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PochhammerBasis

Pochhammer polynomials based at a point

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

PochhammerBasis(k, a, x)

Parameters

k

-

algebraic expression; the index

a

-

algebraic expression; the starting point

x

-

algebraic expression; the argument

Description

• 

PochhammerBasisk,a,x=j=0k1x+a+j defines the kth Pochhammer polynomial of degree n. The degree of the kth Pochhammer polynomial is k.

• 

At present, this can only be evaluated in Maple by prior use of the object-oriented representation obtained by P:=convert(p,MatrixPolynomialObject,x) and subsequent call to P:-Value(<x-value>) , which uses Horner's method to evaluate the polynomial p.

Examples

aa

aa

(1)

p3PochhammerBasis0&comma;a&comma;x+5PochhammerBasis2&comma;a&comma;x+7PochhammerBasis3&comma;a&comma;x

p3PochhammerBasis0&comma;a&comma;x+5PochhammerBasis2&comma;a&comma;x+7PochhammerBasis3&comma;a&comma;x

(2)

This is in effect a NewtonBasis polynomial expression on the nodes a, a+1, and a+2.

Pconvertp&comma;MatrixPolynomialObject&comma;x

PRecordValue=Defaultvalue&comma;Variable=x&comma;Degree=3&comma;Coefficient=coe&comma;Dimension=1&comma;1&comma;Basis=PochhammerBasis&comma;BasisParameters=a&comma;IsMonic=mon&comma;OutputOptions=shape=&comma;storage=rectangular&comma;order=Fortran_order&comma;fill=0&comma;attributes=

(3)

P:-Degree

3

(4)

Note that the result returned by convert...,MatrixPolynomialObject represents a matrix polynomial; hence these results are 1 by 1 matrices.

P:-Valuex1,1

x+ax+a+17x+7a+19+3

(5)

aa

aa

(6)

paddbkPochhammerBasisk&comma;a&comma;x&comma;k=0..3

pb0PochhammerBasis0&comma;a&comma;x+b1PochhammerBasis1&comma;a&comma;x+b2PochhammerBasis2&comma;a&comma;x+b3PochhammerBasis3&comma;a&comma;x

(7)

Pconvertp&comma;MatrixPolynomialObject&comma;x

PRecordValue=Defaultvalue&comma;Variable=x&comma;Degree=3&comma;Coefficient=coe&comma;Dimension=1&comma;1&comma;Basis=PochhammerBasis&comma;BasisParameters=a&comma;IsMonic=mon&comma;OutputOptions=shape=&comma;storage=rectangular&comma;order=Fortran_order&comma;fill=0&comma;attributes=

(8)

collectP:-Valuet1,1&comma;seqbk&comma;k=0..3&comma;factor

b0+t+ab1+t+at+a+1b2+t+at+a+1t+a+2b3

(9)

See Also

BernsteinBasis

convert/MatrixPolynomialObject

LagrangeBasis

LinearAlgebra[CompanionMatrix]

NewtonBasis

OrthogonalSeries

type/MatrixPolynomialObject