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1: 18.37 Classical OP’s in Two or More Variables
18.37.1 R m , n ( α ) ( r e i θ ) = e i ( m - n ) θ r | m - n | P min ( m , n ) ( α , | m - n | ) ( 2 r 2 - 1 ) P min ( m , n ) ( α , | m - n | ) ( 1 ) , r 0 , θ , α > - 1 .
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: …
18.37.5 R m , n ( α ) ( 1 ) = 1 .
18.37.7 P m , n α , β , γ ( x , y ) = P m - n ( α , β + γ + 2 n + 1 ) ( 2 x - 1 ) x n P n ( β , γ ) ( 2 x - 1 y - 1 ) , m n 0 , α , β , γ > - 1 .
2: Roelof Koekoek
Koekoek is mainly a teacher of mathematics and has published a few papers on orthogonal polynomials. He is coauthor of the book Hypergeometric Orthogonal Polynomials and Their q -Analogues). …
  • 3: 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. …
  • 4: 16.7 Relations to Other Functions
    §16.7 Relations to Other Functions
    For orthogonal polynomials see Chapter 18. …
    5: 18.40 Methods of Computation
    Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). … For further information see Clenshaw (1955), Gautschi (2004, §§2.1, 8.1), and Mason and Handscomb (2003, §2.4). …
    6: 18.38 Mathematical Applications
    Quadrature
    Integrable Systems
    Riemann–Hilbert Problems
    Radon Transform
    Group Representations
    7: 18.1 Notation
    x , y real variables.
    OP’s orthogonal polynomials.
    Hahn Class OP’s
    Wilson Class OP’s
  • Disk: R m , n ( α ) ( z ) .

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

  • 8: 32.15 Orthogonal Polynomials
    §32.15 Orthogonal Polynomials
    9: 18 Orthogonal Polynomials
    Chapter 18 Orthogonal Polynomials
    10: 18.39 Physical Applications
    §18.39 Physical Applications
    §18.39(i) Quantum Mechanics
    The corresponding eigenfunctions are … For physical applications of q -Laguerre polynomials see §17.17. …