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11: Bibliography G

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Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.

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ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.18.

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BibTeX

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

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FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

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

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BibTeX

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

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BibTeX

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Bessel functions and representation theory. I. J. Functional Analysis 22 (2), pp. 73–105.

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External Links:

Document, MathReview (E. A. Thieleker), ZentralBlatt

Referenced by:

§35.5(i).

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BibTeX

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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).

Encodings:

BibTeX

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12: 18.3 Definitions

… ►For representations of the polynomials in Table 18.3.1 by Rodrigues formulas, see §18.5(ii). …Explicit power series for Chebyshev, Legendre, Laguerre, and Hermite polynomials for $n=0,1,\mathrm{\dots },6$ are given in §18.5(iv). For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …

13: 27.13 Functions

… ►Each representation of $n$ as a sum of elements of $S$ is called a partition of $n$, and the number $S(n)$ of such partitions is often of great interest. … ►

§27.13(iv) Representation by Squares

… ►In fact, there are four representations, given by $5=2^2+1^2=2^2+{(-1)}^2={(-2)}^2+1^2={(-2)}^2+{(-1)}^2$, and four more with the order of summands reversed. … ►Explicit formulas for $r_k\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …

14: 10.74 Methods of Computation

… ►It should be noted, however, that there is a difficulty in evaluating the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, from the explicit expressions (10.20.10)–(10.20.13) when $z$ is close to $1$ owing to severe cancellation. … ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. … ►

§10.74(iii) Integral Representations

►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). … ►The integral representation used is based on (10.32.8). …

15: 18.38 Mathematical Applications

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

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Group Representations

… ►where $Q_0$ is a constant with explicit expression in terms of $e_1,e_2,e_3,e_4$ and $q$ given in Koornwinder (2007a, (2.8)). … ►Algebraic structures were built of which special representations involve Dunkl type operators. … ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. …