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OreTools

 SetOreRing
 define an Ore polynomial ring

 Calling Sequence SetOreRing(var, 'shift') SetOreRing($\left[\mathrm{var},q\right]$, 'qshift') SetOreRing(var, 'differential') SetOreRing(var, algebra_name, 'sigma' = proc1, 'sigma_inverse' = proc2, 'delta' = proc3, 'theta1' = expr)

Parameters

 var - name; variable q - name; qshift parameter algebra_name - name; algebra to be defined proc1, proc2, proc3 - procedures; define algebra expr - Maple expression

Description

 • The SetOreRing(var, 'shift') calling sequence defines a shift algebra.
 • The SetOreRing([var, q], 'qshift') calling sequence defines a qshift algebra.
 • The SetOreRing(var, 'differential') calling sequence defines a differential algebra.
 • The shift, qshift, and differential algebras are pre-defined. You can use the SetOreRing command to define other Ore polynomial rings. You must specify procedures to compute sigma, sigma_inverse, and delta, and an expression to define theta(1).
 For a brief review of pseudo-linear algebra (also known as Ore algebra), see OreAlgebra.

Examples

 > $\mathrm{with}\left(\mathrm{OreTools}\right):$

Define the shift algebra.

 > $A≔\mathrm{SetOreRing}\left(n,'\mathrm{shift}'\right)$
 ${A}{:=}{\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{shift}}\right)$ (1)

Define the difference algebra.

 > B := SetOreRing(n, 'difference',   'sigma' = proc(p, x) eval(p, x=x+1) end,   'sigma_inverse' = proc(p, x) eval(p, x=x-1) end,   'delta' = proc(p, x) eval(p, x=x+1) - p end,   'theta1' = 0);
 ${B}{:=}{\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{difference}}\right)$ (2)