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1: 27.10 Periodic Number-Theoretic Functions
Another generalization of Ramanujan’s sum is the Gauss sum G ( n , χ ) associated with a Dirichlet character χ ( mod k ) . …In particular, G ( n , χ 1 ) = c k ( n ) . G ( n , χ ) is separable for some n if … For a primitive character χ ( mod k ) , G ( n , χ ) is separable for every n , and … Conversely, if G ( n , χ ) is separable for every n , then χ is primitive (mod k ). …
2: 20.11 Generalizations and Analogs
§20.11(i) Gauss Sum
For relatively prime integers m , n with n > 0 and m n even, the Gauss sum G ( m , n ) is defined by
20.11.1 G ( m , n ) = k = 0 n - 1 e - π i k 2 m / n ;
20.11.2 1 n G ( m , n ) = 1 n k = 0 n - 1 e - π i k 2 m / n = e - π i / 4 m j = 0 m - 1 e π i j 2 n / m = e - π i / 4 m G ( - n , m ) .
3: 5.16 Sums
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).
4: 17.7 Special Cases of Higher ϕ s r Functions
q -Analog of Bailey’s F 1 2 ( - 1 ) Sum
q -Analog of Gauss’s F 1 2 ( - 1 ) Sum
q -Analog of Dixon’s F 2 3 ( 1 ) Sum
Gasper–Rahman q -Analog of Watson’s F 2 3 Sum
Gasper–Rahman q -Analog of Whipple’s F 2 3 Sum
5: 17.6 ϕ 1 2 Function
q -Gauss Sum
6: 16.4 Argument Unity
Pfaff–Saalschütz Balanced Sum
Dixon’s Well-Poised Sum
Watson’s Sum
Whipple’s Sum
Džrbasjan’s Sum
7: 34.2 Definition: 3 j Symbol
where F 2 3 is defined as in §16.2. For alternative expressions for the 3 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 2 3 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
8: 34.4 Definition: 6 j Symbol
For alternative expressions for the 6 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 3 4 of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
9: 15.2 Definitions and Analytical Properties
15.2.1 F ( a , b ; c ; z ) = s = 0 ( a ) s ( b ) s ( c ) s s ! z s = 1 + a b c z + a ( a + 1 ) b ( b + 1 ) c ( c + 1 ) 2 ! z 2 + = Γ ( c ) Γ ( a ) Γ ( b ) s = 0 Γ ( a + s ) Γ ( b + s ) Γ ( c + s ) s ! z s ,
15.2.4 F ( - m , b ; c ; z ) = n = 0 m ( - m ) n ( b ) n ( c ) n n ! z n = n = 0 m ( - 1 ) n ( m n ) ( b ) n ( c ) n z n .
10: 15.16 Products
15.16.3 F ( a , b c ; z ) F ( a , b c ; ζ ) = s = 0 ( a ) s ( b ) s ( c - a ) s ( c - b ) s ( c ) s ( c ) 2 s s ! ( z ζ ) s F ( a + s , b + s c + 2 s ; z + ζ - z ζ ) , | z | < 1 , | ζ | < 1 , | z + ζ - z ζ | < 1 .