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1: Bibliography M
  • W. Magnus, F. Oberhettinger, and R. P. Soni (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edition, Springer-Verlag, New York-Berlin.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.
  • L. Moser and M. Wyman (1958b) Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.
  • Y. Murata (1985) Rational solutions of the second and the fourth Painlevé equations. Funkcial. Ekvac. 28 (1), pp. 1–32.
  • 2: Errata
  • Subsection 17.9(iii)

    The title of the paragraph which was previously “Gasper’s q -Analog of Clausen’s Formula” has been changed to “Gasper’s q -Analog of Clausen’s Formula (16.12.2)”.

  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Version 1.0.9 (August 29, 2014)
  • Chapters 8, 20, 36

    Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

  • Version 1.0.3 (Aug 29, 2011)
    3: Bibliography K
  • A. A. Kapaev and A. V. Kitaev (1993) Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.
  • R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.
  • A. D. Kerr (1978) An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
  • 4: Bibliography O
  • F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.
  • A. B. Olde Daalhuis (1998a) Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
  • F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.
  • M. Onoe (1956) Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.
  • M. L. Overton (2001) Numerical Computing with IEEE Floating Point Arithmetic. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • 5: Bibliography C
  • R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.
  • A. Cayley (1895) An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.
  • M. Chellali (1988) Accélération de calcul de nombres de Bernoulli. J. Number Theory 28 (3), pp. 347–362 (French).
  • E. D. Constantinides and R. J. Marhefka (1993) Efficient and accurate computation of the incomplete Airy functions. Radio Science 28 (4), pp. 441–457.
  • R. Cools (2003) An encyclopaedia of cubature formulas. J. Complexity 19 (3), pp. 445–453.
  • 6: Bibliography G
  • I. Gargantini and P. Henrici (1967) A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • A. Gil, J. Segura, and N. M. Temme (2003b) Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.
  • 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.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
  • 7: 25.11 Hurwitz Zeta Function
    See accompanying text
    Figure 25.11.1: Hurwitz zeta function ζ ( x , a ) , a = 0. …8, 1, 20 x 10 . … Magnify
    §25.11(iii) Representations by the Euler–Maclaurin Formula
    25.11.5 ζ ( s , a ) = n = 0 N 1 ( n + a ) s + ( N + a ) 1 s s 1 s N x x ( x + a ) s + 1 d x , s 1 , s > 0 , a > 0 , N = 0 , 1 , 2 , 3 , .
    25.11.6 ζ ( s , a ) = 1 a s ( 1 2 + a s 1 ) s ( s + 1 ) 2 0 B ~ 2 ( x ) B 2 ( x + a ) s + 2 d x , s 1 , s > 1 , a > 0 .
    25.11.12 ζ ( n + 1 , a ) = ( 1 ) n + 1 ψ ( n ) ( a ) n ! , n = 1 , 2 , 3 , .
    8: Bibliography D
  • M. D’Ocagne (1904) Sur une classe de nombres rationnels réductibles aux nombres de Bernoulli. Bull. Sci. Math. (2) 28, pp. 29–32 (French).
  • B. Deconinck and M. van Hoeij (2001) Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.
  • S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
  • A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
  • J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.