Casoratian

Casoratian[{y1, y2, ...}, n]
gives the Casoratian determinant for the sequences , , ... depending on n.

Casoratian[eqn, y, n]
gives the Casoratian determinant for the basis of the solutions of the linear difference equation eqn involving .

Casoratian[eqns, {y1, y2, ...}, n]
gives the Casoratian determinant for the system of linear difference equations eqns.

DetailsDetails

  • The Casoratian determinant is defined as: Det[Table[DiscreteShift[yi, {n, j}], {i, m}, {j, 0, m-1}]].
  • Linear independence of the functions , , ... is equivalent to the vanishing of the Casoratian.

ExamplesExamplesopen allclose all

Basic Examples (4)Basic Examples (4)

These sequences are linearly independent:

In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=

The following sequences are linearly dependent:

In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=

Here the sequences are linearly dependent only when :

In[1]:=
Click for copyable input
Out[1]=

The Casoratian for a linear equation:

In[1]:=
Click for copyable input
Out[1]=

Except for a constant, the result is the same as for the explicit solution:

In[2]:=
Click for copyable input
Out[2]=
In[3]:=
Click for copyable input
Out[3]=
New in 7
New to Mathematica? Find your learning path »
Have a question? Ask support »