Skew polynomials and linear operators

$\Sigma{}^{{\rm it}}$ provides several types for computing with univariate skew-polynomials or specific linear operators. The most general is UnivariateSkewPolynomial, for which you provide the Automorphism $\sigma$ and the $\sigma$-derivation $\delta$ as parameters. In addition, you also provide a usual polynomial type that is meant to be used as internal representation for the skew-polynomials. This lets you choose between dense or sparse representation by selecting the appropriate representation type. As for polynomials, naming the variable is not necessary since the name is used only for output. If you want to name the variable, pass a polynomial type with a variable name as reprensentation type. There are several ways to create the Automorphism $\sigma$: since Automorphism is of category Monoid, the constant 1 can be used for the identity map, while morphism is used for more general maps.

There are also types for common linear operators:

• LinearOrdinaryDifferentialOperator provides linear ordinary differential operators with respect to an arbitrary Derivation that is given as parameter. The derivation can be omitted if the coefficient type is a DifferentialRing.
• LinearOrdinaryDifferenceOperator provides linear ordinary generalized difference operators with respect to an arbitrary Automorpshism that is given as parameter.
• LinearOrdinaryRecurrence provides linear ordinary difference operators with coefficients in a polynomial ring $R[x]$ and with respect to the shift $x \to x + 1$. LinearOrdinaryRecurrenceQ provides the same operators but over the rational fractions $R(x)$.
• LinearOrdinaryQDifferenceOperator provides linear ordinary $q$-difference operators with coefficients in a polynomial ring $R[x]$ and with respect to the shift $x \to q x$ for a given $q \in R$, which is given as parameter to the type. LinearOrdinaryQDifferenceOperatorQ provides the same operators but over the rational fractions $R(x)$.

All of the above are implemented using a dense representation, and allow an optional Symbol as last parameter if you want to name the variable for output.

As for polynomials, when writing generic code for manipulating skew-polynomials or operators, use a type parameter with the appropriate category selected from Figure 2.