of one variable
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1: Mourad E. H. Ismail
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►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).
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2: 21.8 Abelian Functions
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►In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable.
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3: 37.20 Mathematical Applications
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►For the unit ball and the simplex, these quantities can be written as an one-variable integral involving the Jacobi polynomials.
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4: Bibliography I
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Classical and Quantum Orthogonal Polynomials in One Variable.
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
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Classical and Quantum Orthogonal Polynomials in One Variable.
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
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5: 37.10 Other Orthogonal Polynomials of Two Variables
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§37.10(ii) Orthogonal Polynomials on an Annulus
… ►See §18.32 for Bernstein–Szegő polynomials of one variable. … ►
37.10.7
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6: 37.3 Triangular Region with Weight Function
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§37.3(i) Orthogonal Decomposition
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37.3.1
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►involving one-variable Jacobi polynomials (see Table 18.3.1), are the case , of (37.2.16).
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37.3.22
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7: 1.4 Calculus of One Variable
§1.4 Calculus of One Variable
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1.4.1
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§1.4(vi) Taylor’s Theorem for Real Variables
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8: 37.19 Other Orthogonal Polynomials of Variables
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►These are orthogonal polynomials for an inner product that involves derivatives of functions; see Marcellán and Xu (2015) for the one-variable case.
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►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
variables.
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►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 -ultraspherical, or Askey–Wilson polynomials.
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9: 37.2 General Orthogonal Polynomials of Two Variables
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►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.
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►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)).
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►where is the OP of degree for the weight function
on an interval in .
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►Also, let be a system of OPs on
with respect to the weight function , and put
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