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The ChangeBasis(p, P1, .., Pk) calling sequence creates a series expanded in terms of the polynomials P1, .., Pk that is equal to the polynomial p. The polynomials P1, .., Pk must have distinct indices and variables, and be in the OrthogonalSeries database.

The ChangeBasis(S, P1, .., Pk) calling sequence replaces each polynomial in S by the polynomial Pi that depends on the same variable if it exists. If multiple polynomials Pi, .., Pj share the same variable, the last one is used.

If the series S is infinite, the change of polynomial is not necessarily possible for every Pi. For infinite series, the ChangeBasis routine attempts an iterated derivative representation transform to perform the change of polynomials. If the process fails for some Pi, it is ignored and a warning message is printed.

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

$S≔\mathrm{ChangeBasis}\left(1+2x+3x^3\,\mathrm{GegenbauerC}\left(n\,1\,x\right)\right)$

$S≔\mathrm{GegenbauerC}\left(0\,1\,x\right)+\frac{7\mathrm{GegenbauerC}\left(1\,1\,x\right)}{4}+\frac{3\mathrm{GegenbauerC}\left(3\,1\,x\right)}{8}$

$\mathrm{ChangeBasis}\left(S\,\mathrm{LaguerreL}\left(n\,\frac{2}{3}\,x\right)\right)$

$\frac{479\mathrm{LaguerreL}\left(0\,\frac{2}{3}\,x\right)}{9}-90\mathrm{LaguerreL}\left(1\,\frac{2}{3}\,x\right)+66\mathrm{LaguerreL}\left(2\,\frac{2}{3}\,x\right)-18\mathrm{LaguerreL}\left(3\,\frac{2}{3}\,x\right)$

$\mathrm{S1}≔\mathrm{ChangeBasis}\left(y^2+x^2-1\,\mathrm{ChebyshevU}\left(m\,y\right)\,\mathrm{LaguerreL}\left(n\,1\,x\right)\right)$

$\mathrm{S1}≔\frac{21\mathrm{ChebyshevU}\left(0\,y\right)\mathrm{LaguerreL}\left(0\,1\,x\right)}{4}-6\mathrm{ChebyshevU}\left(0\,y\right)\mathrm{LaguerreL}\left(1\,1\,x\right)+2\mathrm{ChebyshevU}\left(0\,y\right)\mathrm{LaguerreL}\left(2\,1\,x\right)+\frac{\mathrm{ChebyshevU}\left(2\,y\right)\mathrm{LaguerreL}\left(0\,1\,x\right)}{4}$

$\mathrm{ChangeBasis}\left(\mathrm{S1}\,\mathrm{ChebyshevU}\left(n\,x\right)\,\mathrm{LaguerreL}\left(m\,1\,y\right)\right)$

$\frac{21\mathrm{LaguerreL}\left(0\,1\,y\right)\mathrm{ChebyshevU}\left(0\,x\right)}{4}+\frac{\mathrm{LaguerreL}\left(0\,1\,y\right)\mathrm{ChebyshevU}\left(2\,x\right)}{4}-6\mathrm{LaguerreL}\left(1\,1\,y\right)\mathrm{ChebyshevU}\left(0\,x\right)+2\mathrm{LaguerreL}\left(2\,1\,y\right)\mathrm{ChebyshevU}\left(0\,x\right)$

$\mathrm{ChangeBasis}\left(\mathrm{S1}\,\mathrm{Kravchouk}\left(n\,\frac{2}{7}\,N\,x\right)\,\mathrm{LaguerreL}\left(m\,1\,y\right)\right)$

$\left(5+\frac{4}{49}{N}^2+\frac{10}{49}N\right)\mathrm{LaguerreL}\left(0\,1\,y\right)\mathrm{Kravchouk}\left(0\,\frac{2}{7}\,N\,x\right)+\left(\frac{4N}{7}+\frac{3}{7}\right)\mathrm{LaguerreL}\left(0\,1\,y\right)\mathrm{Kravchouk}\left(1\,\frac{2}{7}\,N\,x\right)+2\mathrm{LaguerreL}\left(0\,1\,y\right)\mathrm{Kravchouk}\left(2\,\frac{2}{7}\,N\,x\right)-6\mathrm{LaguerreL}\left(1\,1\,y\right)\mathrm{Kravchouk}\left(0\,\frac{2}{7}\,N\,x\right)+2\mathrm{LaguerreL}\left(2\,1\,y\right)\mathrm{Kravchouk}\left(0\,\frac{2}{7}\,N\,x\right)$

