sums or differences of squares
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11: 6.15 Sums
§6.15 Sums
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6.15.1
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6.15.2
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6.15.3
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►For further sums see Fempl (1960), Hansen (1975, pp. 423–424), Harris (2000), Prudnikov et al. (1986b, vol. 2, pp. 649–650), and Slavić (1974).
12: 1.7 Inequalities
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§1.7(i) Finite Sums
… ►Cauchy–Schwarz Inequality
… ►Minkowski’s Inequality
… ►Cauchy–Schwarz Inequality
… ►Minkowski’s Inequality
…13: 27.6 Divisor Sums
§27.6 Divisor Sums
►Sums of number-theoretic functions extended over divisors are of special interest. … ►
27.6.1
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►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors.
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27.6.6
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14: 24.6 Explicit Formulas
15: 16.20 Integrals and Series
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16: 27.1 Special Notation
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positive integers (unless otherwise indicated). | |
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, | sum, product taken over divisors of . |
sum taken over , and relatively prime to . | |
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, | sum, product extended over all primes. |
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17: 1.2 Elementary Algebra
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►If are positive integers and , then there exist polynomials , , such that
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►If then, depending on , there is either no solution or there are infinitely many solutions, being the sum of a particular solution of (1.2.61) and any solution of .
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►Nonzero vectors are linearly independent if implies that all coefficients are zero.
…The sum of all multiplicities is .
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►Thus is the sum of the (counted according to their multiplicities) eigenvalues of .
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18: 27.7 Lambert Series as Generating Functions
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27.7.1
►If , then the quotient is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series:
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27.7.2
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27.7.5
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27.7.6
19: 34.13 Methods of Computation
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►Methods of computation for and symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981).
►For symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989).
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