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1: 32.16 Physical Applications
§32.16 Physical Applications
Integrable Continuous Dynamical Systems
2: 1.4 Calculus of One Variable
§1.4(ii) Continuity
For an example, see Figure 1.4.1If ϕ ( x ) is continuous or piecewise continuous, then …
3: 18.30 Associated OP’s
In the recurrence relation (18.2.8) assume that the coefficients A n , B n , and C n + 1 are defined when n is a continuous nonnegative real variable, and let c be an arbitrary positive constant. …
18.30.1 A n A n + 1 C n + 1 > 0 , n 0 .
18.30.3 p n + 1 ( x ; c ) = ( A n + c x + B n + c ) p n ( x ; c ) - C n + c p n - 1 ( x ; c ) , n = 0 , 1 , .
18.30.4 P n ( α , β ) ( x ; c ) = p n ( x ; c ) , n = 0 , 1 , ,
18.30.7 P n ( x ; c ) = = 0 n c + c P ( x ) P n - ( x ) .
4: About Color Map
Continuous Phase Mapping
For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. …
Figure 3: Continuous phase mapping
5: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
Ismail (1986) gives asymptotic expansions as n , with x and other parameters fixed, for continuous q -ultraspherical, big and little q -Jacobi, and Askey–Wilson polynomials. … For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). For asymptotic approximations to the largest zeros of the q -Laguerre and continuous q - 1 -Hermite polynomials see Chen and Ismail (1998).
6: 18.1 Notation
  • Continuous Hahn: p n ( x ; a , b , a ¯ , b ¯ ) .

  • Continuous Dual Hahn: S n ( x ; a , b , c ) .

  • Continuous q -Ultraspherical: C n ( x ; β | q ) .

  • Continuous q -Hermite: H n ( x | q ) .

  • Continuous q - 1 -Hermite: h n ( x | q )

  • 7: 18.26 Wilson Class: Continued
    Wilson Continuous Dual Hahn
    Wilson Continuous Hahn
    Continuous Dual Hahn Meixner–Pollaczek
    Continuous Dual Hahn
    Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …
    8: 18.28 Askey–Wilson Class
    §18.28(v) Continuous q -Ultraspherical Polynomials
    §18.28(vi) Continuous q -Hermite Polynomials
    §18.28(vii) Continuous q - 1 -Hermite Polynomials
    18.28.18 h n ( sinh t | q ) = = 0 n q 1 2 ( + 1 ) ( q - n ; q ) ( q ; q ) e ( n - 2 ) t = e n t ϕ 1 1 ( q - n 0 ; q , - q e - 2 t ) = i - n H n ( i sinh t | q - 1 ) .
    For continuous q - 1 -Hermite polynomials the orthogonality measure is not unique. …
    9: 18.25 Wilson Class: Definitions
    Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) . …
    §18.25(ii) Weights and Normalizations: Continuous Cases
    Continuous Dual Hahn
    18.25.6 p n ( x ) = S n ( x ; a 1 , a 2 , a 3 ) ,
    Table 18.25.2 provides the leading coefficients k n 18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …
    10: 18.21 Hahn Class: Interrelations
    Continuous Hahn Meixner–Pollaczek
    18.21.10 lim t t - n p n ( x - t ; λ + i t , - t tan ϕ , λ - i t , - t tan ϕ ) = ( - 1 ) n ( cos ϕ ) n P n ( λ ) ( x ; ϕ ) .
    See accompanying text
    Figure 18.21.1: Askey scheme. …(This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify