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1: 17.16 Mathematical Applications
§17.16 Mathematical Applications
Many special cases of q -series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. In Lie algebras Lepowsky and Milne (1978) and Lepowsky and Wilson (1982) laid foundations for extensive interaction with q -series. …
2: 4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
3: 12.20 Approximations
§12.20 Approximations
Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions U ( a , b , x ) and M ( a , b , x ) 13.2(i)) whose regions of validity include intervals with endpoints x = and x = 0 , respectively. As special cases of these results a Chebyshev-series expansion for U ( a , x ) valid when λ x < follows from (12.7.14), and Chebyshev-series expansions for U ( a , x ) and V ( a , x ) valid when 0 x λ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …
4: 16.20 Integrals and Series
§16.20 Integrals and Series
Series of the Meijer G -function are given in Erdélyi et al. (1953a, §5.5.1), Luke (1975, §5.8), and Prudnikov et al. (1990, §6.11). …
5: 4.33 Maclaurin Series and Laurent Series
§4.33 Maclaurin Series and Laurent Series
6: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
7: 6.20 Approximations
§6.20(ii) Expansions in Chebyshev Series
  • Luke and Wimp (1963) covers Ei ( x ) for x 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein ( a x ) , Si ( a x ) , and Cin ( a x ) for 1 x 1 , a . The coefficients are given in terms of series of Bessel functions.

  • Luke (1969b, pp. 321–322) covers Ein ( x ) and Ein ( x ) for 0 x 8 (the Chebyshev coefficients are given to 20D); E 1 ( x ) for x 5 (20D), and Ei ( x ) for x 8 (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U -function (§13.2(i)) from which Chebyshev expansions near infinity for E 1 ( z ) , f ( z ) , and g ( z ) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If | ph z | < π the scheme can be used in backward direction.

  • 8: 4.11 Sums
    For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).
    9: 30.10 Series and Integrals
    §30.10 Series and Integrals
    10: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of s ζ ( s + 1 ) and ζ ( s + k ) , k = 2 , 3 , 4 , 5 , 8 , for 0 s 1 (23D).

  • Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover ζ ( s ) for 0 s 1 (15D), ζ ( s + 1 ) for 0 s 1 (20D), and ln ξ ( 1 2 + i x ) 25.4) for 1 x 1 (20D). For errata see Piessens and Branders (1972).

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals < x 2 and 2 x < , with s = 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 4 , 10 8 , 10 12 .