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OrthogonalSeries

 ChangeBasis
 change the expansion basis of a series

 Calling Sequence ChangeBasis(p, P1,.., Pk) ChangeBasis(S, P1,.., Pk)

Parameters

 p - polynomial expression S - orthogonal series P1, .., Pk - classical orthogonal polynomials

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $S≔\mathrm{ChangeBasis}\left(1+2x+3{x}^{3},\mathrm{GegenbauerC}\left(n,1,x\right)\right)$
 ${S}{:=}{\mathrm{GegenbauerC}}{}\left({0}{,}{1}{,}{x}\right){+}\frac{{7}}{{4}}{}{\mathrm{GegenbauerC}}{}\left({1}{,}{1}{,}{x}\right){+}\frac{{3}}{{8}}{}{\mathrm{GegenbauerC}}{}\left({3}{,}{1}{,}{x}\right)$ (1)
 > $\mathrm{ChangeBasis}\left(S,\mathrm{LaguerreL}\left(n,\frac{2}{3},x\right)\right)$
 $\frac{{479}}{{9}}{}{\mathrm{LaguerreL}}{}\left({0}{,}\frac{{2}}{{3}}{,}{x}\right){-}{90}{}{\mathrm{LaguerreL}}{}\left({1}{,}\frac{{2}}{{3}}{,}{x}\right){+}{66}{}{\mathrm{LaguerreL}}{}\left({2}{,}\frac{{2}}{{3}}{,}{x}\right){-}{18}{}{\mathrm{LaguerreL}}{}\left({3}{,}\frac{{2}}{{3}}{,}{x}\right)$ (2)
 > $\mathrm{S1}≔\mathrm{ChangeBasis}\left({y}^{2}+{x}^{2}-1,\mathrm{ChebyshevU}\left(m,y\right),\mathrm{LaguerreL}\left(n,1,x\right)\right)$
 ${\mathrm{S1}}{:=}\frac{{21}}{{4}}{}{\mathrm{ChebyshevU}}{}\left({0}{,}{y}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{1}{,}{x}\right){-}{6}{}{\mathrm{ChebyshevU}}{}\left({0}{,}{y}\right){}{\mathrm{LaguerreL}}{}\left({1}{,}{1}{,}{x}\right){+}{2}{}{\mathrm{ChebyshevU}}{}\left({0}{,}{y}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{1}{,}{x}\right){+}\frac{{1}}{{4}}{}{\mathrm{ChebyshevU}}{}\left({2}{,}{y}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{1}{,}{x}\right)$ (3)
 > $\mathrm{ChangeBasis}\left(\mathrm{S1},\mathrm{ChebyshevU}\left(n,x\right),\mathrm{LaguerreL}\left(m,1,y\right)\right)$
 $\frac{{21}}{{4}}{}{\mathrm{LaguerreL}}{}\left({0}{,}{1}{,}{y}\right){}{\mathrm{ChebyshevU}}{}\left({0}{,}{x}\right){+}\frac{{1}}{{4}}{}{\mathrm{LaguerreL}}{}\left({0}{,}{1}{,}{y}\right){}{\mathrm{ChebyshevU}}{}\left({2}{,}{x}\right){-}{6}{}{\mathrm{LaguerreL}}{}\left({1}{,}{1}{,}{y}\right){}{\mathrm{ChebyshevU}}{}\left({0}{,}{x}\right){+}{2}{}{\mathrm{LaguerreL}}{}\left({2}{,}{1}{,}{y}\right){}{\mathrm{ChebyshevU}}{}\left({0}{,}{x}\right)$ (4)
 > $\mathrm{ChangeBasis}\left(\mathrm{S1},\mathrm{Kravchouk}\left(n,\frac{2}{7},N,x\right),\mathrm{LaguerreL}\left(m,1,y\right)\right)$
 $\left({5}{+}\frac{{4}}{{49}}{}{{N}}^{{2}}{+}\frac{{10}}{{49}}{}{N}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{1}{,}{y}\right){}{\mathrm{Kravchouk}}{}\left({0}{,}\frac{{2}}{{7}}{,}{N}{,}{x}\right){+}\left(\frac{{4}}{{7}}{}{N}{+}\frac{{3}}{{7}}\right){}{\mathrm{LaguerreL}}{}\left({0}{,}{1}{,}{y}\right){}{\mathrm{Kravchouk}}{}\left({1}{,}\frac{{2}}{{7}}{,}{N}{,}{x}\right){+}{2}{}{\mathrm{LaguerreL}}{}\left({0}{,}{1}{,}{y}\right){}{\mathrm{Kravchouk}}{}\left({2}{,}\frac{{2}}{{7}}{,}{N}{,}{x}\right){-}{6}{}{\mathrm{LaguerreL}}{}\left({1}{,}{1}{,}{y}\right){}{\mathrm{Kravchouk}}{}\left({0}{,}\frac{{2}}{{7}}{,}{N}{,}{x}\right){+}{2}{}{\mathrm{LaguerreL}}{}\left({2}{,}{1}{,}{y}\right){}{\mathrm{Kravchouk}}{}\left({0}{,}\frac{{2}}{{7}}{,}{N}{,}{x}\right)$ (5)
