sums or differences of squares
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21: 27.5 Inversion Formulas
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27.5.1
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►which, in turn, is the basis for the Möbius inversion formula relating sums over divisors:
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27.5.3
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27.5.4
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27.5.6
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22: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.2
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►The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important.
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22.12.8
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22.12.11
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22.12.12
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23: 25.8 Sums
§25.8 Sums
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25.8.1
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25.8.3
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25.8.9
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►For other sums see Prudnikov et al. (1986b, pp. 648–649), Hansen (1975, pp. 355–357), Ogreid and Osland (1998), and Srivastava and Choi (2001, Chapter 3).
24: 24.14 Sums
§24.14 Sums
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24.14.2
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►In the following two identities, valid for , the sums are taken over all nonnegative integers with .
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►In the next identity, valid for , the sum is taken over all positive integers with .
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►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).
25: 17.5 Functions
26: 27.4 Euler Products and Dirichlet Series
27: 8.15 Sums
28: 25.17 Physical Applications
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►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect).
It has been found possible to perform such regularizations by equating the divergent sums to zeta functions and associated functions (Elizalde (1995)).