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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: 16.7 Relations to Other Functions
    §16.7 Relations to Other Functions
    For orthogonal polynomials see Chapter 18. …
    4: 18.40 Methods of Computation
    Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). … However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …
    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: 18.38 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. …