 > $\mathrm{ChangeBasis}\left(\left(\sum _{k=0}^{5}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{x}^{k}\right)\left(\sum _{k=0}^{5}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{y}^{k}\right),\mathrm{ChebyshevT}\left(n,x\right),\mathrm{ChebyshevT}\left(m,y\right)\right)$
 $\frac{{19}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({1}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({4}{,}{y}\right){+}\frac{{19}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({1}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({5}{,}{y}\right){+}\frac{{15}}{{8}}{}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({0}{,}{y}\right){+}\frac{{19}}{{8}}{}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({1}{,}{y}\right){+}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({2}{,}{y}\right){+}\frac{{9}}{{16}}{}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({3}{,}{y}\right){+}\frac{{1}}{{8}}{}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({4}{,}{y}\right){+}\frac{{1}}{{16}}{}{\mathrm{ChebyshevT}}{}\left({2}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({5}{,}{y}\right){+}\frac{{135}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({0}{,}{y}\right){+}\frac{{171}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({1}{,}{y}\right){+}\frac{{9}}{{16}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({2}{,}{y}\right){+}\frac{{81}}{{256}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({3}{,}{y}\right){+}\frac{{9}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({4}{,}{y}\right){+}\frac{{9}}{{256}}{}{\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({5}{,}{y}\right){+}\frac{{15}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({4}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({0}{,}{y}\right){+}\frac{{19}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({4}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({1}{,}{y}\right){+}\frac{{1}}{{8}}{}{\mathrm{ChebyshevT}}{}\left({4}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({2}{,}{y}\right){+}\frac{{9}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({4}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({3}{,}{y}\right){+}\frac{{1}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({4}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({4}{,}{y}\right){+}\frac{{1}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({4}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({5}{,}{y}\right){+}\frac{{15}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({5}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({0}{,}{y}\right){+}\frac{{19}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({5}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({1}{,}{y}\right){+}\frac{{1}}{{16}}{}{\mathrm{ChebyshevT}}{}\left({5}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({2}{,}{y}\right){+}\frac{{9}}{{256}}{}{\mathrm{ChebyshevT}}{}\left({5}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({3}{,}{y}\right){+}\frac{{1}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({5}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({4}{,}{y}\right){+}\frac{{1}}{{256}}{}{\mathrm{ChebyshevT}}{}\left({5}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({5}{,}{y}\right){+}\frac{{225}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({0}