$\mathrm{ChangeBasis}\left(\left(\mathrm{sum}\left(x^k\,k=0..5\right)\right)\left(\mathrm{sum}\left(y^k\,k=0..5\right)\right)\,\mathrm{ChebyshevT}\left(n\,x\right)\,\mathrm{ChebyshevT}\left(m\,y\right)\right)$

$\frac{15\mathrm{ChebyshevT}\left(0\,x\right)\mathrm{ChebyshevT}\left(2\,y\right)}{8}+\frac{135\mathrm{ChebyshevT}\left(0\,x\right)\mathrm{ChebyshevT}\left(3\,y\right)}{128}+\frac{15\mathrm{ChebyshevT}\left(0\,x\right)\mathrm{ChebyshevT}\left(4\,y\right)}{64}+\frac{15\mathrm{ChebyshevT}\left(0\,x\right)\mathrm{ChebyshevT}\left(5\,y\right)}{128}+\frac{285\mathrm{ChebyshevT}\left(1\,x\right)\mathrm{ChebyshevT}\left(0\,y\right)}{64}+\frac{361\mathrm{ChebyshevT}\left(1\,x\right)\mathrm{ChebyshevT}\left(1\,y\right)}{64}+\frac{19\mathrm{ChebyshevT}\left(1\,x\right)\mathrm{ChebyshevT}\left(2\,y\right)}{8}+\frac{171\mathrm{ChebyshevT}\left(1\,x\right)\mathrm{ChebyshevT}\left(3\,y\right)}{128}+\frac{19\mathrm{ChebyshevT}\left(1\,x\right)\mathrm{ChebyshevT}\left(4\,y\right)}{64}+\frac{19\mathrm{ChebyshevT}\left(1\,x\right)\mathrm{ChebyshevT}\left(5\,y\right)}{128}+\frac{15\mathrm{ChebyshevT}\left(2\,x\right)\mathrm{ChebyshevT}\left(0\,y\right)}{8}+\frac{19\mathrm{ChebyshevT}\left(2\,x\right)\mathrm{ChebyshevT}\left(1\,y\right)}{8}+\mathrm{ChebyshevT}\left(2\,x\right)\mathrm{ChebyshevT}\left(2\,y\right)+\frac{9\mathrm{ChebyshevT}\left(2\,x\right)\mathrm{ChebyshevT}\left(3\,y\right)}{16}+\frac{\mathrm{ChebyshevT}\left(2\,x\right)\mathrm{ChebyshevT}\left(4\,y\right)}{8}+\frac{\mathrm{ChebyshevT}\left(2\,x\right)\mathrm{ChebyshevT}\left(5\,y\right)}{16}+\frac{135\mathrm{ChebyshevT}\left(3\,x\right)\mathrm{ChebyshevT}\left(0\,y\right)}{128}+\frac{171\mathrm{ChebyshevT}\left(3\,x\right)\mathrm{ChebyshevT}\left(1\,y\right)}{128}+\frac{9\mathrm{ChebyshevT}\left(3\,x\right)\mathrm{ChebyshevT}\left(2\,y\right)}{16}+\frac{81\mathrm{ChebyshevT}\left(3\,x\right)\mathrm{ChebyshevT}\left(3\,y\right)}{256}+\frac{9\mathrm{ChebyshevT}\left(3\,x\right)\mathrm{ChebyshevT}\left(4\,y\right)}{128}+\frac{9\mathrm{ChebyshevT}\left(3\,x\right)\mathrm{ChebyshevT}\left(5\,y\right)}{256}+\frac{15\mathrm{ChebyshevT}\left(4\,x\right)\mathrm{ChebyshevT}\left(0\,y\right)}{64}+\frac{19\mathrm{ChebyshevT}\left(4\,x\right)\mathrm{ChebyshevT}\left(1\,y\right)}{64}+\frac{\mathrm{ChebyshevT}\left(4\,x\right)\mathrm{ChebyshevT}\left(2\,y\right)}{8}+\frac{9\mathrm{ChebyshevT}\left(4\,x\right)\mathrm{ChebyshevT}\left(3\,y\right)}{128}+\frac{\mathrm{ChebyshevT}\left(4\,x\right)\mathrm{ChebyshevT}\left(4\,y\right)}{64}+\frac{\mathrm{ChebyshevT}\left(4\,x\right)\mathrm{ChebyshevT}\left(5\,y\right)}{128}+\frac{15\mathrm{ChebyshevT}\left(5\,x\right)\mathrm{ChebyshevT}\left(0\,y\right)}{128}+\frac{19\mathrm{ChebyshevT}\left(5\,x\right)\mathrm{ChebyshevT}\left(1\,y\right)}{128}+\frac{\mathrm{ChebyshevT}\left(5\,x\right)\mathrm{ChebyshevT}\left(2\,y\right)}{16}+\frac{9\mathrm{ChebyshevT}\left(5\,x\right)\mathrm{ChebyshevT}\left(3\,y\right)}{256}+\frac{\mathrm{ChebyshevT}\left(5\,x\right)\mathrm{ChebyshevT}\left(4\,y\right)}{128}+\frac{\mathrm{ChebyshevT}\left(5\,x\right)\mathrm{ChebyshevT}\left(5\,y\right)}{256}+\frac{225\mathrm{ChebyshevT}\left(0\,x\right)\mathrm{ChebyshevT}\left(0\,y\right)}{64}+\frac{285\mathrm{ChebyshevT}\left(0\,x\right)\mathrm{ChebyshevT}\left(1\,y\right)}{64}$

