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1: 37.2 General Orthogonal Polynomials of Two Variables
Then … Then necessarily P k , n , Q k , n W 0 ( k = 0 , 1 , , n ). …
§37.2(v) Product Weight Functions
§37.2(vii) Rotation Invariant Weight Functions
2: 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). …
18.32.2 w ( x ) = | x | α exp ( Q ( x ) ) , x ,  α > 1 ,
3: 37.10 Other Orthogonal Polynomials of Two Variables
For the weight functionThe Bernstein–Szegő weight function is defined by …One example of the weight function is …The OPs for these weight functions can be constructed explicitly and they are studied in Delgado et al. (2009). …
4: 37.19 Other Orthogonal Polynomials of d Variables
These are orthogonal polynomials for a family of reflection invariant weight functions on the unit sphere. …The weight function is invariant under the reflection group G . …
§37.19(ii) OPs on the Ball for Generalized Weight Functions
§37.19(iii) OPs on d for Generalized Weight Functions
Let w κ be the weight function (37.19.5). …
5: 37.13 General Orthogonal Polynomials of d Variables
Define an inner product
§37.13(i) OPs for a Rotation Invariant Weight Function
For each k 0 let p n ( k ) ( x ) denote an OP of degree n for the weight function x k w ( x ) on [ 0 , ) . The space 𝒱 n d for the rotation invariant weight function W can be orthogonally decomposed as … and weight function
6: 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. …
7: null
error generating summary
8: 37.5 Quarter Plane with Weight Function x α y β e x y
§37.5 Quarter Plane with Weight Function x α y β e x y
define the weight functionDefine Laguerre–Jacobi polynomials on the quarter plane by …
§37.5(ii) Differential Equations
§37.5(iii) Limits
9: 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 . …
10: 37.7 Parabolic Biangular Region with Weight Function ( 1 x ) α ( x y 2 ) β
§37.7 Parabolic Biangular Region with Weight Function ( 1 x ) α ( x y 2 ) β
bounded by a parabolic arc and a line segment, define the weight function
§37.7(ii) Quadratic Transformations
§37.7(iii) Differential Equations
The polynomials (37.3.9) and (37.7.16) are related by the quadratic transformations …