The n-th continuant is defined recursively by
- 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 reverting the order of indeterminates:
- , the (n+1)-st Fibonacci number.
- Ratios of continuants represent (convergents to) continued fractions as follows:
- The following matrix identity holds:
- For determinants, it implies that
- and also
An generalized definition takes the continuant with respect to three sequences a, b and c, so that K(n) is a polynomial of a1,...,an, b1,...,bn−1 and c1,...,cn−1. In this case the recurrence relation becomes
Since br and cr enter into K only as a product brcr there is no loss of generality in assuming that the br are all equal to 1.
The extended continuant is precisely the determinant of the tridiagonal matrix
In Muir's book the generalized continuant is simply called continuant.
- Thomas Muir (1960). A treatise on the theory of determinants. Dover Publications. pp. 516–525.
- Cusick, Thomas W.; Flahive, Mary E. (1989). The Markoff and Lagrange Spectra. Mathematical Surveys and Monographs 30. Providence, RI: American Mathematical Society. p. 89. ISBN 0-8218-1531-8. Zbl 0685.10023.
- George Chrystal (1999). Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1. American Mathematical Society. p. 500. ISBN 0-8218-1649-7.