D?rbasjan sum
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1: 4.27 Sums
§4.27 Sums
►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2015, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).2: 4.11 Sums
§4.11 Sums
…3: 4.41 Sums
§4.41 Sums
►For sums of hyperbolic functions see Gradshteyn and Ryzhik (2015, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).4: 7.15 Sums
§7.15 Sums
►For sums involving the error function see Hansen (1975, p. 423) and Prudnikov et al. (1986b, vol. 2, pp. 650–651).5: 5.16 Sums
§5.16 Sums
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5.16.1
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►For further sums involving the psi function see Hansen (1975, pp. 360–367).
For sums of gamma functions see Andrews et al. (1999, Chapters 2 and 3) and §§15.2(i), 16.2.
►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).
6: 27.10 Periodic Number-Theoretic Functions
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►An example is Ramanujan’s sum:
…It can also be expressed in terms of the Möbius function as a divisor sum:
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►More generally, if and are arbitrary, then the sum
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►Another generalization of Ramanujan’s sum is the Gauss sum
associated with a Dirichlet character .
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is separable for some if
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7: 24.20 Tables
§24.20 Tables
►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D. …8: 25.16 Mathematical Applications
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►
§25.16(ii) Euler Sums
►Euler sums have the form … ► is the special case of the function …which satisfies the reciprocity law …when both and are finite. …9: 15.15 Sums
§15.15 Sums
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15.15.1
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►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).