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1: 18.37 Classical OP’s in Two or More Variables
18.37.2 x 2 + y 2 < 1 R m , n ( α ) ( x + i y ) R j , ( α ) ( x i y ) ( 1 x 2 y 2 ) α d x d y = 0 , m j and/or n .
The following three conditions, taken together, determine R m , n ( α ) ( z ) uniquely: …
18.37.4 x 2 + y 2 < 1 R m , n ( α ) ( x + i y ) ( x i y ) m j ( x + i y ) n j ( 1 x 2 y 2 ) α d x d y = 0 , j = 1 , 2 , , min ( m , n ) ;
18.37.5 R m , n ( α ) ( 1 ) = 1 .
18.37.8 0 < y < x < 1 P m , n α , β , γ ( x , y ) P j , α , β , γ ( x , y ) ( 1 x ) α ( x y ) β y γ d x d y = 0 , m j and/or n .
2: René F. Swarttouw
Swarttouw is mainly a teacher of mathematics and has published a few papers on special functions and orthogonal polynomials. He is coauthor of the book Hypergeometric Orthogonal Polynomials and Their q -AnaloguesHypergeometric Orthogonal Polynomials and Their q -Analogues. …
  • 3: 18 Orthogonal Polynomials
    Chapter 18 Orthogonal Polynomials
    4: 16.7 Relations to Other Functions
    §16.7 Relations to Other Functions
    For orthogonal polynomials see Chapter 18. …
    5: Roelof Koekoek
    Koekoek is mainly a teacher of mathematics and has published a few papers on orthogonal polynomials. He is also author of the book Hypergeometric Orthogonal Polynomials and Their q -Analogues (with P. …
  • 6: 18.1 Notation
    x , y , t real variables.
    OP’s orthogonal polynomials.
    EOP’s exceptional orthogonal polynomials.
    Hahn Class OP’s
  • Disk: R m , n ( α ) ( z ) .

  • Triangle: P m , n α , β , γ ( x , y ) .

  • 7: 32.15 Orthogonal Polynomials
    §32.15 Orthogonal Polynomials
    8: Wolter Groenevelt
    Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …
    9: 18.21 Hahn Class: Interrelations
    §18.21 Hahn Class: Interrelations
    §18.21(i) Dualities
    §18.21(ii) Limit Relations and Special Cases
    Hahn Jacobi
    See accompanying text
    Figure 18.21.1: Askey scheme. … Magnify
    10: 18.3 Definitions
    §18.3 Definitions
  • 2.

    With the property that { p n + 1 ( x ) } n = 0 is again a system of OP’s. See §18.9(iii).

  • 3.

    As given by a Rodrigues formula (18.5.5).

  • Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …
    Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
    For explicit power series coefficients up to n = 12 for these polynomials and for coefficients up to n = 6 for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …