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

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

MathSciNet review:
1072106

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**R. Askey and M. Ismail,*Recurrence relations, continued fractions and orthogonal polynomials*, Mem. Amer. Math. Soc., no. 300, 1984. MR**743545 (85g:33008)****[2]**G. A. Baker, Jr. and P. Graves-Morris,*Padé approximants, part*I:*Basic theory*, Encyclopedia of Math. and its Applications, No. 13, Addison-Wesley, Reading, Mass., 1981. MR**635619 (83a:41009a)****[3]**J. Gill,*Enhancing the convergence region of a sequence of bilinear transformations*, Math. Scand.**43**(1978), 74-80. MR**523827 (80d:30004)****[4]**L. Jacobsen,*Functions defined by continued fractions meromorphic continuation*, Rocky Mountain J. Math.**15**(1985), 685-703. MR**813268 (87f:30006)****[5]**-,*Approximants for functions represented by limit periodic continued fractions*, Constructive Theory of Functions (Varna, 1987), Bulgar. Acad. Sci., Sofia, 1988, pp. 242-250. MR**994844 (90j:30006)****[6]**W. B. Jones and W. J. Thron,*Continued fractions, analytic theory and applications*, Encyclopedia of Math. and its Applications, No. 11, Addison-Wesley, Reading, Mass., 1980. MR**595864 (82c:30001)****[7]**P. G. Nevai,*Orthogonal polynomials*, Mem. Amer. Math. Soc., no. 213, 1979. MR**519926 (80k:42025)****[8]**O. Perron,*Die Lehre von den Kettenbrüchen*, Bd. 2, Teubner, Stuttgart, 1957. MR**0085349 (19:25c)****[9]**H.-J. Runckel,*Continuity on the boundary and analytic continuation of continued fractions*, Math. Z.**148**(1976), 189-205. MR**0430223 (55:3230)****[10]**-,*Zeros of complex orthogonal polynomials*, Internat. Sympos. on Orthogonal Polynomials and their Applications for the 150th Anniversary of E. N. Laguerre, (Bar-Le-Duc, France, 1984), Lecture Notes in Math., vol. 1171, Springer-Verlag, Berlin, 1985, pp. 278-282. MR**838994 (87i:30015)****[11]**-,*Meromorphic extension of analytic continued fractions across the line of nonconvergence*, Rocky Mountain J. Math.**21**(1991), 539-556. MR**1113942 (92m:30006)****[12]**H. Schlierf,*Analytische Fortsetzung von limitärperiodischen und*-*limitärperiodischen*-*Kettenbrüchen sowie Integral-darstellungen für spezielle*-*Kettenbrüche*, Dissertation, University of Ulm, 1988.**[13]**W. J. Thron and H. Waadeland,*Accelerating convergence of limit periodic continued fractions*, Numer. Math.**34**(1980), 155-170. MR**566679 (81e:30008)****[14]**-,*Analytic continuation of functions defined by means of continued fractions*, Math. Scand.**47**(1980), 72-90. MR**600079 (82c:30004)****[15]**H. S. Wall,*Analytic theory of continued fractions*, Van Nostrand, New York, 1948. MR**0025596 (10:32d)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30B70,
30B40,
40A15,
42C05

Retrieve articles in all journals with MSC: 30B70, 30B40, 40A15, 42C05

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