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1: 32.8 Rational Solutions
where the Q n ( z ) are monic polynomials (coefficient of highest power of z is 1 ) satisfying …
Q 3 ( z ) = z 6 + 20 z 3 80 ,
For determinantal representations see Kajiwara and Masuda (1999). … For determinantal representations see Kajiwara and Ohta (1998) and Noumi and Yamada (1999). … For determinantal representations see Masuda et al. (2002). …
2: 19.36 Methods of Computation
When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …
19.36.2 1 3 14 E 2 + 1 6 E 3 + 9 88 E 2 2 3 22 E 4 9 52 E 2 E 3 + 3 26 E 5 1 16 E 2 3 + 3 40 E 3 2 + 3 20 E 2 E 4 + 45 272 E 2 2 E 3 9 68 ( E 3 E 4 + E 2 E 5 ) .
For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. … Faster convergence of power series for K ( k ) and E ( k ) can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …
3: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2004c) Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
  • P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.
  • E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.