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OrthogonalSeries[Create] - create a series

Calling Sequence

Create(expr, P1,..., Pn)

Create({expr, k=a..b}, P1)

Create(L, P1,..., Pn)

Create({L, expr}, P1,..., Pn)

Create({L, expr, k=a..b}, P1)

Parameters

expr

-

algebraic expression

P1, ..., Pn

-

orthogonal polynomials

k

-

name

a, b

-

integers (or infinity for b)

L

-

list of equalities

Description

• 

The Create(arguments) function creates an orthogonal series. The series is expanded in terms of the polynomials P1,..,Pn, which must have distinct indices and variables, and be in the OrthogonalSeries database.

• 

The Create(expr, P1,..., Pn) calling sequence creates an infinite series expanded in terms of P1,..,Pn with the coefficient expr depending on the indices of the Pis. The indices of each Pi run from 0 to infinity. For univariate series, the range [a,b] can be specified by using the Create({expr, k=a..b}, P1) calling sequence where k is the index of P1.

• 

The Create(L, P1,..., Pn) calling sequence creates a finite series, namely, a polynomial expanded in the basis P1,...,Pn. The elements of L must have the form (k1,.., kn) = val. That is, the coefficient of index k1,..,kn of the created series is equal to val. In the case of a univariate series, the input L=v0=0,v1=1,..,vN=N can be abbreviated as L=[v0,v1,...,vN].

• 

Series with both a finite (particular) part and infinite (general) part can be created by using the Create({L, expr}, P1,..., Pn) calling sequence. For a range different from 0,, use the Create({L, expr, k=a..b}, P1) calling sequence.

Examples

withOrthogonalSeries:

Create1nn!,ChebyshevTn,x

n=0∞1nChebyshevTn,xn!

(1)

Create1=2,3=4,4=α,LaguerreLn,β,x

2LaguerreL1,β,x+4LaguerreL3,β,x+αLaguerreL4,β,x

(2)

Create1,2,3,4,LaguerreLn,β,x

LaguerreL0,β,x+2LaguerreL1,β,x+3LaguerreL2,β,x+4LaguerreL3,β,x

(3)

Create1,2,3,4,1m2+1,m=3..100,Discrete_qHermite1m,q,y

Discrete_qHermite10,q,y+2Discrete_qHermite11,q,y+3Discrete_qHermite12,q,y+4Discrete_qHermite13,q,y+m=3100Discrete_qHermite1m,q,ym2+1

(4)

Create1,1=2,n2+m2+1,HermiteHn,x,HermiteHm,z

2HermiteH1,xHermiteH1,z+m=0∞n=0∞m2+n2+1HermiteHn,xHermiteHm,z

(5)

Create1,12,13,un,n=2..∞,Kravchoukn,p,N,x

Kravchouk0,p,N,x+12Kravchouk1,p,N,x+13Kravchouk2,p,N,x+n=2∞unKravchoukn,p,N,x

(6)

Create1,2=3,2,1=a2,100,100=13,ChebyshevUn,x,LaguerreLm,1,y

3ChebyshevU1,xLaguerreL2,1,y+a2ChebyshevU2,xLaguerreL1,1,y+13ChebyshevU100,xLaguerreL100,1,y

(7)

See Also

ChebyshevT, ChebyshevU, LaguerreL, OrthogonalSeries


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