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1: 18.3 Definitions
§18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
Shifted Chebyshev of second kind U n * ( x ) ( 0 , 1 ) ( x - x 2 ) 1 2 1 8 π 2 2 n - 1 2 n
In 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 ) : … Note that in this reference the definitions of the Chebyshev polynomials of the third and fourth kinds V n ( x ) and W n ( x ) are the converse of the definitions in this chapter. For another version of the discrete orthogonality property of the polynomials T n ( x ) see (3.11.9). …
2: 18.41 Tables
For P n ( x ) ( = P n ( x ) ) see §14.33. Abramowitz 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: 18.1 Notation
  • Chebyshev of first, second, third, and fourth kinds: T n ( x ) , U n ( x ) , V n ( x ) , W n ( x ) .

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

  • C n ( x ) = 2 T n ( 1 2 x ) ,
    S n ( x ) = U n ( 1 2 x ) .
    In Mason and Handscomb (2003), the definitions of the Chebyshev polynomials of the third and fourth kinds V n ( x ) and W n ( x ) are the converse of the definitions in this chapter.
    4: 18.9 Recurrence Relations and Derivatives
    Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).
    p n ( x ) A n B n C n
    U n ( x ) 2 0 1
    18.9.9 T n ( x ) = 1 2 ( U n ( x ) - U n - 2 ( x ) ) ,
    18.9.10 ( 1 - x 2 ) U n ( x ) = - 1 2 ( T n + 2 ( x ) - T n ( x ) ) .
    18.9.11 W n ( x ) + W n - 1 ( x ) = 2 T n ( x ) ,
    18.9.12 T n + 1 ( x ) + T n ( x ) = ( 1 + x ) W n ( x ) .
    5: 18.6 Symmetry, Special Values, and Limits to Monomials
    Table 18.6.1: Classical OP’s: symmetry and special values.
    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 ) 2 n + 1 ( - 1 ) n ( - 1 ) n ( 2 n + 2 )
    W n ( x ) ( - 1 ) n V n ( x ) 1 ( - 1 ) n ( - 1 ) n ( 2 n + 2 )
    6: 18.5 Explicit Representations
    Chebyshev
    T 0 ( x ) = 1 ,
    T 1 ( x ) = x ,
    U 0 ( x ) = 1 ,
    U 1 ( x ) = 2 x ,
    7: 18.7 Interrelations and Limit Relations
    Chebyshev, Ultraspherical, and Jacobi
    18.7.7 T n * ( x ) = T n ( 2 x - 1 ) ,
    18.7.17 U 2 n ( x ) = V n ( 2 x 2 - 1 ) ,
    18.7.18 T 2 n + 1 ( x ) = x W n ( 2 x 2 - 1 ) .
    8: 3.11 Approximation Techniques
    The Chebyshev polynomials T n are given by …
    3.11.7 T n + 1 ( x ) - 2 x T n ( x ) + T n - 1 ( x ) = 0 , n = 1 , 2 , ,
    with initial values T 0 ( x ) = 1 , T 1 ( x ) = x . … For the expansion (3.11.11), numerical values of the Chebyshev polynomials T n ( x ) can be generated by application of the recurrence relation (3.11.7). …Let c n T n ( x ) be the last term retained in the truncated series. …
    9: 18.8 Differential Equations
    Table 18.8.1: Classical OP’s: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
    f ( x ) A ( x ) B ( x ) C ( x ) λ n
    T n ( x ) 1 - x 2 - x 0 n 2
    U n ( x ) 1 - x 2 - 3 x 0 n ( n + 2 )
    10: 16.26 Approximations
    For discussions of the approximation of generalized hypergeometric functions and the Meijer G -function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).