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1: 18.3 Definitions
§18.3 Definitions
β–Ί
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
β–Ί β–Ίβ–Ίβ–Ίβ–Ί
Name p n ⁑ ( x ) ( a , b ) w ⁑ ( x ) h n k n k ~ n / k n Constraints
Chebyshev of second kind U n ⁑ ( x ) ( 1 , 1 ) ( 1 x 2 ) 1 2 1 2 ⁒ Ο€ 2 n 0
Shifted Chebyshev of second kind U n * ⁑ ( x ) ( 0 , 1 ) ( x x 2 ) 1 2 1 8 ⁒ Ο€ 2 2 ⁒ n 1 2 ⁒ n
β–Ί
β–ΊIn addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials T n ⁑ ( x ) , n = 0 , 1 , , N , are orthogonal on the discrete point set comprising the zeros x N + 1 , n , n = 1 , 2 , , N + 1 , of T N + 1 ⁑ ( x ) : … β–ΊFor another version of the discrete orthogonality property of the polynomials T n ⁑ ( x ) see (3.11.9). …
2: 18.41 Tables
β–ΊFor P n ⁑ ( x ) ( = 𝖯 n ⁑ ( x ) ) see §14.33. β–ΊAbramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates T n ⁑ ( x ) , U n ⁑ ( x ) , L n ⁑ ( x ) , and H n ⁑ ( x ) for n = 0 ⁒ ( 1 ) ⁒ 12 . The ranges of x are 0.2 ⁒ ( .2 ) ⁒ 1 for T n ⁑ ( x ) and U n ⁑ ( x ) , and 0.5 , 1 , 3 , 5 , 10 for L n ⁑ ( x ) and H n ⁑ ( x ) . …
3: 18.9 Recurrence Relations and Derivatives
β–Ί
Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).
β–Ί β–Ίβ–Ίβ–Ίβ–Ί
p n ⁑ ( x ) A n B n C n
U n ⁑ ( x ) 2 0 1
U n * ⁑ ( x ) 4 2 1
β–Ί
β–Ί
18.9.9 T n ⁑ ( x ) = 1 2 ⁒ ( U n ⁑ ( x ) U n 2 ⁑ ( x ) ) ,
β–Ί
18.9.11 V n ⁑ ( x ) + V n 1 ⁑ ( x ) = 2 ⁒ T n ⁑ ( x ) ,
β–Ί
18.9.12 T n + 1 ⁑ ( x ) + T n ⁑ ( x ) = ( 1 + x ) ⁒ V n ⁑ ( x ) .
4: 18.7 Interrelations and Limit Relations
β–Ί
Chebyshev, Ultraspherical, and Jacobi
β–Ί
18.7.7 T n * ⁑ ( x ) = T n ⁑ ( 2 ⁒ x 1 ) ,
β–Ί
18.7.8 U n * ⁑ ( x ) = U n ⁑ ( 2 ⁒ x 1 ) .
β–Ί
18.7.17 U 2 ⁒ n ⁑ ( x ) = W n ⁑ ( 2 ⁒ x 2 1 ) ,
β–Ί
18.7.18 T 2 ⁒ n + 1 ⁑ ( x ) = x ⁒ V n ⁑ ( 2 ⁒ x 2 1 ) .
5: 18.5 Explicit Representations
β–Ί
T 0 ⁑ ( x ) = 1 ,
β–Ί
T 1 ⁑ ( x ) = x ,
β–Ί
T 2 ⁑ ( x ) = 2 ⁒ x 2 1 ,
β–Ί
T 3 ⁑ ( x ) = 4 ⁒ x 3 3 ⁒ x ,
β–Ί
U 0 ⁑ ( x ) = 1 ,
6: 18.1 Notation
β–Ί
  • Chebyshev of first, second, third, and fourth kinds: T n ⁑ ( x ) , U n ⁑ ( x ) , V n ⁑ ( x ) , W n ⁑ ( x ) .

  • β–Ί
  • Shifted Chebyshev of first and second kinds: T n * ⁑ ( x ) , U n * ⁑ ( x ) .

  • β–ΊNor do we consider the shifted Jacobi polynomials: … β–Ί
    C n ⁑ ( x ) = 2 ⁒ T n ⁑ ( 1 2 ⁒ x ) ,
    β–Ί
    S n ⁑ ( x ) = U n ⁑ ( 1 2 ⁒ x ) .
    7: 18.6 Symmetry, Special Values, and Limits to Monomials
    β–Ί
    Table 18.6.1: Classical OP’s: symmetry and special values.
    β–Ί β–Ίβ–Ίβ–Ίβ–Ίβ–Ίβ–Ί
    p n ⁑ ( x ) p n ⁑ ( x ) p n ⁑ ( 1 ) p 2 ⁒ n ⁑ ( 0 ) p 2 ⁒ n + 1 ⁑ ( 0 )
    T n ⁑ ( x ) ( 1 ) n ⁒ T n ⁑ ( x ) 1 ( 1 ) n ( 1 ) n ⁒ ( 2 ⁒ n + 1 )
    U n ⁑ ( x ) ( 1 ) n ⁒ U n ⁑ ( x ) n + 1 ( 1 ) n ( 1 ) n ⁒ ( 2 ⁒ n + 2 )
    V n ⁑ ( x ) ( 1 ) n ⁒ W n ⁑ ( x ) 1 ( 1 ) n ( 1 ) n ⁒ ( 2 ⁒ n + 2 )
    W n ⁑ ( x ) ( 1 ) n ⁒ V n ⁑ ( x ) 2 ⁒ n + 1 ( 1 ) n ( 1 ) n ⁒ ( 2 ⁒ n + 2 )
    β–Ί
    8: 18.13 Continued Fractions
    β–Ί
    Chebyshev
    β–Ί T n ⁑ ( x ) is the denominator of the n th approximant to: …and U n ⁑ ( x ) is the denominator of the n th approximant to: …
    9: 29.15 Fourier Series and Chebyshev Series
    β–Ί
    §29.15(ii) Chebyshev Series
    β–ΊThe Chebyshev polynomial T of the first kind (§18.3) satisfies cos ⁑ ( p ⁒ Ο• ) = T p ⁑ ( cos ⁑ Ο• ) . … β–Ί β–ΊUsing also sin ⁑ ( ( p + 1 ) ⁒ Ο• ) = ( sin ⁑ Ο• ) ⁒ U p ⁑ ( cos ⁑ Ο• ) , with U denoting the Chebyshev polynomial of the second kind (§18.3), we obtain β–Ί
    10: 18.18 Sums
    β–Ί
    Chebyshev
    β–Ί
    18.18.21 T m ⁑ ( x ) ⁒ T n ⁑ ( x ) = 1 2 ⁒ ( T m + n ⁑ ( x ) + T m n ⁑ ( x ) ) .
    β–Ί
    18.18.32 2 ⁒ β„“ = 0 n T 2 ⁒ β„“ ⁑ ( x ) = 1 + U 2 ⁒ n ⁑ ( x ) ,
    β–Ί
    18.18.33 2 ⁒ β„“ = 0 n T 2 ⁒ β„“ + 1 ⁑ ( x ) = U 2 ⁒ n + 1 ⁑ ( x ) .
    β–Ί
    18.18.34 2 ⁒ ( 1 x 2 ) ⁒ β„“ = 0 n U 2 ⁒ β„“ ⁑ ( x ) = 1 T 2 ⁒ n + 2 ⁑ ( x ) ,