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• W2032085599 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." @default.
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• W2032085599 date "1992-01-01" @default.
• W2032085599 modified "2023-10-18" @default.
• W2032085599 title "Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials" @default.
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• W2032085599 doi "https://doi.org/10.1090/s0002-9947-1992-1072106-8" @default.
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