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1: 29.21 Tables
  • Arscott and Khabaza (1962) tabulates the coefficients of the polynomials P in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues h for k 2 = 0.1 ( .1 ) 0.9 , n = 1 ( 1 ) 30 . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

  • 2: 29.20 Methods of Computation
    The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … §29.15(i) includes formulas for normalizing the eigenvectors. …
    3: 35.9 Applications
    In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument F q p , with p 2 and q 1 . … For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). …
    4: 7.1 Special Notation
    The notations P ( z ) , Q ( z ) , and Φ ( z ) are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions.
    5: 33.13 Complex Variable and Parameters
    33.13.2 R = ( 2 + 1 ) C ( η ) / C - 1 ( η ) .
    6: 8.23 Statistical Applications
    The function B x ( a , b ) and its normalization I x ( a , b ) play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …
    7: 18.3 Definitions
    Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). …
    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
    8: 23.23 Tables
    2 in Abramowitz and Stegun (1964) gives values of ( z ) , ( z ) , and ζ ( z ) to 7 or 8D in the rectangular and rhombic cases, normalized so that ω 1 = 1 and ω 3 = i a (rectangular case), or ω 1 = 1 and ω 3 = 1 2 + i a (rhombic case), for a = 1. …
    9: 28.7 Analytic Continuation of Eigenvalues
    The branch points are called the exceptional values, and the other points normal values. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). All real values of q are normal values. …
    10: 8.2 Definitions and Basic Properties
    8.2.3 γ ( a , z ) + Γ ( a , z ) = Γ ( a ) , a 0 , - 1 , - 2 , .
    Normalized functions are: …
    8.2.6 γ * ( a , z ) = z - a P ( a , z ) = z - a Γ ( a ) γ ( a , z ) .