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1: Mourad E. H. Ismail
His well-known book Classical and Quantum Orthogonal Polynomials in One Variable was published by Cambridge University Press in 2005 and reprinted with corrections in paperback in Ismail (2009). …
2: 21.8 Abelian Functions
In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …
3: 37.20 Mathematical Applications
For the unit ball and the simplex, these quantities can be written as an one-variable integral involving the Jacobi polynomials. …
4: Bibliography I
  • M. E. H. Ismail (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
  • M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
  • 5: 37.10 Other Orthogonal Polynomials of Two Variables
    §37.10(ii) Orthogonal Polynomials on an Annulus
    See §18.32 for Bernstein–Szegő polynomials of one variable. …
    37.10.7 lim N Q k , n ( N x , N y ; α , β , γ , N ) = P k , n α , β , γ ( x , y ) P k , n α , β , γ ( 0 , 0 ) .
    6: 37.3 Triangular Region with Weight Function x α y β ( 1 x y ) γ
    §37.3(i) Orthogonal Decomposition
    37.3.1 W α , β , γ ( x , y ) = x α y β ( 1 x y ) γ
    involving one-variable Jacobi polynomials P n ( α , β ) ( x ) (see Table 18.3.1), are the case w 1 ( x ) = x α ( 1 x ) β + γ , w 2 ( x ) = x β ( 1 x ) γ of (37.2.16). …
    37.3.22 D y ( W α , β + 1 , γ + 1 ( x , y ) P k 1 , n 1 α , β + 1 , γ + 1 ( x , y ) ) = k W α , β , γ ( x , y ) P k , n α , β , γ ( x , y ) ,
    7: 1.4 Calculus of One Variable
    §1.4 Calculus of One Variable
    1.4.1 f ( c + ) lim x c + f ( x ) = f ( c ) ,
    §1.4(vi) Taylor’s Theorem for Real Variables
    See accompanying text
    Figure 1.4.2: Convex function f ( x ) . … Magnify
    8: 37.19 Other Orthogonal Polynomials of d Variables
    These are orthogonal polynomials for an inner product that involves derivatives of functions; see Marcellán and Xu (2015) for the one-variable case. … Just as the classical OPs fit into the Askey scheme (see §18.19 and Figure 18.21.1) with Wilson and Racah polynomials on top, the Jacobi polynomials on the simplex fit into a scheme of OPs defined as products of one-variable OPs belonging to the Askey scheme by formulas somewhat resembling (37.14.7). However, when the one-variable OPs are taken from a higher level in the Askey scheme, the analogues of the denominators in the arguments in (37.14.7) will be parameters depending on x variables. … Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials of several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. In one variable they are essentially ultraspherical, Jacobi, continuous q -ultraspherical, or Askey–Wilson polynomials. …
    9: 37.2 General Orthogonal Polynomials of Two Variables
    In the other direction, as an analogue of Favard’s theorem (see §18.2(viii) for the one-variable case), any polynomial system that satisfies the three-term relations (37.2.7), together with the conditions (37.2.10) and (37.2.8) of the coefficient matrices, must be orthonormal with respect to a positive definite linear functional. … If the equality holds in (37.2.13), then the common zeros are nodes of Gaussian cubature rules that are a complete analogue of Gaussian quadrature in one variable (see §3.5(v)). … where p k ( w , x ) is the OP of degree k for the weight function w ( x ) on an interval in . … Also, let { q n ( x ) } n = 0 , 1 , 2 , be a system of OPs on ( 0 , 1 ) with respect to the weight function w 2 ( x ) , and put …
    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
    37.7.8 P k , n 1 2 , γ ( 1 x 2 , y ) = ( 1 ) n k ( γ + 1 ) k ( 1 2 ) n k ( 2 γ + 1 ) k ( γ + k + 1 ) n k C k , 2 n k ( γ + 1 2 ) ( x , y ) ,
    37.7.19 y R k , n α , 1 2 , γ ( 1 x , y 2 ) = ( 1 ) n ( 1 + k ) k + 1 ( α + 1 + k ) k + 1 R 2 k + 1 , n + k + 1 α , γ ( x , y ) .
    For the Laguerre and Hermite polynomials in one variable see Table 18.3.1. …
    37.7.23 x D x x + 1 2 D y y + ( α + 1 x ) D x y D y .