representations%20by%20Euler%E2%80%93Maclaurin%20formula
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1: 24.1 Special Notation
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►Unless otherwise noted, the formulas in this chapter hold for all values of the variables and , and for all nonnegative integers .
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Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: Wolter Groenevelt
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►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.
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3: 20 Theta Functions
Chapter 20 Theta Functions
…4: 25.12 Polylogarithms
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►The notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):
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Integral Representation
… ►Sometimes the factor is omitted. …5: 32.8 Rational Solutions
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►In the general case assume , so that as in §32.2(ii) we may set and .
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►For determinantal representations see Kajiwara and Masuda (1999).
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►For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999).
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(c)
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►For determinantal representations see Masuda et al. (2002).
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, , and , with even.
6: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Integral representation of Kelvin functions and their derivatives with respect to the order.
Z. Angew. Math. Phys. 42 (5), pp. 708–714.
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7: 25.5 Integral Representations
§25.5 Integral Representations
… ►For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339). ►In (25.5.15)–(25.5.19), , is the digamma function, and is Euler’s constant (§5.2). … ►
25.5.19
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§25.5(iii) Contour Integrals
…8: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Zhang and Jin (1996, Table 3.8) tabulates for , to 8D or 8S.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
9: 8 Incomplete Gamma and Related
Functions
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10: 28 Mathieu Functions and Hill’s Equation
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