In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.
The n-th continuant K n ( x 1 , x 2 , … , x n ) is defined recursively by
K 0 = 1 ; K 1 ( x 1 ) = x 1 ; K n ( x 1 , x 2 , … , x n ) = x n K n − 1 ( x 1 , x 2 , … , x n − 1 ) + K n − 2 ( x 1 , x 2 , … , x n − 2 ) . The continuant K n ( x 1 , x 2 , … , x n ) 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, K 5 ( x 1 , x 2 , x 3 , x 4 , x 5 ) = x 1 x 2 x 3 x 4 x 5 + x 3 x 4 x 5 + x 1 x 4 x 5 + x 1 x 2 x 5 + x 1 x 2 x 3 + x 1 + x 3 + x 5 . It follows that continuants are invariant with respect to reversing the order of indeterminates:
K n ( x 1 , … , x n ) = K n ( x n , … , x 1 ) . The continuant can be computed as the determinant of a tridiagonal matrix: K n ( 1 , … , 1 ) = F n + 1 , the (n+1)-st Fibonacci number. K n ( x 1 , … , x n ) K n − 1 ( x 2 , … , x n ) = x 1 + K n − 2 ( x 3 , … , x n ) K n − 1 ( x 2 , … , x n ) . Ratios of continuants represent (convergents to) continued fractions as follows:The following matrix identity holds:For determinants, it implies thatand alsoA 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
K 0 = 1 ; K 1 = a 1 ; K n = a n K n − 1 − b n − 1 c n − 1 K n − 2 . 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
( a 1 b 1 0 … 0 0 c 1 a 2 b 2 … 0 0 0 c 2 a 3 … 0 0 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 … a n − 1 b n − 1 0 0 0 … c n − 1 a n ) . In Muir's book the generalized continuant is simply called continuant.