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Transactions of the American Mathematical Society

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Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials

Author: Hans-J. Runckel
Journal: Trans. Amer. Math. Soc. 334 (1992), 183-212
MSC: Primary 30B70; Secondary 30B40, 40A15, 42C05
MathSciNet review: 1072106
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Abstract: For the limit periodic $ J$-fraction $ K( - {a_n}/(\lambda + {b_n}))$, $ {a_n}$, $ {b_n} \in \mathbb{C}$, $ n \in \mathbb{N}$, which is normalized such that it converges and represents a meromorphic function $ f(\lambda )$ on $ {\mathbb{C}^{\ast} }: = \mathbb{C}\backslash [ - 1,1]$, the numerators $ {A_n}$ and denominators $ {B_n}$ of its $ n$th approximant are explicitly determined for all $ n \in \mathbb{N}$. Under natural conditions on the speed of convergence of $ {a_n}$, $ {b_n}$, $ n \to \infty $, the asymptotic behaviour of the orthogonal polynomials $ {B_n}$, $ {A_{n + 1}}$ (of first and second kind) is investigated on $ {\mathbb{C}^{\ast} }$ and $ [ - 1,1]$. An explicit representation for $ f(\lambda )$ yields continuous extension of $ f$ from $ {\mathbb{C}^{\ast} }$ onto upper and lower boundary of the cut $ ( - 1,1)$. Using this and a determinant relation, which asymptotically connects both sequences $ {A_n}$, $ {B_n}$, one obtains nontrivial explicit formulas for the absolutely continuous part (weight function) of the distribution functions for the orthogonal polynomial sequences $ {B_n}$, $ {A_{n + 1}}$, $ n \in \mathbb{N}$. This leads to short proofs of results which generalize and supplement results obtained by P. G. Nevai [7]. Under a stronger condition the explicit representation for $ f(\lambda )$ yields meromorphic extension of $ f$ from $ {\mathbb{C}^{\ast} }$ across $ ( - 1,1)$ onto a region of a second copy of $ \mathbb{C}$ which there is bounded by an ellipse, whose focal points $ \pm 1$ are first order algebraic branch points for $ f$. Then, by substitution, analogous results on continuous and meromorphic extension are obtained for limit periodic continued fractions $ K( - {a_n}(z)/(\lambda (z) + {b_n}(z)))$, where $ {a_n}(z)$, $ {b_n}(z)$, $ \lambda (z)$ are holomorphic on a region in $ \mathbb{C}$. Finally, for $ T$-fractions $ T(z) = K( - {c_n}z/(1 + {d_n}z))$ with $ {c_n} \to c$, $ {d_n} \to d$, $ n \to \infty $, the exact convergence regions are determined for all $ c$, $ d \in \mathbb{C}$. Again, explicit representations for $ T(z)$ yield continuous and meromorphic extension results. For all $ c$, $ d \in \mathbb{C}$ the regions (on Riemann surfaces) onto which $ T(z)$ can be extended meromorphically, are described explicitly.

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Keywords: Limit periodic analytic continued fraction, $ J$-fraction, $ T$-fraction, meromorphic extension, continuous extension onto boundary, asymptotic behaviour and weight function of orthogonal polynomials, explicit solution of second order linear difference equations
Article copyright: © Copyright 1992 American Mathematical Society