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Casoratian

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The Casoratian of sequences x_n^((1)), x_n^((2)), ..., x_n^((k)) is defined by the k×k determinant

C(x_n^((1)),x_n^((2)),...,x_n^((k))) =|x_n^((1)) x_n^((2)) ... x_n^((k)); x_(n+1)^((1)) x_(n+1)^((2)) ... x_(n+1)^((k)); | | ... ...; x_(n+k-1)^((1)) x_(n+k-1)^((2)) ... x_(n+k-1)^((k))|.

The Casoratian is implemented in the Wolfram Language as Casoratian[{y1, y2, ...}, n].

The solutions x_n^((1)), x_n^((2)), ..., x_n^((k)) of the linear difference equation

x_(n+k)+b_n^((k-1))x_(n+(k-1))+...+b_n^((1))x_(n+1)+b_n^((0))x_n=0

for n=0, 1, ..., are linearly independent sequences iff their Casoratian is nonzero for n=0 (Zwillinger 1995).

See also

Linearly Dependent Sequences

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References

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 229, 1995.

Referenced on Wolfram|Alpha

Casoratian

Cite this as:

Weisstein, Eric W. "Casoratian." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Casoratian.html

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