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  ruled by the following commutation relations:

for .  Any other pair of indeterminates commute.

A q-difference algebra is an algebra of noncommutative polynomials in the indeterminates  ruled by the following commutation relations:

for .  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  (difference case) or  (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 , 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  can therefore be printed either  or .

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  (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](..).