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1: 18.32 OP’s with Respect to Freud Weights
§18.32 OP’s with Respect to Freud Weights
A Freud weight is a weight function of the form
18.32.1 w ( x ) = exp ( - Q ( x ) ) , - < x < ,
For asymptotic approximations to OP’s that correspond to Freud weights with more general functions Q ( x ) see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999).
2: 12.15 Generalized Parabolic Cylinder Functions
This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …
3: 3.5 Quadrature
An interpolatory quadrature rule
Gauss Formula for a Logarithmic Weight Function
Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.
x k w k
Table 3.5.18: Nodes and weights for the 5-point complex Gauss quadrature formula with s = 1 .
ζ k w k
Table 3.5.21: Cubature formulas for disk and square.
Diagram ( x j , y j ) w j R
4: 18.31 Bernstein–Szegő Polynomials
The Bernstein–Szegő polynomials { p n ( x ) } , n = 0 , 1 , , are orthogonal on ( - 1 , 1 ) with respect to three types of weight function: ( 1 - x 2 ) - 1 2 ( ρ ( x ) ) - 1 , ( 1 - x 2 ) 1 2 ( ρ ( x ) ) - 1 , ( 1 - x ) 1 2 ( 1 + x ) - 1 2 ( ρ ( x ) ) - 1 . …
5: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
6: 18.2 General Orthogonal Polynomials
A system (or set) of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on ( a , b ) with respect to the weight function w ( x ) ( 0 ) if
18.2.1 a b p n ( x ) p m ( x ) w ( x ) d x = 0 , n m .
18.2.5 h n = a b ( p n ( x ) ) 2 w ( x ) d x  or  x X ( p n ( x ) ) 2 w x ,
18.2.6 h ~ n = a b x ( p n ( x ) ) 2 w ( x ) d x  or  x X x ( p n ( x ) ) 2 w x ,
7: 18.36 Miscellaneous Polynomials
These are OP’s on the interval ( - 1 , 1 ) with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at - 1 and 1 to the weight function for the Jacobi polynomials. …
8: 18.19 Hahn Class: Definitions
Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.
p n ( x ) X w x h n
18.19.2 w ( z ; a , b , a ¯ , b ¯ ) = Γ ( a + i z ) Γ ( b + i z ) Γ ( a ¯ - i z ) Γ ( b ¯ - i z ) ,
18.19.3 w ( x ) = w ( x ; a , b , a ¯ , b ¯ ) = | Γ ( a + i x ) Γ ( b + i x ) | 2 ,
18.19.7 w ( λ ) ( z ; ϕ ) = Γ ( λ + i z ) Γ ( λ - i z ) e ( 2 ϕ - π ) z ,
9: 18.33 Polynomials Orthogonal on the Unit Circle
A system of polynomials { ϕ n ( z ) } , n = 0 , 1 , , where ϕ n ( z ) is of proper degree n , is orthonormal on the unit circle with respect to the weight function w ( z ) ( 0 ) if Let { p n ( x ) } and { q n ( x ) } , n = 0 , 1 , , be OP’s with weight functions w 1 ( x ) and w 2 ( x ) , respectively, on ( - 1 , 1 ) . …
18.33.16 w ( z ) = | ( q z ; q 2 ) / ( a q z ; q 2 ) | 2 , a 2 q 2 < 1 .
10: 18.25 Wilson Class: Definitions
§18.25(ii) Weights and Normalizations: Continuous Cases
18.25.2 0 p n ( x ) p m ( x ) w ( x ) d x = h n δ n , m .
18.25.4 w ( y 2 ) = 1 2 y | j Γ ( a j + i y ) Γ ( 2 i y ) | 2 ,
18.25.7 w ( y 2 ) = 1 2 y | j Γ ( a j + i y ) Γ ( 2 i y ) | 2 ,
18.25.15 h n = n ! ( N - n ) ! ( γ + δ + 2 ) N N ! ( γ + 1 ) n ( δ + 1 ) N - n .