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 -fraction , , , , which is normalized such that it converges and represents a meromorphic function on , the numerators and denominators of its th approximant are explicitly determined for all . Under natural conditions on the speed of convergence of , , , the asymptotic behaviour of the orthogonal polynomials , (of first and second kind) is investigated on and . An explicit representation for yields continuous extension of from onto upper and lower boundary of the cut . Using this and a determinant relation, which asymptotically connects both sequences , , one obtains nontrivial explicit formulas for the absolutely continuous part (weight function) of the distribution functions for the orthogonal polynomial sequences , , . 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 yields meromorphic extension of from across onto a region of a second copy of which there is bounded by an ellipse, whose focal points are first order algebraic branch points for . Then, by substitution, analogous results on continuous and meromorphic extension are obtained for limit periodic continued fractions , where , , are holomorphic on a region in . Finally, for -fractions with , , , the exact convergence regions are determined for all , . Again, explicit representations for yield continuous and meromorphic extension results. For all , the regions (on Riemann surfaces) onto which can be extended meromorphically, are described explicitly.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1072106-8

Keywords:
Limit periodic analytic continued fraction,
-fraction,
-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