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Continuant (mathematics)

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In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in continued fractions.

Definition

The n-th continuant K n ( x 1 , x 2 , … , x n ) {\displaystyle K_{n}(x_{1},\;x_{2},\;\ldots ,\;x_{n})} is defined recursively by

K 0 = 1 ; {\displaystyle K_{0}=1;\,}
K 1 ( x 1 ) = x 1 ; {\displaystyle 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 ) . {\displaystyle K_{n}(x_{1},\;x_{2},\;\ldots ,\;x_{n})=x_{n}K_{n-1}(x_{1},\;x_{2},\;\ldots ,\;x_{n-1})+K_{n-2}(x_{1},\;x_{2},\;\ldots ,\;x_{n-2}).\,}

Properties

  • The continuant K n ( x 1 , x 2 , … , x n ) {\displaystyle K_{n}(x_{1},\;x_{2},\;\ldots ,\;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 . {\displaystyle 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 ) . {\displaystyle K_{n}(x_{1},\;\ldots ,\;x_{n})=K_{n}(x_{n},\;\ldots ,\;x_{1}).}
  • The continuant can be computed as the determinant of a tridiagonal matrix:
    K n ( x 1 , x 2 , … , x n ) = det ( x 1 1 0 ⋯ 0 − 1 x 2 1 ⋱ ⋮ 0 − 1 ⋱ ⋱ 0 ⋮ ⋱ ⋱ ⋱ 1 0 ⋯ 0 − 1 x n ) . {\displaystyle K_{n}(x_{1},\;x_{2},\;\ldots ,\;x_{n})=\det {\begin{pmatrix}x_{1}&1&0&\cdots &0\\-1&x_{2}&1&\ddots &\vdots \\0&-1&\ddots &\ddots &0\\\vdots &\ddots &\ddots &\ddots &1\\0&\cdots &0&-1&x_{n}\end{pmatrix}}.}
  • K n ( 1 , … , 1 ) = F n + 1 {\displaystyle K_{n}(1,\;\ldots ,\;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 ) . {\displaystyle {\frac {K_{n}(x_{1},\;\ldots ,\;x_{n})}{K_{n-1}(x_{2},\;\ldots ,\;x_{n})}}=x_{1}+{\frac {K_{n-2}(x_{3},\;\ldots ,\;x_{n})}{K_{n-1}(x_{2},\;\ldots ,\;x_{n})}}.}
  • Ratios of continuants represent (convergents to) continued fractions as follows:
    K n ( x 1 , … , x n ) K n − 1 ( x 2 , … , x n ) = [ x 1 ; x 2 , … , x n ] = x 1 + 1 x 2 + 1 x 3 + … . {\displaystyle {\frac {K_{n}(x_{1},\;\ldots ,x_{n})}{K_{n-1}(x_{2},\;\ldots ,\;x_{n})}}==x_{1}+{\frac {1}{\displaystyle {x_{2}+{\frac {1}{x_{3}+\ldots }}}}}.}
  • The following matrix identity holds:
    ( K n ( x 1 , … , x n ) K n − 1 ( x 1 , … , x n − 1 ) K n − 1 ( x 2 , … , x n ) K n − 2 ( x 2 , … , x n − 1 ) ) = ( x 1 1 1 0 ) × … × ( x n 1 1 0 ) {\displaystyle {\begin{pmatrix}K_{n}(x_{1},\;\ldots ,\;x_{n})&K_{n-1}(x_{1},\;\ldots ,\;x_{n-1})\\K_{n-1}(x_{2},\;\ldots ,\;x_{n})&K_{n-2}(x_{2},\;\ldots ,\;x_{n-1})\end{pmatrix}}={\begin{pmatrix}x_{1}&1\\1&0\end{pmatrix}}\times \ldots \times {\begin{pmatrix}x_{n}&1\\1&0\end{pmatrix}}} .
    • For determinants, it implies that
      K n ( x 1 , … , x n ) ⋅ K n − 2 ( x 2 , … , x n − 1 ) − K n − 1 ( x 1 , … , x n − 1 ) ⋅ K n − 1 ( x 2 , … , x n ) = ( − 1 ) n . {\displaystyle K_{n}(x_{1},\;\ldots ,\;x_{n})\cdot K_{n-2}(x_{2},\;\ldots ,\;x_{n-1})-K_{n-1}(x_{1},\;\ldots ,\;x_{n-1})\cdot K_{n-1}(x_{2},\;\ldots ,\;x_{n})=(-1)^{n}.}
    • and also
      K n − 1 ( x 2 , … , x n ) ⋅ K n + 2 ( x 1 , … , x n + 2 ) − K n ( x 1 , … , x n ) ⋅ K n + 1 ( x 2 , … , x n + 2 ) = ( − 1 ) n + 1 x n + 2 . {\displaystyle K_{n-1}(x_{2},\;\ldots ,\;x_{n})\cdot K_{n+2}(x_{1},\;\ldots ,\;x_{n+2})-K_{n}(x_{1},\;\ldots ,\;x_{n})\cdot K_{n+1}(x_{2},\;\ldots ,\;x_{n+2})=(-1)^{n+1}x_{n+2}.}

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,...,an, b1,...,bn−1 and c1,...,cn−1. In this case the recurrence relation becomes

K 0 = 1 ; {\displaystyle K_{0}=1;\,}
K 1 = a 1 ; {\displaystyle K_{1}=a_{1};\,}
K n = a n K n − 1 − b n − 1 c n − 1 K n − 2 . {\displaystyle 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 generalized 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 ) . {\displaystyle {\begin{pmatrix}a_{1}&b_{1}&0&\ldots &0&0\\c_{1}&a_{2}&b_{2}&\ldots &0&0\\0&c_{2}&a_{3}&\ldots &0&0\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\ldots &a_{n-1}&b_{n-1}\\0&0&0&\ldots &c_{n-1}&a_{n}\end{pmatrix}}.}

In Muir's book the generalized continuant is simply called continuant.

References

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