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Continuant

The continuant K_n(x_1,...,x_n) is the determinant of the tridiagonal matrix

K_n(x_1,...,x_n)=det[x_1 1 0; -1 x_2 ... ; ... ... 1; 0 -1 x_n].
(1)

It is characterized by the recurrence equation

K_0=1
(2)
K_1(x_1)=x_1
(3)
K_n(x_1,...,x_n)=x_nK_(n-1)(x_1,...,x_(n-1))+K_(n-2)(x_1,...,x_(n-2)).
(4)

Continuants give the numerators and denominators of the convergents of a continued fraction. A version with the signs of the two off-diagonals equal gives the corresponding recurrence equation for the determinant with a minus sign and is used for general tridiagonal matrices.

See also

Continued Fraction, Convergent, Tridiagonal Matrix

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References

Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.

Cite this as:

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

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