Continuant (mathematics)

Definition

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 ) .

Properties

The continuant K n ( x 1 , x 2 , … , x n ) can be computed by taking the sum of all possible products of x₁,...,x_n, 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 that

and also

Generalizations

A generalized definition takes the continuant with respect to three sequences a, b and c, so that K(n) is a polynomial of a₁,...,a_n, b₁,...,b_n−1 and c₁,...,c_n−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 b_r and c_r enter into K only as a product b_rc_r there is no loss of generality in assuming that the b_r 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.