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In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.
Definition
The n-th continuant is defined recursively by
Properties
- The continuant can be computed by taking the sum of all possible products of x1,...,xn, in which any number of disjoint pairs of consecutive terms are deleted (Euler's rule). For example,
- It follows that continuants are invariant with respect to reversing the order of indeterminates:
- The continuant can be computed as the determinant of a tridiagonal matrix:
- , the (n+1)-st Fibonacci number.
- Ratios of continuants represent (convergents to) continued fractions as follows:
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