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OrthogonalSeries

  

DerivativeRepresentation

  

take differential representation transform of a series

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DerivativeRepresentation(S, x, optional_root)

DerivativeRepresentation(S, x1,.., xn, optional_root)

DerivativeRepresentation(S, [x1,.., xn], optional_root)

Parameters

S

-

orthogonal series

x, x1, .., xn

-

names

optional_root

-

(optional) equation of the form root = val where val is a symbol representing a root of the polynomial associated with the expansion family

Description

• 

The DerivativeRepresentation(S, x) calling sequence returns a series equal to S written in terms of the family of polynomials produced by differentiating the S polynomials with respect to x.

• 

The DerivativeRepresentation(S, x1,.., xn) and DerivativeRepresentation(S, [x1,.., xn]) calling sequences are equivalent to the recursive calling sequence DerivativeRepresentation(...DerivativeRepresentation(S, x1),..., xn).

• 

The partial differential representation can be used for continuous hypergeometric polynomials with a degree 2 sigma polynomial. The partial differential representation (with respect to the root xi for the polynomials poly(n, x) depending on x in the series S) is obtained by using the DerivativeRepresentation(S, x, root=val) calling sequence. If val is not a root of the sigma associated with poly(n, x), an error message is returned. The DerivativeRepresentation(S, x1,.., xn, root=val) and DerivativeRepresentation(S, [x1,.., xn], root=val) calling sequences assume that all polynomials depending on x1,.., xn share the common root val. Otherwise, an error is returned.

Examples

withOrthogonalSeries:

S1Createan,m,LaguerreLn,α,x,LaguerreLm,β,y

S1:=m=0∞n=0∞an,mLaguerreLn,α,xLaguerreLm,β,y

(1)

S2DerivativeRepresentationS1,x

S2:=m=0∞n=0∞an,man+1,mLaguerreLn,α+1,xLaguerreLm,β,y

(2)

DerivativeRepresentationS2,y;DerivativeRepresentationS1,x,y

m=0∞n=0∞an,man+1,man,m+1+an+1,m+1LaguerreLn,α+1,xLaguerreLm,β+1,y

m=0∞n=0∞an,mLaguerreLn,α,xLaguerreLm,β,y

(3)

Find the partial differential representation for Jacobi polynomials. In this case, sigma(x) = x^2-1.

S5Create1n+1,JacobiPn,1,2,x

S5:=n=0∞JacobiPn,1,2,xn+1

(4)

DerivativeRepresentationS5,x,root=1

n=0∞n2+5n+7JacobiPn,1,3,xn+3n+2n+1

(5)

DerivativeRepresentationS5,x,root=1

n=0∞32JacobiPn,2,2,xn+2n+1

(6)

DerivativeRepresentationS5,x,root=2

Error, (in OrthogonalSeries:-DerivativeRepresentation) -2 is not a root of x^2-1

See Also

JacobiP

LaguerreL

OrthogonalSeries

OrthogonalSeries[Create]

 


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