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

open allclose all

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:

Wolfram Research (2008), Casoratian, Wolfram Language function, https://reference.wolfram.com/language/ref/Casoratian.html.

Text

Wolfram Research (2008), Casoratian, Wolfram Language function, https://reference.wolfram.com/language/ref/Casoratian.html.

CMS

Wolfram Language. 2008. "Casoratian." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Casoratian.html.

APA

Wolfram Language. (2008). Casoratian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Casoratian.html

BibTeX

@misc{reference.wolfram_2024_casoratian, author="Wolfram Research", title="{Casoratian}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Casoratian.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_casoratian, organization={Wolfram Research}, title={Casoratian}, year={2008}, url={https://reference.wolfram.com/language/ref/Casoratian.html}, note=[Accessed: 21-December-2024 ]}