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({0}{,}{y}\right){+}\frac{{285}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({0}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({1}{,}{y}\right){+}\frac{{15}}{{8}}{}{\mathrm{ChebyshevT}}{}\left({0}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({2}{,}{y}\right){+}\frac{{135}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({0}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({3}{,}{y}\right){+}\frac{{15}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({0}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({4}{,}{y}\right){+}\frac{{15}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({0}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({5}{,}{y}\right){+}\frac{{285}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({1}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({0}{,}{y}\right){+}\frac{{361}}{{64}}{}{\mathrm{ChebyshevT}}{}\left({1}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({1}{,}{y}\right){+}\frac{{19}}{{8}}{}{\mathrm{ChebyshevT}}{}\left({1}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({2}{,}{y}\right){+}\frac{{171}}{{128}}{}{\mathrm{ChebyshevT}}{}\left({1}{,}{x}\right){}{\mathrm{ChebyshevT}}{}\left({3}{,}{y}\right)$ (6)
 > $\mathrm{S2}≔\mathrm{Create}\left(u\left(n\right),\mathrm{LaguerreL}\left(n,a,x\right)\right)$
 ${\mathrm{S2}}{:=}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{a}{,}{x}\right)$ (7)
 > $\mathrm{ChangeBasis}\left(\mathrm{S2},\mathrm{ChebyshevT}\left(n,x\right)\right)$
 ${"Warning : impossible change of basis for this infinite series"}$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{a}{,}{x}\right)$ (8)
 > $\mathrm{ChangeBasis}\left(\mathrm{S2},\mathrm{LaguerreL}\left(n,a+1,x\right)\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({u}{}\left({n}\right){-}{u}{}\left({n}{+}{1}\right)\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{a}{+}{1}{,}{x}\right)$ (9)
 > $\mathrm{S3}≔\mathrm{Create}\left(\left\{\left[2=1\right],\frac{1}{n!},n=3..\mathrm{∞}\right\},\mathrm{JacobiP}\left(n,2,2,x\right)\right)$
 ${\mathrm{S3}}{:=}{\mathrm{JacobiP}}{}\left({2}{,}{2}{,}{2}{,}{x}\right){+}{\sum }_{{n}{=}{3}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{JacobiP}}{}\left({n}{,}{2}{,}{2}{,}{x}\right)}{{n}{!}}$ (10)
 > $\mathrm{C1}≔\mathrm{ChangeBasis}\left(\mathrm{S3},\mathrm{JacobiP}\left(n,2,2+1,x\right)\right)$
 ${\mathrm{C1}}{:=}\frac{{4}}{{9}}{}{\mathrm{JacobiP}}{}\left({1}{,}{2}{,}{3}{,}{x}\right){+}\frac{{169}}{{198}}{}{\mathrm{JacobiP}}{}\left({2}{,}{2}{,}{3}{,}{x}\right){+}{\sum }_{{n}{=}{3}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({2}{}\left({n}{+}{1}\right){!}{}{{n}}^{{2}}{+}{2}{}{n}{!}{}{{n}}^{{2}}{+}{17}{}\left({n}{+}{1}\right){!}{}{n}{+}{11}{}{n}{!}{}{n}{+}{35}{}\left({n}{+}{1}\right){!}{+}{15}{}{n}{!}\right){}{\mathrm{JacobiP}}{}\left({n}{,}{2}{,}{3}{,}{x}\right)}{\left({2}{}{n}{+}{7}\right){}\left({n}{+}{1}\right){!}{}\left({2}{}{n}{+}{5}\right){}{n}{!}}$ (11)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{C1},\mathrm{simplify}\right)$
