shift returns the expression equivalent to $\mathrm{subs}\left(\mathrm{var}=\mathrm{var}+i\,e\right)$, where i is assumed to be 1 in the two argument case.

This procedure knows about various Maple constructs, like diff, where the variable must not be simply substituted.  Currently shift knows about diff, Diff, int, Int, sum, Sum, product, and Product.

Additional constructs can be added by the user.  If the procedure `LREtools/shift/f` is defined then the function call LREtools[shift](f(n,x), n, 3) will invoke `LREtools/shift/f`(f(n,x), n, 3) to compute the shift.

delta(e, var) is defined to be shift(e, var, 1)-e and delta(e, var, n) is defined to be delta(shift(e, var, 1)-e, var, n-1).  Thus differencing is also user-extensible by providing the extension for shift.

$\mathrm{with}\left(\mathrm{LREtools}\right)\:$

$\mathrm{shift}\left(x^5\,x\right)$

${\left(x+1\right)}^5$

$\mathrm{shift}\left(x^5\,x\,3\right)$

${\left(x+3\right)}^5$

$\mathrm{shift}\left(f\left(x\right)\,x\,-2\right)$

$f\left(x-2\right)$

$\mathrm{shift}\left(\mathrm{diff}\left(f\left(x\right)\,x\right)\,x\right)$

$\mathrm{D}\left(f\right)\left(x+1\right)$

$\mathrm{\delta }\left(\sin \left(n\right)\,n\right)$

$\sin \left(n+1\right)-\sin \left(n\right)$

$\mathrm{\delta }\left(\cos \left(n\right)\,n\,2\right)$

$\cos \left(n+2\right)-2\cos \left(n+1\right)+\cos \left(n\right)$

$\mathrm{\delta }\left(\mathrm{Product}\left(g\left(x\right)\,x\right)\,x\right)$

$\left(\prod _{x}g\left(x+1\right)\right)-\left(\prod _{x}g\left(x\right)\right)$