Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials
HTML articles powered by AMS MathViewer
- by Hans-J. Runckel
- Trans. Amer. Math. Soc. 334 (1992), 183-212
- DOI: https://doi.org/10.1090/S0002-9947-1992-1072106-8
- PDF | Request permission
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
- Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545, DOI 10.1090/memo/0300
- George A. Baker Jr. and Peter Graves-Morris, Padé approximants. Part I, Encyclopedia of Mathematics and its Applications, vol. 13, Addison-Wesley Publishing Co., Reading, Mass., 1981. Basic theory; With a foreword by Peter A. Carruthers. MR 635619
- John Gill, Enhancing the convergence region of a sequence of bilinear transformations, Math. Scand. 43 (1978/79), no. 1, 74â80. MR 523827, DOI 10.7146/math.scand.a-11765
- Lisa Jacobsen, Functions defined by continued fractions. Meromorphic continuation, Rocky Mountain J. Math. 15 (1985), no. 3, 685â703. MR 813268, DOI 10.1216/RMJ-1985-15-3-685
- Lisa Jacobsen, Approximants for functions represented by limit periodic continued fractions, Constructive theory of functions (Varna, 1987) Publ. House Bulgar. Acad. Sci., Sofia, 1988, pp. 242â250. MR 994844
- William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
- Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
- Oskar Perron, Die Lehre von den KettenbrĂŒchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische KettenbrĂŒche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957 (German). MR 0085349
- Hans-J. Runckel, Continuity on the boundary and analytic continuation of continued fractions, Math. Z. 148 (1976), no. 2, 189â205. MR 430223, DOI 10.1007/BF01214708
- Hans-J. Runckel, Zeros of complex orthogonal polynomials, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 278â282. MR 838994, DOI 10.1007/BFb0076554
- Hans-J. Runckel, Meromorphic extension of analytic continued fractions across the line of nonconvergence, Proceedings of the U.S.-Western Europe Regional Conference on PadĂ© Approximants and Related Topics (Boulder, CO, 1988), 1991, pp. 539â556. MR 1113942, DOI 10.1216/rmjm/1181073022 H. Schlierf, Analytische Fortsetzung von limitĂ€rperiodischen und $(2,1)$-limitĂ€rperiodischen $\delta$-KettenbrĂŒchen sowie Integral-darstellungen fĂŒr spezielle $\delta$-KettenbrĂŒche, Dissertation, University of Ulm, 1988.
- W. J. Thron and H. Waadeland, Accelerating convergence of limit periodic continued fractions $K(a_{n}/1)$, Numer. Math. 34 (1980), no. 2, 155â170. MR 566679, DOI 10.1007/BF01396057
- W. J. Thron and Haakon Waadeland, Analytic continuation of functions defined by means of continued fractions, Math. Scand. 47 (1980), no. 1, 72â90. MR 600079, DOI 10.7146/math.scand.a-11875
- H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR 0025596
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- 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