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OrthogonalSeries

  

ChangeBasis

  

change the expansion basis of a series

 

Calling Sequence

Parameters

Description

Examples

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

withOrthogonalSeries:

SChangeBasis1+2x+3x3,GegenbauerCn,1,x

S:=GegenbauerC0,1,x+74GegenbauerC1,1,x+38GegenbauerC3,1,x

(1)

ChangeBasisS,LaguerreLn,23,x

4799LaguerreL0,23,x90LaguerreL1,23,x+66LaguerreL2,23,x18LaguerreL3,23,x

(2)

S1ChangeBasisy2+x21,ChebyshevUm,y,LaguerreLn,1,x

S1:=214ChebyshevU0,yLaguerreL0,1,x6ChebyshevU0,yLaguerreL1,1,x+2ChebyshevU0,yLaguerreL2,1,x+14ChebyshevU2,yLaguerreL0,1,x

(3)

ChangeBasisS1,ChebyshevUn,x,LaguerreLm,1,y

214LaguerreL0,1,yChebyshevU0,x+14LaguerreL0,1,yChebyshevU2,x6LaguerreL1,1,yChebyshevU0,x+2LaguerreL2,1,yChebyshevU0,x

(4)

ChangeBasisS1,Kravchoukn,27,N,x,LaguerreLm,1,y

5+449N2+1049NLaguerreL0,1,yKravchouk0,27,N,x+47N+37LaguerreL0,1,yKravchouk1,27,N,x+2LaguerreL0,1,yKravchouk2,27,N,x6LaguerreL1,1,yKravchouk0,27,N,x+2LaguerreL2,1,yKravchouk0,27,N,x

(5)

ChangeBasisk=05xkk=05yk,ChebyshevTn,x,ChebyshevTm,y

1964ChebyshevT1,xChebyshevT4,y+19128ChebyshevT1,xChebyshevT5,y+158ChebyshevT2,xChebyshevT0,y+198ChebyshevT2,xChebyshevT1,y+ChebyshevT2,xChebyshevT2,y+916ChebyshevT2,xChebyshevT3,y+18ChebyshevT2,xChebyshevT4,y+116ChebyshevT2,xChebyshevT5,y+135128ChebyshevT3,xChebyshevT0,y+171128ChebyshevT3,xChebyshevT1,y+916ChebyshevT3,xChebyshevT2,y+81256ChebyshevT3,xChebyshevT3,y+9128ChebyshevT3,xChebyshevT4,y+9256ChebyshevT3,xChebyshevT5,y+1564ChebyshevT4,xChebyshevT0,y+1964ChebyshevT4,xChebyshevT1,y+18ChebyshevT4,xChebyshevT2,y+9128ChebyshevT4,xChebyshevT3,y+164ChebyshevT4,xChebyshevT4,y+1128ChebyshevT4,xChebyshevT5,y+15128ChebyshevT5,xChebyshevT0,y+19128ChebyshevT5,xChebyshevT1,y+116ChebyshevT5,xChebyshevT2,y+9256ChebyshevT5,xChebyshevT3,y+1128ChebyshevT5,xChebyshevT4,y+1256ChebyshevT5,xChebyshevT5,y+22564ChebyshevT0,xChebyshevT0,y+28564ChebyshevT0,xChebyshevT1,y+158ChebyshevT0,xChebyshevT2,y+135128ChebyshevT0,xChebyshevT3,y+1564ChebyshevT0,xChebyshevT4,y+15128ChebyshevT0,xChebyshevT5,y+28564ChebyshevT1,xChebyshevT0,y+36164ChebyshevT1,xChebyshevT1,y+198ChebyshevT1,xChebyshevT2,y+171128ChebyshevT1,xChebyshevT3,y

(6)

S2Createun,LaguerreLn,a,x

S2:=n=0∞unLaguerreLn,a,x

(7)

ChangeBasisS2,ChebyshevTn,x

Warning : impossible change of basis for this infinite series

n=0∞unLaguerreLn,a,x

(8)

ChangeBasisS2,LaguerreLn,a+1,x

n=0∞unun+1LaguerreLn,a+1,x

(9)

S3Create2=1,1n!,n=3..∞,JacobiPn,2,2,x

S3:=JacobiP2,2,2,x+n=3∞JacobiPn,2,2,xn!

(10)

C1ChangeBasisS3,JacobiPn,2,2+1,x

C1:=49JacobiP1,2,3,x+169198JacobiP2,2,3,x+n=3∞2n+1!n2+2n!n2+17n+1!n+11n!n+35n+1!+15n!JacobiPn,2,3,x2n+7n+1!2n+5n!

(11)

SimplifyCoefficientsC1,simplify

49JacobiP1,2,3,x+169198JacobiP2,2,3,x+n=3∞2n3+21n2+63n+50JacobiPn,2,3,x2n+52n+7n+1!

(12)

C2ChangeBasisS3,GegenbauerCn,2+12,x

C2:=25GegenbauerC2,52,x+n=3∞12GegenbauerCn,52,xn!2n2nn+34+n

(13)

SimplifyCoefficientsC2,simplify

25GegenbauerC2,52,x+n=3∞12GegenbauerCn,52,xn!n+34+n

(14)

S5Createun,m,JacobiPn,1,1,x,LaguerreLm,2,y

S5:=m=0∞n=0∞un,mJacobiPn,1,1,xLaguerreLm,2,y

(15)

C3ChangeBasisS5,LaguerreLn,2+1,y,JacobiPm,2,1,x

C3:=m=0∞n=0∞2un+1,m+1n22un,m+1n22un+1,mn2+2un,mn2+7un+1,m+1n11un,m+1n7un+1,mn+11un,mn+6un+1,m+115un,m+16un+1,m+15un,mJacobiPn,2,1,x2n+52n+3LaguerreLm,3,y

(16)

SimplifyCoefficientsC3,collect,u

m=0∞n=0∞2n2+11n+15un,m2n+52n+3+2n211n15un,m+12n+52n+3+2n27n6un+1,m2n+52n+3+2n2+7n+6un+1,m+12n+52n+3JacobiPn,2,1,xLaguerreLm,3,y

(17)

See Also

ChebyshevT

ChebyshevU

GegenbauerC

JacobiP

LaguerreL

OrthogonalSeries

OrthogonalSeries[Create]

OrthogonalSeries[SimplifyCoefficients]

 


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