DVR (discrete variable representations)

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11: 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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External Links:

Document, ZentralBlatt

Referenced by:

§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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Referenced by:

§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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Notes:

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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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

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Representations of Integers as Sums of Squares. Springer-Verlag, New York.

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ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt

Referenced by:

§27.13(iv).

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12: 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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Lie groups, Lie algebras, and representations. Second edition, Graduate Texts in Mathematics, Vol. 222, Springer, Cham.

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Notes:

An elementary introduction

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ISBN 978-3-319-13466-6; 978-3-319-13467-3, Document, Link, MathReview Entry

Referenced by:

§1.2(vi).

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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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Notes:

Reprinted in 1993.

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

Referenced by:

§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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Variable precision exponential function. ACM Trans. Math. Software 12 (2), pp. 79–91.

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ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(iii).

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13: 18.1 Notation

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$x,y,t$	real variables.
$z(=x+\mathrm{i}y)$	complex variable.
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Discrete $q$-Hermite I: $h_n(x;q)$.

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Discrete $q$-Hermite II: ${\tilde{h}}_n(x;q)$.

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Classical OP’s in Two Variables

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14: 10.64 Integral Representations

§10.64 Integral Representations

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10.64.1 $${\mathrm{ber}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\cos \left(x\sin t-nt\right)\cosh \left(x\sin t\right)dt,$$

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${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E1

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pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

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10.64.2 $${\mathrm{bei}}_n\left(x\sqrt{2}\right)=\frac{{(-1)}^n}{\pi }{\int }_0^{\pi }\sin \left(x\sin t-nt\right)\sinh \left(x\sin t\right)dt.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/10.64.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.64, §10.64 and Ch.10

►See Apelblat (1991) for these results, and also for similar representations for ${\mathrm{ber}}_{\nu }\left(x\sqrt{2}\right)$, ${\mathrm{bei}}_{\nu }\left(x\sqrt{2}\right)$, and their $\nu$-derivatives. …

15: 18.3 Definitions

… ►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). … ►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 $T_{N+1}\left(x\right)$: … ►

18.3.2 $$x_{N+1,n}=\cos \left((n-\frac{1}{2})\pi /(N+1)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.3.E2

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See also:

Annotations for §18.3, §18.3 and Ch.18

… ►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). …

16: 18.28 Askey–Wilson Class

… ►The Askey–Wilson polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. … ►In the remainder of this section the Askey–Wilson class OP’s are defined by their $q$-hypergeometric representations, followed by their orthogonal properties. … ►More generally, if $|ab|\le 1$ instead of $|a|,|b|\le 1$, discrete terms need to be added to the right-hand side of (18.28.8); see Koekoek et al. (2010, Eq. (14.8.3)). … ►Leonard (1982) classified all (finite or infinite) discrete systems of OP’s $p_n(x)$ on a set $\{x(m)\}$ for which there is a system of discrete OP’s $q_m(y)$ on a set $\{y(n)\}$ such that $p_n(x(m))=q_m(y(n))$. …

17: Bibliography B

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Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.

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External Links:

ISSN 0236-5294, Document, FullText, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§11.7(v), §11.7(v).

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Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

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ISBN 0-8218-0324-7, MathReview, ZentralBlatt

Referenced by:

§26.19.

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Discrete ordinate solution of Fokker-Planck equations with non-linear coefficients. Phys. Rev. A 31 (3), pp. 1855–1868.

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Referenced by:

§18.39(iii).

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Discrete ordinate solution of a Fokker-Planck equation in laser physics. Transport Theory Statist. Phys. 15 (1-2), pp. 181–210.

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Referenced by:

§18.39(iii).

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Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.

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External Links:

ISBN 978-0-12-373624-6; 0-12-373624-2, MathReview (Gerd Teschke)

Referenced by:

§18.3.

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18: Bibliography L

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The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

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Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

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ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

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Discrete-variable representations and their utilization. In Advances in Chemical Physics, pp. 263–310.

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External Links:

ISBN 9780470141731, Document, Link

Referenced by:

§18.38(i).

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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.

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ISSN 0022-247X, Document, MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.34(iii), §24.11, §24.16(i), §24.16(i).

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

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19: 26.22 Software

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GAP (website). A system for computational discrete algebra.

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20: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

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Handbook of Discrete and Combinatorial Mathematics. CRC Press, Boca Raton, FL.

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External Links:

ISBN 0-8493-0149-1, MathReview, ZentralBlatt

Referenced by:

§26.1, §26.1, §26.11, §26.19, Ch.26, Notations, Notations.

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