Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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 $ 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.


References [Enhancements On Off] (What's this?)

  • [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 $ (2,1)$-limitärperiodischen $ \delta $-Kettenbrüchen sowie Integral-darstellungen für spezielle $ \delta $-Kettenbrüche, Dissertation, University of Ulm, 1988.
  • [13] W. J. Thron and H. Waadeland, Accelerating convergence of limit periodic continued fractions $ K({a_n}/1)$, 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)

Similar Articles

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, $ J$-fraction, $ T$-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

American Mathematical Society