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Euler%20polynomials

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1: 25.6 Integer Arguments
§25.6(i) Function Values
25.6.3 ζ ( n ) = B n + 1 n + 1 , n = 1 , 2 , 3 , .
25.6.6 ζ ( 2 k + 1 ) = ( 1 ) k + 1 ( 2 π ) 2 k + 1 2 ( 2 k + 1 ) ! 0 1 B 2 k + 1 ( t ) cot ( π t ) d t , k = 1 , 2 , 3 , .
25.6.12 ζ ′′ ( 0 ) = 1 2 ( ln ( 2 π ) ) 2 + 1 2 γ 2 1 24 π 2 + γ 1 ,
where γ 1 is given by (25.2.5). …
2: Bibliography L
  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright ω function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • J. L. López and N. M. Temme (1999b) Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials. J. Math. Anal. Appl. 239 (2), pp. 457–477.
  • J. L. López and N. M. Temme (1999c) Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.
  • J. L. López and N. M. Temme (2010b) Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.
  • 3: Software Index
    4: 24.2 Definitions and Generating Functions
    §24.2(i) Bernoulli Numbers and Polynomials
    §24.2(ii) Euler Numbers and Polynomials
    §24.2(iii) Periodic Bernoulli and Euler Functions
    Table 24.2.2: Bernoulli and Euler polynomials.
    n B n ( x ) E n ( x )
    5: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • R. Askey and J. Wilson (1985) Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
  • 6: 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.

  • 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 .

  • 7: 5.11 Asymptotic Expansions
    The scaled gamma function Γ ( z ) is defined in (5.11.3) and its main property is Γ ( z ) 1 as z in the sector | ph z | π δ . Wrench (1968) gives exact values of g k up to g 20 . …
    5.11.8 Ln Γ ( z + h ) ( z + h 1 2 ) ln z z + 1 2 ln ( 2 π ) + k = 2 ( 1 ) k B k ( h ) k ( k 1 ) z k 1 ,
    where h ( ) is fixed, and B k ( h ) is the Bernoulli polynomial defined in §24.2(i). … In terms of generalized Bernoulli polynomials B n ( ) ( x ) 24.16(i)), we have for k = 0 , 1 , , …
    8: 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
    25.11.6 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 .
    25.11.7 ζ ( s , a ) = 1 a s + 1 ( 1 + a ) s ( 1 2 + 1 + a s 1 ) + k = 1 n ( s + 2 k 2 2 k 1 ) B 2 k 2 k 1 ( 1 + a ) s + 2 k 1 ( s + 2 n 2 n + 1 ) 1 B ~ 2 n + 1 ( x ) ( x + a ) s + 2 n + 1 d x , s 1 , a > 0 , n = 1 , 2 , 3 , , s > 2 n .
    For B ~ n ( x ) see §24.2(iii). …
    9: 32.8 Rational Solutions
    where the Q n ( z ) are monic polynomials (coefficient of highest power of z is 1 ) satisfying … Next, let p m ( z ) be the polynomials defined by p m ( z ) = 0 for m < 0 , and … In the general case assume γ δ 0 , so that as in §32.2(ii) we may set γ = 1 and δ = 1 . …where P m ( z ) and Q m ( z ) are polynomials of degree m , with no common zeros. … where λ , μ are constants, and P n 1 ( z ) , Q n ( z ) are polynomials of degrees n 1 and n , respectively, with no common zeros. …
    10: Bibliography P
  • A. M. Parkhurst and A. T. James (1974) Zonal Polynomials of Order 1 Through 12 . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.
  • P. I. Pastro (1985) Orthogonal polynomials and some q -beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
  • J. Patera and P. Winternitz (1973) A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.
  • K. Pearson (Ed.) (1965) Tables of the Incomplete Γ -function. Biometrika Office, Cambridge University Press, Cambridge.
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.