Padé table

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1: 3.11 Approximation Techniques

… ►is called a Padé table. … … ►Any five approximants arranged in the Padé table as …Starting with the first column ${[n/0]}_f$, $n=0,1,2,\mathrm{\dots }$, and initializing the preceding column by ${[n/-1]}_f=\mathrm{\infty }$, $n=1,2,\mathrm{\dots }$, we can compute the lower triangular part of the table via (3.11.25). …

2: Bibliography W

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Upon systems of recursions which obtain among the quotients of the Padé table. Numer. Math. 8 (3), pp. 264–269.

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External Links:

ISSN 0029-599X, Document, MathReview (F. L. Bauer), ZentralBlatt

Referenced by:

§3.11(iv).

Encodings:

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3: 3.9 Acceleration of Convergence

… ►If $s_n$ is the $n$th partial sum of a power series $f$, then $t_{n,2k}={\epsilon }_{2k}^{(n)}$ is the Padé approximant ${[(n+k)/k]}_f$ (§3.11(iv)). … ►In Table 3.9.1 values of the transforms $t_{n,2k}$ are supplied for … ►

Table 3.9.1: Shanks’ transformation for $s_n={\sum }_{j=1}^n{(-1)}^{j+1}{j}^{-2}$.

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$n$	$t_{n,2}$	$t_{n,4}$	$t_{n,6}$	$t_{n,8}$	$t_{n,10}$
…

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4: 33.23 Methods of Computation

… ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …

5: Bibliography

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Tables of $\ln \mathrm{\Gamma }(z)$ for Complex Argument. Pergamon Press, New York.

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External Links:

ZentralBlatt

Referenced by:

§5.22(iii).

Encodings:

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Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

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External Links:

ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt

Referenced by:

§17.18.

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Table of Definite and Infinite Integrals. Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.

External Links:

ISBN 0-444-42151-3, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§1.4(iv), §10.22(vi), §10.43(vi), §11.7(v), §11.9(iv), §12.5(iv), §13.10(vi), §13.23(v), §15.14, §16.20, §16.5, §18.17(ix), §4.10(iii), §4.26(v), §4.40(v), §5.13, §6.14(iii), §7.14(iii), §8.14, §8.19(x).

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Numerical Tables for Angular Correlation Computations in $\alpha$-, $\beta$- and $\gamma$-Spectroscopy: $3j$-, $6j$-, $9j$-Symbols, F- and $\mathrm{\Gamma }$-Coefficients. Landolt-Börnstein Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag.

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External Links:

ISBN 978-3-540-04218-1

Referenced by:

§34.14.

Encodings:

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Tables of Lamé Polynomials. Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§29.15(ii), 2nd item.

Encodings:

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6: 18.36 Miscellaneous Polynomials

… ►They are related to Hermite–Padé approximation and can be used for proofs of irrationality or transcendence of interesting numbers. … ►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, $A_n{A}_{n-1}{C}_n>0$ for $n\ge 1$ as per (18.2.9_5). … ►The $y(x)={\widehat{L}}_n^{(k)}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients: …

7: Bibliography B

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Padé Approximants. 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 59, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-45007-1, MathReview (Annie A. M. Cuyt), ZentralBlatt

Referenced by:

§3.11(iv), §3.11(iv).

Encodings:

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Tables of Normalized Associated Legendre Polynomials. Pergamon Press, The Macmillan Co., Oxford-New York.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

3rd item.

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Nouvelles Tables d’Intégrales Définies. P. Engels, Leide.

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Notes:

Corrected edition printed by Hafner Press, 1965.

Referenced by:

§1.4(iv).

Encodings:

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Nouvelles Tables d’Intégrales Définies. G.E. Stechert & Co., New York.

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Referenced by:

§4.10(iii), §4.26(v), §4.40(v), §6.14(iii), §6.7(iv).

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Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, Vol. 50, Birkhäuser Verlag, Basel.

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External Links:

ISBN 3-7643-1100-2, MathReview (A. Magnus), ZentralBlatt

Referenced by:

§3.11(iv).

Encodings:

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