$\mathrm{S2}≔\mathrm{Create}\left(u\left(n\right)\,\mathrm{LaguerreL}\left(n\,a\,x\right)\right)$

$\mathrm{S2}≔\sum _{n=0}^{\mathrm{\infty }}u\left(n\right)\mathrm{LaguerreL}\left(n\,a\,x\right)$

$\mathrm{ChangeBasis}\left(\mathrm{S2}\,\mathrm{ChebyshevT}\left(n\,x\right)\right)$

$''Warning\; :\; impossible\; change\; of\; basis\; for\; this\; infinite\; series''$

$\sum _{n=0}^{\mathrm{\infty }}u\left(n\right)\mathrm{LaguerreL}\left(n\,a\,x\right)$

$\mathrm{ChangeBasis}\left(\mathrm{S2}\,\mathrm{LaguerreL}\left(n\,a+1\,x\right)\right)$

$\sum _{n=0}^{\mathrm{\infty }}\left(u\left(n\right)-u\left(n+1\right)\right)\mathrm{LaguerreL}\left(n\,a+1\,x\right)$

$\mathrm{S3}≔\mathrm{Create}\left(\left\{\frac{1}{n!}\,n=3..\mathrm{\infty }\,\left[2=1\right]\right\}\,\mathrm{JacobiP}\left(n\,2\,2\,x\right)\right)$

$\mathrm{S3}≔\mathrm{JacobiP}\left(2\,2\,2\,x\right)+\left(\sum _{n=3}^{\mathrm{\infty }}\frac{\mathrm{JacobiP}\left(n\,2\,2\,x\right)}{n!}\right)$

$\mathrm{C1}≔\mathrm{ChangeBasis}\left(\mathrm{S3}\,\mathrm{JacobiP}\left(n\,2\,2+1\,x\right)\right)$

$\mathrm{C1}≔\frac{4\mathrm{JacobiP}\left(1\,2\,3\,x\right)}{9}+\frac{169\mathrm{JacobiP}\left(2\,2\,3\,x\right)}{198}+\left(\sum _{n=3}^{\mathrm{\infty }}\frac{\left(2\left(n+1\right)!n^2+2n!n^2+17\left(n+1\right)!n+11n!n+35\left(n+1\right)!+15n!\right)\mathrm{JacobiP}\left(n\,2\,3\,x\right)}{\left(2n+7\right)\left(n+1\right)!\left(2n+5\right)n!}\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{C1}\,\mathrm{simplify}\right)$

