiP}\left(2\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{39{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2\left(a+1\right)}$

$\mathrm{SimplifyCoefficients}\left(\mathrm{res}\,\mathrm{normal}\right)$

$-\frac{144^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{\pi }}\sqrt{3}{a}^2\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{JacobiP}\left(0\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{45{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2}+\frac{4^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{\pi }}\sqrt{3}\mathrm{\Gamma }\left(\frac{5}{6}\right)\left(12a^3+12a^2-25\right)\mathrm{JacobiP}\left(1\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{45{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2\left(a+1\right)}+\frac{204^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\sqrt{\mathrm{\pi }}\sqrt{3}\mathrm{\Gamma }\left(\frac{5}{6}\right)\mathrm{JacobiP}\left(2\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{39{\mathrm{\Gamma }\left(\frac{2}{3}\right)}^2\left(a+1\right)}$

$\mathrm{SimplifyCoefficients}\left(\mathrm{res}\,\mathrm{simplify}\right)$

$-\frac{14a^2\mathrm{JacobiP}\left(0\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{15}+\frac{\left(12a^3+12a^2-25\right)\mathrm{JacobiP}\left(1\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{15a+15}+\frac{20\mathrm{JacobiP}\left(2\,\frac{1}{6}\,\frac{7}{6}\,x\right)}{13a+13}$

$\mathrm{coef}≔\frac{u\left(n\right)-u\left(n+1\right)}{n^2}+\frac{u\left(n+2\right)-u\left(n\right)}{n\left(n+1\right)}$

$\mathrm{coef}≔\frac{u\left(n\right)-u\left(n+1\right)}{n^2}+\frac{u\left(n+2\right)-u\left(n\right)}{n\left(n+1\right)}$

$\mathrm{R1}≔\mathrm{Create}\left(\left\{\mathrm{coef}\,n=1..\mathrm{\infty }\right\}\,\mathrm{GegenbauerC}\left(n\,a\,x\right)\right)$

$\mathrm{R1}≔\sum _{n=1}^{\mathrm{\infty }}\left(\frac{u\left(n\right)-u\left(n+1\right)}{n^2}+\frac{u\left(n+2\right)-u\left(n\right)}{n\left(n+1\right)}\right)\mathrm{GegenbauerC}\left(n\,a\,x\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{R1}\,\mathrm{collect}\,u\right)$

$\sum _{n=1}^{\mathrm{\infty }}\left(\left(\frac{1}{n^2}-\frac{1}{n\left(n+1\right)}\right)u\left(n\right)-\frac{u\left(n+1\right)}{n^2}+\frac{u\left(n+2\right)}{n\left(n+1\right)}\right)\mathrm{GegenbauerC}\left(n\,a\,x\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{R1}\,\mathrm{collect}\,u\,\mathrm{normal}\right)$

$\sum _{n=1}^{\mathrm{\infty }}\left(\frac{u\left(n\right)}{n^2\left(n+1\right)}-\frac{u\left(n+1\right)}{n^2}+\frac{u\left(n+2\right)}{n\left(n+1\right)}\right)\mathrm{GegenbauerC}\left(n\,a\,x\right)$