Continuant

2020 Mathematics Subject Classification: Primary: 11Y65 [MSN][ZBL]

An algebraic function of a sequence of variables which has applications to continued fractions and as the determinant of a tridiagonal matrix.

The $n$-th continuant, $K(n)$, of a sequence $\mathbf{a} = a_1,\ldots,a_n,\ldots$ is defined recursively by $$ K(0) = 1 ; $$ $$ K(1) = a_1 ; $$ $$ K(n) = a_n K(n-1) + K(n-2) \ . $$ It may also be obtained by taking the sum of all possible products of the $a_1,\ldots,a_n$ in which any pairs of consecutive terms are deleted.

An extended definition takes the continuant with respect to three sequences $\mathbf a$, $\mathbf b$, $\mathbf c$, so that $K(n)$ is a function of $a_1,\ldots,a_n$, $b_1,\ldots,b_{n-1}$, $c_1,\ldots,c_{n-1}$. 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 the $b_r$ and $c_r$ enter into $K$ only as the product $b_r c_r$ there is no loss of generality in assuming that the $b_r$ are all equal to 1.

The simple continuant gives the value of a continued fraction of the form $[a_0;a_1,a_2,\ldots]$. The $n$-th convergent is $$ \frac{K(n+1,(a_0,\ldots,a_n))}{K(n,(a_1,\ldots,a_n))} \ . $$

The extended continuant is the determinant of the tridiagonal matrix $$ \begin{pmatrix} a_1 & b_1 & 0 & 0 & \ldots & 0 & 0 \\ c_1 & a_2 & b_2 & 0 & \ldots & 0 & 0 \\ 0 & c_2 & a_3 & b_3 & \ldots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & c_{n-1} & a_n \end{pmatrix} \ . $$

References

• Thomas Muir. A treatise on the theory of determinants. (Dover Publications, 1960 [1933]), pp. 516-525.