Abstract
We give a simple construction of the Padé approximants toq analogues of exp and log. The construction is based on the functional relations they satisfy. The Padé approximants for the ordinary exp and log are then limiting cases.
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G. E. Andrews, I. P. Goulden, D. M. Jackson (1986):Shank's convergence acceleration transform, Padé approximates and partitions. J. Combin. Theory Ser. A,43:70–84.
R.Askey, M. E. H.Ismail (1984):Recurrence relations, continued fractions and orthogonal polynomials. Mem. Amer. Math. Soc.,49.
G. A. Baker, P. R. Graves-Morris (1981): Padé Approximatifs, vols. I and II. Reading, MA: Addison-Wesley.
D. V. Chudnovsky, G. V. Chudnovsky (1984): Padé and Rational Approximation to Systems of Functions and Their Arithmetic Applications. Springer Lecture Notes in Mathematics, vol. 1052. Berlin: Springer-Verlag.
M. Exton (1983): g-Hypergeometric Functions and Applications. Chichester: Ellis Horwood.
F. H. Jackson (1904):A basic sine and cosine with symbolic solutions of certain differential equations. Proc. Edinburgh Math. Soc.,22:28–39.
D. S.Lubinsky, E. B.Saff (preprint):Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials.
K. Mahler (1931):Zur Approximation der Expontialfunkion und des Logarithmus. J. Reine Angew. Math.,166:118–150.
E. B. Saff, R. S. Varga (1978):On the zeros and poles of Padé approximants to e z, III. Numer. Math.,30:241–266.
G.Szegö (1926):Ein Beitrag zur Theorie der Thetafunktionen. Sitzungsberichte der Preussichen Akademie der Wissenschaften. Phys. Math. Kl., 242–251. (Reprinted (1982) in: Collected Papers, vol. 1. Boston: Birkhauser, pp. 795–805).
L. Trefethen (1984):The asymptotic accuracy of rational best approximations to e z on a disk. J. Approx. Theory,40:380–383.
R. Wallisser (1985):Rationale Approximation des q-Analogons der Exponential funktion und Irrationalitätsaussagen für diese Funktion. Arch. Math. (Basel),44:59–64.
P. Wynn,A general system of orthogonal polynomials 1967): Quart. J. Math.,18:81–96.
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Communicated by Edward B. Saff.
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Borvein, P.B. Padé approximants for theq-elementary functions. Constr. Approx 4, 391–402 (1988). https://doi.org/10.1007/BF02075469
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DOI: https://doi.org/10.1007/BF02075469