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

Runckel, Hans-J.

doi:10.1090/S0002-9947-1992-1072106-8

Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials
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by Hans-J. Runckel PDF

Trans. Amer. Math. Soc. 334 (1992), 183-212 Request permission

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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Analytische Fortsetzung von limitärperiodischen und

limitärperiodischen

Kettenbrüchen sowie Integral-darstellungen für spezielle

Kettenbrüche

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