representations by Euler?Maclaurin formula

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1: Wolter Groenevelt

… ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

2: 18.10 Integral Representations

§18.10 Integral Representations

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Ultraspherical

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Legendre

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Jacobi

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Ultraspherical

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3: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(ii) Integral Representations

… ►Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation … ►

Sine and Cosine Functions

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§1.17(iii) Series Representations

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4: 21.10 Methods of Computation

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Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

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Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.

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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

5: Bibliography V

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Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii).

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Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.27.

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Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).

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External Links:

ZentralBlatt

Referenced by:

§27.13(ii).

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

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Fourier series representation of Ferrers function $𝖯$ .

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External Links:

FullText

Referenced by:

(14.13.1).

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6: 26.19 Mathematical Applications

§26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). …

7: 35.9 Applications

… ►These references all use results related to the integral formulas (35.4.7) and (35.5.8). ►For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). …

8: 18.5 Explicit Representations

§18.5 Explicit Representations

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Chebyshev

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§18.5(ii) Rodrigues Formulas

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9: 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. … ►A general formula states that … ►

§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. … ►Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …

10: 14.21 Definitions and Basic Properties

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). … ►

§14.21(iii) Properties

… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …