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11: 1.7 Inequalities
§1.7(i) Finite Sums
Cauchy–Schwarz Inequality
Minkowski’s Inequality
Cauchy–Schwarz Inequality
Minkowski’s Inequality
12: 27.6 Divisor Sums
§27.6 Divisor Sums
Sums of number-theoretic functions extended over divisors are of special interest. …
27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
27.6.6 d | n ϕ k ( d ) ( n d ) k = 1 k + 2 k + + n k ,
13: 24.6 Explicit Formulas
24.6.1 B 2 n = k = 2 2 n + 1 ( 1 ) k 1 k ( 2 n + 1 k ) j = 1 k 1 j 2 n ,
24.6.2 B n = 1 n + 1 k = 1 n j = 1 k ( 1 ) j j n ( n + 1 k j ) / ( n k ) ,
24.6.3 B 2 n = k = 1 n ( k 1 ) ! k ! ( 2 k + 1 ) ! j = 1 k ( 1 ) j 1 ( 2 k k + j ) j 2 n .
24.6.4 E 2 n = k = 1 n 1 2 k 1 j = 1 k ( 1 ) j ( 2 k k j ) j 2 n ,
24.6.9 B n = k = 0 n 1 k + 1 j = 0 k ( 1 ) j ( k j ) j n ,
14: 16.20 Integrals and Series
15: 27.1 Special Notation
d , k , m , n positive integers (unless otherwise indicated).
d | n , d | n sum, product taken over divisors of n .
( m , n ) = 1 sum taken over m , 1 m n and m relatively prime to n .
p , p sum, product extended over all primes.
n x n = 1 x .
16: 27.7 Lambert Series as Generating Functions
27.7.1 n = 1 f ( n ) x n 1 x n .
If | x | < 1 , then the quotient x n / ( 1 x n ) is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series:
27.7.2 n = 1 f ( n ) x n 1 x n = n = 1 d | n f ( d ) x n .
27.7.5 n = 1 n α x n 1 x n = n = 1 σ α ( n ) x n ,
27.7.6 n = 1 λ ( n ) x n 1 x n = n = 1 x n 2 .
17: 34.13 Methods of Computation
Methods of computation for 3 j and 6 j 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 9 j 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). …
18: 10.44 Sums
§10.44 Sums
§10.44(i) Multiplication Theorem
§10.44(ii) Addition Theorems
§10.44(iii) Neumann-Type Expansions
§10.44(iv) Compendia
19: 27.5 Inversion Formulas
27.5.1 h ( n ) = d | n f ( d ) g ( n d ) ,
which, in turn, is the basis for the Möbius inversion formula relating sums over divisors:
27.5.3 g ( n ) = d | n f ( d ) f ( n ) = d | n g ( d ) μ ( n d ) .
27.5.4 n = d | n ϕ ( d ) ϕ ( n ) = d | n d μ ( n d ) ,
27.5.6 G ( x ) = n x F ( x n ) F ( x ) = n x μ ( n ) G ( x n ) ,
20: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.2 2 K k sn ( 2 K t , k ) = n = π sin ( π ( t ( n + 1 2 ) τ ) ) = n = ( m = ( 1 ) m t m ( n + 1 2 ) τ ) ,
The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important. …
22.12.8 2 K dc ( 2 K t , k ) = n = π sin ( π ( t + 1 2 n τ ) ) = n = ( m = ( 1 ) m t + 1 2 m n τ ) ,
22.12.11 2 K ns ( 2 K t , k ) = n = π sin ( π ( t n τ ) ) = n = ( m = ( 1 ) m t m n τ ) ,
22.12.12 2 K ds ( 2 K t , k ) = n = ( 1 ) n π sin ( π ( t n τ ) ) = n = ( m = ( 1 ) m + n t m n τ ) ,