Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials
Author:
HansJ. Runckel
Journal:
Trans. Amer. Math. Soc. 334 (1992), 183212
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:
http://dx.doi.org/10.1090/S00029947199210721068
PII:
S 00029947(1992)10721068
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
