PochhammerBasis - Maple Help

PochhammerBasis

Pochhammer polynomials based at a point

 Calling Sequence PochhammerBasis(k, a, x)

Parameters

 k - algebraic expression; the index a - algebraic expression; the starting point x - algebraic expression; the argument

Description

 • $\mathrm{PochhammerBasis}\left(k,a,x\right)=\prod _{j=0}^{k-1}\left(x+a+j\right)$ defines the $k$th Pochhammer polynomial of degree $n$. The degree of the $k$th 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() , which uses Horner's method to evaluate the polynomial p.

Examples

 > $a≔'a'$
 ${a}{≔}{a}$ (1)
 > $p≔3\mathrm{PochhammerBasis}\left(0,a,x\right)+5\mathrm{PochhammerBasis}\left(2,a,x\right)+7\mathrm{PochhammerBasis}\left(3,a,x\right)$
 ${p}{≔}{3}{}{\mathrm{PochhammerBasis}}{}\left({0}{,}{a}{,}{x}\right){+}{5}{}{\mathrm{PochhammerBasis}}{}\left({2}{,}{a}{,}{x}\right){+}{7}{}{\mathrm{PochhammerBasis}}{}\left({3}{,}{a}{,}{x}\right)$ (2)

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

 > $P≔\mathrm{convert}\left(p,\mathrm{MatrixPolynomialObject},x\right)$
 ${P}{≔}{\mathrm{Record}}{}\left({\mathrm{Value}}{=}{{\mathrm{Default}}}_{{\mathrm{value}}}{,}{\mathrm{Variable}}{=}{x}{,}{\mathrm{Degree}}{=}{3}{,}{\mathrm{Coefficient}}{=}{\mathrm{coe}}{,}{\mathrm{Dimension}}{=}\left[{1}{,}{1}\right]{,}{\mathrm{Basis}}{=}{\mathrm{PochhammerBasis}}{,}{\mathrm{BasisParameters}}{=}\left[{a}\right]{,}{\mathrm{IsMonic}}{=}{\mathrm{mon}}{,}{\mathrm{OutputOptions}}{=}\left[{\mathrm{shape}}{=}\left[\right]{,}{\mathrm{storage}}{=}{\mathrm{rectangular}}{,}{\mathrm{order}}{=}{\mathrm{Fortran_order}}{,}{\mathrm{fill}}{=}{0}{,}{\mathrm{attributes}}{=}\left[\right]\right]\right)$ (3)
 > $P:-\mathrm{Degree}\left(\right)$
 ${3}$ (4)

Note that the result returned by $\mathrm{convert}\left(...,\mathrm{MatrixPolynomialObject}\right)$ represents a matrix polynomial; hence these results are 1 by 1 matrices.

 > $P:-\mathrm{Value}\left(x\right)\left[1,1\right]$
 $\left({x}{+}{a}\right){}\left({x}{+}{a}{+}{1}\right){}\left({7}{}{x}{+}{7}{}{a}{+}{19}\right){+}{3}$ (5)
 > $a≔'a'$
 ${a}{≔}{a}$ (6)
 > $p≔\mathrm{add}\left(b\left[k\right]\mathrm{PochhammerBasis}\left(k,a,x\right),k=0..3\right)$
 ${p}{≔}{{b}}_{{0}}{}{\mathrm{PochhammerBasis}}{}\left({0}{,}{a}{,}{x}\right){+}{{b}}_{{1}}{}{\mathrm{PochhammerBasis}}{}\left({1}{,}{a}{,}{x}\right){+}{{b}}_{{2}}{}{\mathrm{PochhammerBasis}}{}\left({2}{,}{a}{,}{x}\right){+}{{b}}_{{3}}{}{\mathrm{PochhammerBasis}}{}\left({3}{,}{a}{,}{x}\right)$ (7)
 > $P≔\mathrm{convert}\left(p,\mathrm{MatrixPolynomialObject},x\right)$
 ${P}{≔}{\mathrm{Record}}{}\left({\mathrm{Value}}{=}{{\mathrm{Default}}}_{{\mathrm{value}}}{,}{\mathrm{Variable}}{=}{x}{,}{\mathrm{Degree}}{=}{3}{,}{\mathrm{Coefficient}}{=}{\mathrm{coe}}{,}{\mathrm{Dimension}}{=}\left[{1}{,}{1}\right]{,}{\mathrm{Basis}}{=}{\mathrm{PochhammerBasis}}{,}{\mathrm{BasisParameters}}{=}\left[{a}\right]{,}{\mathrm{IsMonic}}{=}{\mathrm{mon}}{,}{\mathrm{OutputOptions}}{=}\left[{\mathrm{shape}}{=}\left[\right]{,}{\mathrm{storage}}{=}{\mathrm{rectangular}}{,}{\mathrm{order}}{=}{\mathrm{Fortran_order}}{,}{\mathrm{fill}}{=}{0}{,}{\mathrm{attributes}}{=}\left[\right]\right]\right)$ (8)
 > $\mathrm{collect}\left(P:-\mathrm{Value}\left(t\right)\left[1,1\right],\left[\mathrm{seq}\left(b\left[k\right],k=0..3\right)\right],\mathrm{factor}\right)$
 ${{b}}_{{0}}{+}\left({t}{+}{a}\right){}{{b}}_{{1}}{+}\left({t}{+}{a}\right){}\left({t}{+}{a}{+}{1}\right){}{{b}}_{{2}}{+}\left({t}{+}{a}\right){}\left({t}{+}{a}{+}{1}\right){}\left({t}{+}{a}{+}{2}\right){}{{b}}_{{3}}$ (9)