 $\frac{{4}}{{9}}{}{\mathrm{JacobiP}}{}\left({1}{,}{2}{,}{3}{,}{x}\right){+}\frac{{169}}{{198}}{}{\mathrm{JacobiP}}{}\left({2}{,}{2}{,}{3}{,}{x}\right){+}{\sum }_{{n}{=}{3}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({2}{}{{n}}^{{3}}{+}{21}{}{{n}}^{{2}}{+}{63}{}{n}{+}{50}\right){}{\mathrm{JacobiP}}{}\left({n}{,}{2}{,}{3}{,}{x}\right)}{\left({2}{}{n}{+}{5}\right){}\left({2}{}{n}{+}{7}\right){}\left({n}{+}{1}\right){!}}$ (12)
 > $\mathrm{C2}≔\mathrm{ChangeBasis}\left(\mathrm{S3},\mathrm{GegenbauerC}\left(n,2+\frac{1}{2},x\right)\right)$
 ${\mathrm{C2}}{:=}\frac{{2}}{{5}}{}{\mathrm{GegenbauerC}}{}\left({2}{,}\frac{{5}}{{2}}{,}{x}\right){+}{\sum }_{{n}{=}{3}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{12}{}{\mathrm{GegenbauerC}}{}\left({n}{,}\frac{{5}}{{2}}{,}{x}\right)}{{n}{!}{}{{2}}^{{n}}{}{{2}}^{{-}{n}}{}\left({n}{+}{3}\right){}\left({4}{+}{n}\right)}$ (13)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{C2},\mathrm{simplify}\right)$
 $\frac{{2}}{{5}}{}{\mathrm{GegenbauerC}}{}\left({2}{,}\frac{{5}}{{2}}{,}{x}\right){+}{\sum }_{{n}{=}{3}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{12}{}{\mathrm{GegenbauerC}}{}\left({n}{,}\frac{{5}}{{2}}{,}{x}\right)}{{n}{!}{}\left({n}{+}{3}\right){}\left({4}{+}{n}\right)}$ (14)
 > $\mathrm{S5}≔\mathrm{Create}\left(u\left(n,m\right),\mathrm{JacobiP}\left(n,1,1,x\right),\mathrm{LaguerreL}\left(m,2,y\right)\right)$
 ${\mathrm{S5}}{:=}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}{,}{m}\right){}{\mathrm{JacobiP}}{}\left({n}{,}{1}{,}{1}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{2}{,}{y}\right)$ (15)
 > $\mathrm{C3}≔\mathrm{ChangeBasis}\left(\mathrm{S5},\mathrm{LaguerreL}\left(n,2+1,y\right),\mathrm{JacobiP}\left(m,2,1,x\right)\right)$
 ${\mathrm{C3}}{:=}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({2}{}{u}{}\left({n}{+}{1}{,}{m}{+}{1}\right){}{{n}}^{{2}}{-}{2}{}{u}{}\left({n}{,}{m}{+}{1}\right){}{{n}}^{{2}}{-}{2}{}{u}{}\left({n}{+}{1}{,}{m}\right){}{{n}}^{{2}}{+}{2}{}{u}{}\left({n}{,}{m}\right){}{{n}}^{{2}}{+}{7}{}{u}{}\left({n}{+}{1}{,}{m}{+}{1}\right){}{n}{-}{11}{}{u}{}\left({n}{,}{m}{+}{1}\right){}{n}{-}{7}{}{u}{}\left({n}{+}{1}{,}{m}\right){}{n}{+}{11}{}{u}{}\left({n}{,}{m}\right){}{n}{+}{6}{}{u}{}\left({n}{+}{1}{,}{m}{+}{1}\right){-}{15}{}{u}{}\left({n}{,}{m}{+}{1}\right){-}{6}{}{u}{}\left({n}{+}{1}{,}{m}\right){+}{15}{}{u}{}\left({n}{,}{m}\right)\right){}{\mathrm{JacobiP}}{}\left({n}{,}{2}{,}{1}{,}{x}\right)}{\left({2}{}{n}{+}{5}\right){}\left({2}{}{n}{+}{3}\right)}\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{3}{,}{y}\right)$ (16)
 > $\mathrm{SimplifyCoefficients}\left(\mathrm{C3},\mathrm{collect},u\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{\left({2}{}{{n}}^{{2}}{+}{11}{}{n}{+}{15}\right){}{u}{}\left({n}{,}{m}\right)}{\left({2}{}{n}{+}{5}\right){}\left({2}{}{n}{+}{3}\right)}{+}\frac{\left({-}{2}{}{{n}}^{{2}}{-}{11}{}{n}{-}{15}\right){}{u}{}\left({n}{,}{m}{+}{1}\right)}{\left({2}{}{n}{+}{5}\right){}\left({2}{}{n}{+}{3}\right)}{+}\frac{\left({-}{2}{}{{n}}^{{2}}{-}{7}{}{n}{-}{6}\right){}{u}{}\left({n}{+}{1}{,}{m}\right)}{\left({2}{}{n}{+}{5}\right){}\left({2}{}{n}{+}{3}\right)}{+}\frac{\left({2}{}{{n}}^{{2}}{+}{7}{}{n}{+}{6}\right){}{u}{}\left({n}{+}{1}{,}{m}{+}{1}\right)}{\left({2}{}{n}{+}{5}\right){}\left({2}{}{n}{+}{3}\right)}\right){}{\mathrm{JacobiP}}{}\left({n}{,}{2}{,}{1}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{3}{,}{y}\right)$ (17)