Continuant (mathematics)




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




Contents






  • 1 Definition


  • 2 Properties


  • 3 Generalizations


  • 4 References





Definition


The n-th continuant Kn(x1,x2,…,xn){displaystyle K_{n}(x_{1},;x_{2},;ldots ,;x_{n})}K_n(x_1,;x_2,;ldots,;x_n) is defined recursively by



K0=1;{displaystyle K_{0}=1;,} K_0 = 1 ; ,

K1(x1)=x1;{displaystyle K_{1}(x_{1})=x_{1};,} K_1(x_1) = x_1 ; ,

Kn(x1,x2,…,xn)=xnKn−1(x1,x2,…,xn−1)+Kn−2(x1,x2,…,xn−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}).,} 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 Kn(x1,x2,…,xn){displaystyle K_{n}(x_{1},;x_{2},;ldots ,;x_{n})}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,
    K5(x1,x2,x3,x4,x5)=x1x2x3x4x5+x3x4x5+x1x4x5+x1x2x5+x1x2x3+x1+x3+x5.{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}.}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: Kn(x1,…,xn)=Kn(xn,…,x1).{displaystyle K_{n}(x_{1},;ldots ,;x_{n})=K_{n}(x_{n},;ldots ,;x_{1}).}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:
    Kn(x1,x2,…,xn)=det(x110⋯0−1x21⋱⋮0−1⋱⋱0⋮⋱⋱⋱10⋯0−1xn).{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(x_1,;x_2,;ldots,;x_n)=<br /> det begin{pmatrix} <br /> x_1 & 1 & 0 &cdots & 0 \ <br /> -1 & x_2 & 1 & ddots & vdots\<br /> 0 & -1 & ddots &ddots & 0 \<br /> vdots & ddots & ddots &ddots & 1 \ <br /> 0 & cdots & 0 & -1 &x_n<br /> end{pmatrix}.




  • Kn(1,…,1)=Fn+1{displaystyle K_{n}(1,;ldots ,;1)=F_{n+1}}K_n(1,;ldots,;1) = F_{n+1}, the (n+1)-st Fibonacci number.

  • Kn(x1,…,xn)Kn−1(x2,…,xn)=x1+Kn−2(x3,…,xn)Kn−1(x2,…,xn).{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})}}.}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:
    Kn(x1,…,xn)Kn−1(x2,…,xn)=[x1;x2,…,xn]=x1+1x2+1x3+….{displaystyle {frac {K_{n}(x_{1},;ldots ,x_{n})}{K_{n-1}(x_{2},;ldots ,;x_{n})}}=[x_{1};;x_{2},;ldots ,;x_{n}]=x_{1}+{frac {1}{displaystyle {x_{2}+{frac {1}{x_{3}+ldots }}}}}.}frac{K_n(x_1,;ldots,x_n)}{K_{n-1}(x_2,;ldots,;x_n)} = [x_1;;x_2,;ldots,;x_n] = x_1 + frac{1}{displaystyle{x_2 + frac{1}{x_3 + ldots}}}.


  • The following matrix identity holds:

    (Kn(x1,…,xn)Kn−1(x1,…,xn−1)Kn−1(x2,…,xn)Kn−2(x2,…,xn−1))=(x1110)×…×(xn110){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&0end{pmatrix}}times ldots times {begin{pmatrix}x_{n}&1\1&0end{pmatrix}}}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} =<br /> begin{pmatrix} x_1 & 1 \ 1 & 0 end{pmatrix}timesldotstimesbegin{pmatrix} x_n & 1 \ 1 & 0 end{pmatrix}.


    • For determinants, it implies that
      Kn(x1,…,xn)⋅Kn−2(x2,…,xn−1)−Kn−1(x1,…,xn−1)⋅Kn−1(x2,…,xn)=(−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}.}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
      Kn−1(x2,…,xn)⋅Kn+2(x1,…,xn+2)−Kn(x1,…,xn)⋅Kn+1(x2,…,xn+2)=(−1)n+1xn+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}.}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



K0=1;{displaystyle K_{0}=1;,} K_0 = 1 ; ,

K1=a1;{displaystyle K_{1}=a_{1};,} K_1 = a_1 ; ,

Kn=anKn−1−bn−1cn−1Kn−2.{displaystyle K_{n}=a_{n}K_{n-1}-b_{n-1}c_{n-1}K_{n-2}.,} 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[citation needed] continuant is precisely the determinant of the tridiagonal matrix


(a1b10…00c1a2b2…000c2a3…00⋮⋮⋮⋱⋮⋮000…an−1bn−1000…cn−1an).{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}}.}{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




  • Thomas Muir (1960). A treatise on the theory of determinants. Dover Publications. pp. 516–525..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • 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.




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