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1: 4.27 Sums
§4.27 Sums
For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, 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 (2000, 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
5.16.1 k = 1 ( - 1 ) k ψ ( k ) = - π 2 8 ,
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: 8.15 Sums
§8.15 Sums
8.15.1 γ ( a , λ x ) = λ a k = 0 γ ( a + k , x ) ( 1 - λ ) k k ! .
For sums of infinite series whose terms include incomplete gamma functions, see Prudnikov et al. (1986b, §5.2).
7: 27.10 Periodic Number-Theoretic Functions
An example is Ramanujan’s sum: …It can also be expressed in terms of the Möbius function as a divisor sum: … More generally, if f and g are arbitrary, then the sumAnother generalization of Ramanujan’s sum is the Gauss sum G ( n , χ ) associated with a Dirichlet character χ ( mod k ) . … G ( n , χ ) is separable for some n if …
8: 24.20 Tables
§24.20 Tables
Abramowitz and Stegun (1964, Chapter 23) includes exact values of k = 1 m k n , m = 1 ( 1 ) 100 , n = 1 ( 1 ) 10 ; k = 1 k - n , k = 1 ( - 1 ) k - 1 k - n , k = 0 ( 2 k + 1 ) - n , n = 1 , 2 , , 20D; k = 0 ( - 1 ) k ( 2 k + 1 ) - n , n = 1 , 2 , , 18D. …
9: 25.16 Mathematical Applications
§25.16(ii) Euler Sums
Euler sums have the form … H ( s ) is the special case H ( s , 1 ) of the function …which satisfies the reciprocity law …when both H ( s , z ) and H ( z , s ) are finite. …
10: 15.15 Sums
§15.15 Sums
15.15.1 F ( a , b c ; 1 z ) = ( 1 - z 0 z ) - a s = 0 ( a ) s s ! F ( - s , b c ; 1 z 0 ) ( 1 - z z 0 ) - s .
For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975).