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11: 26.12 Plane Partitions
26.12.9 ( h = 1 r j = 1 s h + j + t 1 h + j 1 ) 2 ;
26.12.10 ( h = 1 r j = 1 s h + j + t 1 h + j 1 ) ( h = 1 r + 1 j = 1 s h + j + t 1 h + j 1 ) ;
26.12.11 ( h = 1 r + 1 j = 1 s h + j + t 1 h + j 1 ) ( h = 1 r j = 1 s + 1 h + j + t 1 h + j 1 ) .
26.12.13 h = 1 r j = 1 r h + j + t 1 h + j 1 ;
26.12.14 h = 1 r j = 1 r + 1 h + j + t 1 h + j 1 .
12: 1.2 Elementary Algebra
§1.2(v) Matrices, Vectors, Scalar Products, and Norms
The transpose of the product is … Column vectors 𝐮 and 𝐯 of the same length n have a scalar product …The dot product notation 𝐮 𝐯 is reserved for the physical three-dimensional vectors of (1.6.2). The scalar product has properties …
13: 5.8 Infinite Products
§5.8 Infinite Products
5.8.2 1 Γ ( z ) = z e γ z k = 1 ( 1 + z k ) e z / k ,
5.8.3 | Γ ( x ) Γ ( x + i y ) | 2 = k = 0 ( 1 + y 2 ( x + k ) 2 ) , x 0 , 1 , .
5.8.5 k = 0 ( a 1 + k ) ( a 2 + k ) ( a m + k ) ( b 1 + k ) ( b 2 + k ) ( b m + k ) = Γ ( b 1 ) Γ ( b 2 ) Γ ( b m ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a m ) ,
14: 1.1 Special Notation
x , y real variables.
f , g inner, or scalar, product for real or complex vectors or functions.
15: 5.14 Multidimensional Integrals
5.14.2 V n ( 1 k = 1 n t k ) z n + 1 1 k = 1 n t k z k 1 d t k = Γ ( z 1 ) Γ ( z 2 ) Γ ( z n + 1 ) Γ ( z 1 + z 2 + + z n + 1 ) .
5.14.3 Δ ( t 1 , t 2 , , t n ) = 1 j < k n ( t j t k ) .
5.14.4 [ 0 , 1 ] n t 1 t 2 t m | Δ ( t 1 , , t n ) | 2 c k = 1 n t k a 1 ( 1 t k ) b 1 d t k = 1 ( Γ ( 1 + c ) ) n k = 1 m a + ( n k ) c a + b + ( 2 n k 1 ) c k = 1 n Γ ( a + ( n k ) c ) Γ ( b + ( n k ) c ) Γ ( 1 + k c ) Γ ( a + b + ( 2 n k 1 ) c ) ,
5.14.5 [ 0 , ) n t 1 t 2 t m | Δ ( t 1 , , t n ) | 2 c k = 1 n t k a 1 e t k d t k = k = 1 m ( a + ( n k ) c ) k = 1 n Γ ( a + ( n k ) c ) Γ ( 1 + k c ) ( Γ ( 1 + c ) ) n ,
5.14.6 1 ( 2 π ) n / 2 ( , ) n | Δ ( t 1 , , t n ) | 2 c k = 1 n exp ( 1 2 t k 2 ) d t k = k = 1 n Γ ( 1 + k c ) ( Γ ( 1 + c ) ) n , c > 1 / n .
16: 21.8 Abelian Functions
For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
17: 27.1 Special Notation
d , k , m , n positive integers (unless otherwise indicated).
d | n , d | n sum, product taken over divisors of n .
p , p sum, product extended over all primes.
18: 10.6 Recurrence Relations and Derivatives
§10.6(iii) Cross-Products
10.6.10 p ν s ν q ν r ν = 4 / ( π 2 a b ) .
19: 9.11 Products
§9.11 Products
§9.11(i) Differential Equation
§9.11(ii) Wronskian
§9.11(iv) Indefinite Integrals
§9.11(v) Definite Integrals
20: 23.17 Elementary Properties
§23.17(iii) Infinite Products
23.17.7 λ ( τ ) = 16 q n = 1 ( 1 + q 2 n 1 + q 2 n 1 ) 8 ,
23.17.8 η ( τ ) = q 1 / 12 n = 1 ( 1 q 2 n ) ,