adjoint

Usage

adjoint L

Parameter	Type	Description
L	${\mathbb{Q}}[n,E]$	A difference operator

Returns

Returns $L^\ast E^{m+d}$ where

$$L^\ast = \sum_{i=d}^m E^{-i} \cdot a_i$$

is the adjoint of

$$L = \sum_{i=d}^m a_i E^i$$

and $\cdot$ is the product in ${\mathbb{Q}}[n,E]$.

Remarks

Because adjoint(L) returns $L^\ast E^{m+d}$ rather than the adjoint $L^\ast$, it follows that adjoint(adjoint(L)) returns $E^{-(m+d)} L E^{m+d}$ rather than L.

Example

It can happen that $L^\ast$ has a nontrivial rational kernel, while $L$ has a trivial one:

1 --> L := (n+2)*E^2 + (n^2+3*n+2)*E - n^2 - 3*n - 2;
2 --> tex(kernel(L));

$$[~]$$

3 --> La := adjoint(L);
4 --> tex(La);

$$\left(-n^{2}-3\,n-2\right)\,E^{2}+\left(n^{2}+n\right)\,E+n$$

5 --> tex(kernel(La));

$$[ {{1} \over {n}} ]$$

We also verify that $L^{\ast\ast} = E^{-2} L E^{2}$:

6 --> tex(L * E^(degree(L)) - E^(degree(L)) * adjoint(La));

$$0$$