This paper considers periodic continued fractions of the form \[ \phi(\lambda)=b_ 0+{\lambda-\alpha_ 1\over{b_ 1 + {\lambda-\alpha_ 2\over {\ddots b_{N-1}+{\lambda-\alpha_N\over b_N-b_0+\phi}}}}} \] which occur in the study of integrable systems and associated periodic dressing chains. Here \(\{\alpha_i\}\) is a fixed \(N\)-periodic sequence \((\alpha_{N+i}=\alpha_i,\quad i \geq 1)\) which is the fundamental datum, \(\{b_i\}\) is a second sequence of complex numbers, \(N\)-periodic from \(i=1\), and \(\lambda\) is a formal parameter. The authors call this a “periodic \(\alpha\)-fraction”. Convergence is not discussed but, quite formally, periodicity of the fraction implies that \(\phi\) satisfies a quadratic equation \(A(\lambda)\phi^2+2B(\lambda)\phi+ C(\lambda)=0\), where \(A,B,C\) are polynomials in \(\lambda\) with coefficients depending polynomially on the \(b_i,\alpha_i\). Thus \(\phi = -B+\sqrt{R(\lambda}/A\) where \(R=B^2-AC\), so that \(\phi\) is an algebraic function on the hyperelliptic curve \(y^2=R(\lambda)\).
The following questions are addressed:
(1) Which algebraic functions of the form \(\phi\) admit \(N\)-periodic \(\alpha\)-fraction expansions?
(2) How many such expansions are there for given \(\phi\) and how does one find them?
(3) What is the geometry of the set of functions \(\phi\) from a given hyperelliptic extension (i.e. with \(R\) fixed) which admit periodic \(\alpha\)-fraction expansions?
Answers are given assuming odd \(N\), and all \(\alpha_i\) distinct. Briefly, let \(\mathcal{A}=\prod (\lambda-\alpha_i) \). A polynomial \(R(\lambda)\) of degree \(N=2g+1\) is called \(\mathcal A\)-admissible if there exists a polynomial \(S\) of degree \(\leq g\) such that \(R=S^2+\mathcal A\). Then, if \(\phi\) has an \(N\)-periodic \(\alpha\)-expansion, the polynomials \(A,B,C\) introduced above satisfy: \(\deg B\leq g\), \(A\) is monic of degree \(g\), \(-C\) is monic of degree \(g+1\) and \(R=B^2-AC\) is \(\mathcal A\)-admissible.
Conversely, for an open dense subset of such triples the corresponding function has exactly two \(\alpha\)-periodic expansions, which can be found by an effective matrix factorization procedure.(The pure periodic case (\(b_N=b_0)\) is exceptional in that there is a unique expansion). This answers questions 1 and 2. An action of \({\mathbb Z}_2\times S_n\) on the fractions is also discussed. Question 3 is answered by relating the set of triples \((A,B,C)\) with \(R\) fixed and \(\mathcal A\)-admissible to the Jacobian of the hyperelliptic curve \(y^2=R\).
Reviewer’s remark. Three types of periodic continued fraction have made appearances in connection with periodicity phenomena in integrable systems: Stieltjes fractions, in [P. van Moerbeke, Invent. Math. 37, 45–81 (1976; Zbl 0361.15010)], the “usual” type (i.e. defined by analogy with \(\mathbb R\) for finite-tailed Laurent series with uniformising parameter \(1/\lambda\)) in [S. A. Andrea and T. G. Berry, Linear Algebra Appl. 161, 117–134 (1992; Zbl 0760.65036)] and finally the fractions of the present paper. Is there some unifying principle to be found?