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Cauchy determinant

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1: 1.3 Determinants, Linear Operators, and Spectral Expansions
§1.3(i) Determinants: Elementary Properties
Relationships Between Determinants
§1.3(ii) Special Determinants
Vandermonde Determinant or Vandermondian
Cauchy Determinant
2: 1.2 Elementary Algebra
which for p = q = 2 is the Cauchy-Schwartz inequality
The Determinant
The matrix 𝐀 has a determinant, det ( 𝐀 ) , explored further in §1.3, denoted, in full index form, as …where det ( 𝐀 ) is defined by the Leibniz formula …
Relation of Eigenvalues to the Determinant and Trace
3: 1.7 Inequalities
Cauchy–Schwarz Inequality
Cauchy–Schwarz Inequality
4: 24.14 Sums
These identities can be regarded as higher-order recurrences. Let det [ a r + s ] denote a Hankel (or persymmetric) determinant, that is, an ( n + 1 ) × ( n + 1 ) determinant with element a r + s in row r and column s for r , s = 0 , 1 , , n . …
24.14.11 det [ B r + s ] = ( 1 ) n ( n + 1 ) / 2 ( k = 1 n k ! ) 6 / ( k = 1 2 n + 1 k ! ) ,
24.14.12 det [ E r + s ] = ( 1 ) n ( n + 1 ) / 2 ( k = 1 n k ! ) 2 .
5: 21.5 Modular Transformations
21.5.3 det 𝚪 = 1 ,
21.5.4 θ ( [ [ 𝐂 𝛀 + 𝐃 ] 1 ] T 𝐳 | [ 𝐀 𝛀 + 𝐁 ] [ 𝐂 𝛀 + 𝐃 ] 1 ) = ξ ( 𝚪 ) det [ 𝐂 𝛀 + 𝐃 ] e π i 𝐳 [ [ 𝐂 𝛀 + 𝐃 ] 1 𝐂 ] 𝐳 θ ( 𝐳 | 𝛀 ) .
θ ( 𝛀 1 𝐳 | 𝛀 1 ) = det [ i 𝛀 ] e π i 𝐳 𝛀 1 𝐳 θ ( 𝐳 | 𝛀 ) ,
21.5.9 θ [ 𝐃 𝜶 𝐂 𝜷 + 1 2 diag [ 𝐂 𝐃 T ] 𝐁 𝜶 + 𝐀 𝜷 + 1 2 diag [ 𝐀 𝐁 T ] ] ( [ [ 𝐂 𝛀 + 𝐃 ] 1 ] T 𝐳 | [ 𝐀 𝛀 + 𝐁 ] [ 𝐂 𝛀 + 𝐃 ] 1 ) = κ ( 𝜶 , 𝜷 , 𝚪 ) det [ 𝐂 𝛀 + 𝐃 ] e π i 𝐳 [ [ 𝐂 𝛀 + 𝐃 ] 1 𝐂 ] 𝐳 θ [ 𝜶 𝜷 ] ( 𝐳 | 𝛀 ) ,
6: 1.1 Special Notation
x , y real variables.
det ( 𝐀 ) determinant of the square matrix 𝐀
7: 19.31 Probability Distributions
19.31.2 n ( 𝐱 T 𝐀 𝐱 ) μ exp ( 𝐱 T 𝐁 𝐱 ) d x 1 d x n = π n / 2 Γ ( μ + 1 2 n ) det 𝐁 Γ ( 1 2 n ) R μ ( 1 2 , , 1 2 ; λ 1 , , λ n ) , μ > 1 2 n .
8: 3.9 Acceleration of Convergence
3.9.9 t n , 2 k = H k + 1 ( s n ) H k ( Δ 2 s n ) , n = 0 , 1 , 2 , ,
where H m is the Hankel determinant
3.9.10 H m ( u n ) = | u n u n + 1 u n + m 1 u n + 1 u n + 2 u n + m u n + m 1 u n + m u n + 2 m 2 | .
The ratio of the Hankel determinants in (3.9.9) can be computed recursively by Wynn’s epsilon algorithm: …
9: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions
Cauchy’s Sum
10: 32.8 Rational Solutions
P II P VI  possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. … where τ n ( z ) is the n × n Wronskian determinantFor determinantal representations see Kajiwara and Masuda (1999). … For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999). … For determinantal representations see Masuda et al. (2002). …