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11: 35.10 Methods of Computation
Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …
12: David M. Bressoud
His books are Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, published in Memoirs of the American Mathematical Society 24, No. …
13: 1.1 Special Notation
x , y real variables.
𝐈 identity matrix
14: 4.8 Identities
§4.8 Identities
§4.8(i) Logarithms
§4.8(ii) Powers
15: 20.11 Generalizations and Analogs
This is the discrete analog of the Poisson identity1.8(iv)). … In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … Similar identities can be constructed for F 1 2 ( 1 3 , 2 3 ; 1 ; k 2 ) , F 1 2 ( 1 4 , 3 4 ; 1 ; k 2 ) , and F 1 2 ( 1 6 , 5 6 ; 1 ; k 2 ) . …
16: 21.7 Riemann Surfaces
§21.7(ii) Fay’s Trisecant Identity
where again all integration paths are identical for all components. Generalizations of this identity are given in Fay (1973, Chapter 2). …
§21.7(iii) Frobenius’ Identity
17: 16.23 Mathematical Applications
Many combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. In Petkovšek et al. (1996) tools are given for automated proofs of these identities.
18: 21.6 Products
§21.6(i) Riemann Identity
Then …This is the Riemann identity. On using theta functions with characteristics, it becomes …Many identities involving products of theta functions can be established using these formulas. …
19: 25.10 Zeros
25.10.1 Z ( t ) exp ( i ϑ ( t ) ) ζ ( 1 2 + i t ) ,
25.10.2 ϑ ( t ) ph Γ ( 1 4 + 1 2 i t ) 1 2 t ln π
20: 27.8 Dirichlet Characters
27.8.6 r = 1 ϕ ( k ) χ r ( m ) χ ¯ r ( n ) = { ϕ ( k ) , m n ( mod k ) , 0 , otherwise .
A Dirichlet character χ ( mod k ) is called primitive (mod k ) if for every proper divisor d of k (that is, a divisor d < k ), there exists an integer a 1 ( mod d ) , with ( a , k ) = 1 and χ ( a ) 1 . …
27.8.7 χ ( a ) = 1  for all  a 1  (mod  d ) , ( a , k ) = 1 .