The shift_algebra(l_1, ..., l_n) and qshift_algebra(lq_1, ..., lq_n) functions each declare an Ore algebra and return a table that is used by other functions of the Ore_algebra package.

A difference algebra is an algebra of noncommutative polynomials in the indeterminates $n_1,...,n_p,S_1,...,S_p$ ruled by the following commutation relations:

for $i=1,...,p$.  Any other pair of indeterminates commute.

A q-difference algebra is an algebra of noncommutative polynomials in the indeterminates $q,n_1,...,n_p,S_1,...,S_p$ ruled by the following commutation relations:

for $i=1,...,p$.  q is a constant and any other pair of indeterminates commute.

Note: Difference and q-difference algebras are special cases of Ore algebras. For more information, see Ore_algebra.

The name n_i can be unassigned.

The name S_i can be unassigned.  It is used to denote the difference or q-difference indeterminate S_i associated to the base indeterminate n_i, that is, the operator of shift or q-shift with respect to n_i.

When the list l_i is of the form $\left[S_i\,n_i\right]$ (difference case) or $\left[S_i\,n_i\,q_i\right]$ (q-difference case), the names n_i and S_i can be unassigned.  Both indeterminates commute with any other indeterminate of the algebra.

When the list l_i is of the form $\left[\mathrm{comm}\,a_i\right]$, the name a_i can be unassigned.  It denotes a parameter that commutes with any other indeterminate of the algebra.

Though difference and q-difference algebras are noncommutative algebras, their elements are represented with the standard commutative Maple product.  Every Ore_algebra function dealing with elements of a difference of q-difference algebra uses its normal form where all S_i appear on the right of the corresponding n_i.  A monomial $n^a{\mathrm{Sn}}^b$ can therefore be printed either $n^a{\mathrm{Sn}}^b$ or $n^a{\mathrm{Sn}}^b$.

The sum in difference or q-difference algebras is performed by simply using the Maple `+`, while the product is performed by the Ore_algebra function skew_product (see examples below).

It is also possible to declare a difference or a q-difference algebra by using Ore_algebra[skew_algebra].  Moreover, the algebras declared by Ore_algebra[shift_algebra] and Ore_algebra[qshift_algebra] are difference and q-difference algebras based on shift and q-shift operators S_i, but it is also possible to declare algebras based on finite difference and q-difference operators ${\mathrm{D}}_i=S_i+1$ (see Ore_algebra[skew_algebra], predefined types delta and qdelta).

Options are available to control the ground ring of the algebra and the action of the operators on Maple objects.  See Ore_algebra[declaration_options].

These function are part of the Ore_algebra package, and so can be used in the form shift_algebra(..) and qshift_algebra(..) only after performing the command with(Ore_algebra) or with(Ore_algebra,<function>).  The functions can always be accessed in the long form Ore_algebra[shift_algebra](..) and Ore_algebra[qshift_algebra](..).

with(Ore_algebra):

A:=shift_algebra([Sn,n],[Sm,m]);

$A≔\mathrm{Ore\_algebra}$

skew_product(Sn,n,A),skew_product(Sm,m,A);

$\left(n+1\right)\mathrm{Sn},\left(m+1\right)\mathrm{Sm}$

skew_product(Sn*Sm,n*m,A);

$\left(nm+m+n+1\right)\mathrm{Sn}\mathrm{Sm}$

skew_product(Sn,n^10,A);

$\left(n^{10}+10n^9+45n^8+120n^7+210n^6+252n^5+210n^4+120n^3+45n^2+10n+1\right)\mathrm{Sn}$

skew_algebra(comm={a,b,c,d,e,f,g,h},shift=[Sn,n]);

$\mathrm{Ore\_algebra}$

shift_algebra([comm,a],[comm,b],[comm,c],[comm,d],[comm,e],[comm,f],
    [comm,g],[comm,h],[Sn,n]);

$\mathrm{Ore\_algebra}$

evalb((5)=(6));

$\mathrm{true}$

shift_algebra([Sn,n]);

$\mathrm{Ore\_algebra}$

skew_algebra(delta=[Dn,n]);

$\mathrm{Ore\_algebra}$

A:=qshift_algebra([Sn,q^n],[Sm,q^m]);

$A≔\mathrm{Ore\_algebra}$

skew_product(Sn,q^n,A),skew_product(Sm,q^m,A);

$q{q}^n\mathrm{Sn},q{q}^m\mathrm{Sm}$

skew_product(Sn*Sm,q^n*q^m,A);

$q^2{q}^n{q}^m\mathrm{Sn}\mathrm{Sm}$

skew_product(Sn,(q^n)^10,A);

$q^{10}{\left(q^n\right)}^{10}\mathrm{Sn}$

A:=qshift_algebra([Sn,q^n],[Sm,p^m]);

$A≔\mathrm{Ore\_algebra}$

skew_product(Sn,q^n,A),skew_product(Sm,p^m,A);

$q{q}^n\mathrm{Sn},p{p}^m\mathrm{Sm}$

skew_product(Sn*Sm,q^n*p^m,A);

$q{q}^{n}p{p}^m\mathrm{Sn}\mathrm{Sm}$

skew_product(Sn,(q^n)^10,A);

$q^{10}{\left(q^n\right)}^{10}\mathrm{Sn}$