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11: 24.19 Methods of Computation

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A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

12: Bibliography F

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

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§2.10(iv).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

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§32.7(ii).

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Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

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ISSN 0010-3616, Link, MathReview (Nikolai A. Kostov), ZentralBlatt

Referenced by:

§32.15.

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Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

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Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.15.

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13: 3.11 Approximation Techniques

… ►Now suppose that $X_{k\mathrm{\ell }}=0$ when $k\ne \mathrm{\ell }$, that is, the functions ${\varphi }_k(x)$ are orthogonal with respect to weighted summation on the discrete set $x_1,x_2,\mathrm{\dots },x_J$. … ►

Example. The Discrete Fourier Transform

… ►is called a discrete Fourier transform pair. … ►The direct computation of the discrete Fourier transform (3.11.38), that is, of …

14: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . …

15: Bibliography N

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

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Affine Weyl groups, discrete dynamical systems and Painlevé equations. Comm. Math. Phys. 199 (2), pp. 281–295.

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ISSN 0010-3616, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

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§32.2(v).

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16: Bibliography H

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A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)

Referenced by:

§18.16(iii).

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Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.

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Reprinted in 1993.

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ISBN 0-471-08703-3; 0-471-58986-1, MathReview (A. E. Heins), ZentralBlatt

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§1.14(v), Ch.3.

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Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(i).

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17: 18.3 Definitions

… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: … ►For another version of the discrete orthogonality property of the polynomials $T_n\left(x\right)$ see (3.11.9). … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). …

18: 18.25 Wilson Class: Definitions

… ►The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn). … ►

§18.25(iii) Weights and Normalizations: Discrete Cases

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19: 18.38 Mathematical Applications

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Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …

20: Bibliography G

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A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.

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Document, ZentralBlatt

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§10.74(vii).

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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§18.39(iii).

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The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.

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Groups, Algorithms, Programming: A system for computational discrete algebra with emphasis on computational group theory.

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GAP

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2nd item.

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