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kernel L
kernel A

Parameter Type Description
L ${\mathbb{Q}}[n,E]$ A difference operator
A ${\mathbb{Q}}(n)^{m,m}$ A matrix of fractions


kernel L returns a basis for $\mbox{Ker}~L \cap {\mathbb{Q}}(n)$, while kernel A returns a basis for $\mbox{Ker}(Y(n+1) = A Y(n)) \cap {\mathbb{Q}}(x)^m$.


The equation

L = (n+2)(n+4)y(n+2)-(2(n+1)(n+3)+1)y(n+1)+(n+1)^2 y(n) = 0

has the following rational solutions:
1 --> L := (n+2)*(n+4)*E^2-(2*(n+1)*(n+3)+1)*E+(n+1)^2;
2 --> K := kernel(L);
3 --> tex(K);

\begin{displaymath}[{{1} \over {n^{2}+3\,n+2}}]

Usage within MAPLE

In order not to conflict with the linalg[kernel] function in MAPLE, kernel is available under MAPLE under the names polynomialKernel and rationalKernel. So the above examples in MAPLE would be:
> L := (n+2)*(n+4)*E^2-(2*(n+1)*(n+3)+1)*E+(n+1)^2;
> rationalKernel(L,E,n);

\left[{{1} \over {n^{2}+3\,n+2}}\right]

See Also

polynomialSolution, rationalSolution

Manuel Bronstein 2002-09-04