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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
Ultraspherical
Legendre
Jacobi
Ultraspherical
3: 1.17 Integral and Series Representations of the Dirac Delta
§1.17 Integral and Series Representations of the Dirac Delta
§1.17(ii) Integral Representations
Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation
Sine and Cosine Functions
§1.17(iii) Series Representations
4: 21.10 Methods of Computation
  • 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).

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

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

  • 5: Bibliography V
  • N. Ja. Vilenkin and A. U. Klimyk (1993) 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.
  • N. Ja. Vilenkin (1968) Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.
  • I. M. Vinogradov (1937) Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).
  • H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.
  • H. Volkmer (2021) Fourier series representation of Ferrers function 𝖯 .
  • 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
    Chebyshev
    §18.5(ii) Rodrigues Formulas
    Related formula: …
    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 ν ± μ ( z ) , Q ν μ ( z ) , and 𝑸 ν μ ( z ) 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 ( 1 , ) , and by continuity elsewhere in the z -plane with a cut along the interval ( , 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). …