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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

a&apos;a&apos;

aa

(1)

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

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

(2)

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

Pconvertp&comma;MatrixPolynomialObject&comma;x

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

(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&comma;1

x&plus;ax&plus;a&plus;17x&plus;7a&plus;19&plus;3

(5)

a&apos;a&apos;

aa

(6)

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

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

(7)

Pconvertp&comma;MatrixPolynomialObject&comma;x

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

(8)

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

b0&plus;t&plus;ab1&plus;t&plus;at&plus;a&plus;1b2&plus;t&plus;at&plus;a&plus;1t&plus;a&plus;2b3

(9)

See Also

BernsteinBasis

convert/MatrixPolynomialObject

LagrangeBasis

LinearAlgebra[CompanionMatrix]

NewtonBasis

OrthogonalSeries

type/MatrixPolynomialObject