$\frac{4\mathrm{JacobiP}\left(1\,2\,3\,x\right)}{9}+\frac{169\mathrm{JacobiP}\left(2\,2\,3\,x\right)}{198}+\left(\sum _{n=3}^{\mathrm{\infty }}\frac{\left(2n^3+21n^2+63n+50\right)\mathrm{JacobiP}\left(n\,2\,3\,x\right)}{\left(4n^2+24n+35\right)\left(n+1\right)!}\right)$

$\mathrm{C2}≔\mathrm{ChangeBasis}\left(\mathrm{S3}\,\mathrm{GegenbauerC}\left(n\,2+\frac{1}{2}\,x\right)\right)$

$\mathrm{C2}≔\frac{2\mathrm{GegenbauerC}\left(2\,\frac{5}{2}\,x\right)}{5}+\left(\sum _{n=3}^{\mathrm{\infty }}\frac{12\mathrm{GegenbauerC}\left(n\,\frac{5}{2}\,x\right)}{n!2^n{2}^{-n}\left(n+3\right)\left(4+n\right)}\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{C2}\,\mathrm{simplify}\right)$

$\frac{2\mathrm{GegenbauerC}\left(2\,\frac{5}{2}\,x\right)}{5}+\left(\sum _{n=3}^{\mathrm{\infty }}\frac{12\mathrm{GegenbauerC}\left(n\,\frac{5}{2}\,x\right)}{n!\left(n+3\right)\left(4+n\right)}\right)$

$\mathrm{S5}≔\mathrm{Create}\left(u\left(n\,m\right)\,\mathrm{JacobiP}\left(n\,1\,1\,x\right)\,\mathrm{LaguerreL}\left(m\,2\,y\right)\right)$

$\mathrm{S5}≔\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}u\left(n\,m\right)\mathrm{JacobiP}\left(n\,1\,1\,x\right)\right)\mathrm{LaguerreL}\left(m\,2\,y\right)$

$\mathrm{C3}≔\mathrm{ChangeBasis}\left(\mathrm{S5}\,\mathrm{LaguerreL}\left(n\,2+1\,y\right)\,\mathrm{JacobiP}\left(m\,2\,1\,x\right)\right)$

$\mathrm{C3}≔\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\left(2u\left(n+1\,m+1\right)n^2-2u\left(n\,m+1\right)n^2-2u\left(n+1\,m\right)n^2+2u\left(n\,m\right)n^2+7u\left(n+1\,m+1\right)n-11u\left(n\,m+1\right)n-7u\left(n+1\,m\right)n+11u\left(n\,m\right)n+6u\left(n+1\,m+1\right)-15u\left(n\,m+1\right)-6u\left(n+1\,m\right)+15u\left(n\,m\right)\right)\mathrm{JacobiP}\left(n\,2\,1\,x\right)}{\left(2n+5\right)\left(2n+3\right)}\right)\mathrm{LaguerreL}\left(m\,3\,y\right)$

$\mathrm{SimplifyCoefficients}\left(\mathrm{C3}\,\mathrm{collect}\,u\right)$

$\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}\left(\frac{\left(2n^2+11n+15\right)u\left(n\,m\right)}{\left(2n+5\right)\left(2n+3\right)}+\frac{\left(-2n^2-11n-15\right)u\left(n\,m+1\right)}{\left(2n+5\right)\left(2n+3\right)}+\frac{\left(-2n^2-7n-6\right)u\left(n+1\,m\right)}{\left(2n+5\right)\left(2n+3\right)}+\frac{\left(2n^2+7n+6\right)u\left(n+1\,m+1\right)}{\left(2n+5\right)\left(2n+3\right)}\right)\mathrm{JacobiP}\left(n\,2\,1\,x\right)\right)\mathrm{LaguerreL}\left(m\,3\,y\right)$