×

Convergence of Padé approximants for a \(q\)-hypergeometric series (Wynn’s power series III). (English) Zbl 0767.41019

Summary: We investigate the convergence of sequences of Padé approximants for the power series \(f(z)=1+\sum_{j=1}^ \infty a_ j z^ j\) where \[ a_ j=\prod_{k=0}^{j-1} {{(A-q^{k+\alpha})} \over {(C- q^{k+\gamma})}}, \qquad j\geq 1; \quad \alpha,\gamma\in\mathbb{R}; \quad A,C,q\in\mathbb{C}. \] For “most” \(A\), and \(| C|\neq 1\), we show that, if \(q=e^{o\theta}\) where \(\theta\in[0,2\pi)\) and \(\theta/2\pi\) is irrational, \(f(z)\) has a natural boundary on its circle of convergence. We show that diagonal and other sequences of Padé approximants converge in capacity of \(f\) and further obtain subsequences of the diagonal sequences \(\{[n/n](z)\}_{n=1}^ \infty\) that converge locally uniformly.

MSC:

41A21 Padé approximation
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
30E05 Moment problems and interpolation problems in the complex plane
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Andrews, G. E.,q-Series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra. [Regional Conference Series in Mathematics, No. 66]. Amer. Math. Soc., Providence, RI, 1986. · Zbl 0594.33001
[2] Baker, G. A. Jr.,Essentials of Padé approximants. Academic Press, New York, 1975.
[3] Borwein, P. B.,Padé approximants for the q-elementary functions. Constr. Approx.4 (1988), 391–402. · Zbl 0685.41015 · doi:10.1007/BF02075469
[4] de Bruin, M. G.,Simultaneous rational approximation to some q-hypergeometric functions (in) non-linear numerical methods and rational approximation (A. Cuyt, ed.). Reidel, Dordrecht, 1988, pp. 135–142.
[5] Driver, K. A.,Convergence of Padé approximants for some q-hypergeometric series (Wynn’s Power Series I, II and III). Thesis. University of the Witwatersrand, 1991. · Zbl 0746.41021
[6] Driver, K. A. andLubinsky, D. S.,Convergence of Padé approximants for some q-hypergeometric series (Wynn’s Power Series I). Aequationes Math.42 (1991), 85–106. · Zbl 0746.41021 · doi:10.1007/BF01818481
[7] Driver, K. A. andLubinsky, D. S.,Convergence of Padé approximants for Wynn’s Power Series II. To appear in Proc. of the International Conference on Approx. Theory, Kecskemét, Hungary, 1990. J. Bolyai Math. Soc., Budapest.
[8] Fine, N. J.,Basic hypergeometric series and their applications. [Mathematical surveys and monographs, No. 27]. Amer. Math. Soc., Providence, RI, 1988.
[9] Gammel, J. L. andNuttall, J.,Convergence of Padé approximants to quasi-analytic functions beyond natural boundaries. J. Math. Anal. Appl.43 (1973), 694–696. · Zbl 0265.30037 · doi:10.1016/0022-247X(73)90284-9
[10] Gasper, G. andRahman, M.,Basic hypergeometric series. Cambridge University Press, Cambridge, 1990. · Zbl 0695.33001
[11] Ismail, M., Perline, R. andWimp, J.,Padé approximants for some q-hypergeometric functions. To appear in Proc. 1st US-USSR Conference on Approx. Theory.
[12] Ismail, M. andRahman, M.,Associated Askey–Wilson polynomias. To appear in Trans. Amer. Math. Soc.
[13] Jones, W. B. andThron, W. J.,Continued fractions: Analytic theory and applications. [Encyclopaedia of Mathematics and its Applications, Vol. 11]. Cambridge University Press, Cambridge–New York, 1980.
[14] Kuipers, L. andNiederreiter, H.,Uniform distribution of sequences. Wiley, New York, 1974.
[15] Landkof, N. S.,Foundations of Modern Potential Theory. [Grundlehren der Mathematischen Wissenschaften, Vol. 190]. Springer, New York, 1972. · Zbl 0253.31001
[16] Lubinsky, D. S.,Diagonal Padé approximants and capacity. J. Math. Anal. Appl.78 (1980), 58–67. · Zbl 0456.30035 · doi:10.1016/0022-247X(80)90210-3
[17] Lubinsky, D. S. andSaff, E. B.,Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials. Constr. Approx.3 (1987), 331–361. · Zbl 0634.41013 · doi:10.1007/BF01890574
[18] Rahmanov, E. A.,On the convergence of Padé approximants in classes of holomorphic functions. Math. USSR-Sb.40 (1981), 149–155. · Zbl 0466.30033 · doi:10.1070/SM1981v040n02ABEH001794
[19] Wynn, P.,A general system of orthogonal polynomials. Quart. J. Math. Oxford Ser.18 (1967), 81–96. · Zbl 0185.30001 · doi:10.1093/qmath/18.1.81
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.