Casoratian

Casoratian[{y1,y2,},n]

gives the Casoratian determinant for the sequences y1, y2, depending on n.

Casoratian[eqn,y,n]

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

Casoratian[eqns,{y1,y2,},n]

gives the Casoratian determinant for the system of linear difference equations eqns.

Details and Options

  • The Casoratian determinant is defined as: Det[Table[DiscreteShift[yi,{n,j}],{i,m},{j,0,m-1}]].
  • If the sequences y1,y2, are linearly dependent, the Casoratian vanishes everywhere.

Examples

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Basic Examples  (4)

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:

Scope  (9)

Sequences  (5)

Polynomials:

Rational functions:

Exponentials and polynomial exponentials:

Trigonometric and polynomial trigonometric:

Hypergeometric term sequences:

Difference Equations  (4)

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 OΔE are linearly independent:

Properties & Relations  (6)

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:

Introduced in 2008
 (7.0)