This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Casoratian

 Casoratian gives the Casoratian determinant for the sequences , , ... depending on n. Casoratiangives the Casoratian determinant for the basis of the solutions of the linear difference equation eqn involving . Casoratiangives the Casoratian determinant for the system of linear difference equations eqns.
• Linear independence of the functions , , ... is equivalent to the vanishing of the Casoratian.
These sequences are linearly independent:
The following sequences are linearly dependent:
Here the sequences are linearly dependent only when :
The Casoratian for a linear equation:
Except for a constant, the result is the same as for the explicit solution:
These sequences are linearly independent:
 Out[1]=
 Out[2]=

The following sequences are linearly dependent:
 Out[1]=
 Out[2]=

Here the sequences are linearly dependent only when :
 Out[1]=

The Casoratian for a linear equation:
 Out[1]=
Except for a constant, the result is the same as for the explicit solution:
 Out[2]=
 Out[3]=
 Scope   (9)
Polynomials:
Rational functions:
Exponentials and polynomial exponentials:
Trigonometric and polynomial trigonometric:
Hypergeometric term sequences:
Constant coefficient linear equation:
The Casoratian for a difference equation is usually simpler than for its solutions:
Polynomial coefficient linear equation:
The corresponding Casoratian from the general solution:
Factorial coefficient linear equation:
Periodic coefficients:
Periodic products:
 Applications   (2)
Variation of parameters for second-order inhomogeneous equations:
Verify that the components of the general solution for an OE are linearly independent:
Casoratian is equivalent to a determinant:
Casoratian detects linear dependence:
Wronskian performs linear dependence for functions of a continuous argument:
Functions of continuous arguments may be independent:
But sampling those functions may generate dependent sequences or aliasing:
Use Orthogonalize to generate a set of linearly independent sequences:
Express a sequence in the basis:
The last component is linearly dependent on the previous ones:
Use Reduce to express polynomials and rational sequences in terms of each other:
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