About the Project
NIST

explicit formulas

AdvancedHelp

(0.001 seconds)

1—10 of 22 matching pages

1: 24.6 Explicit Formulas
§24.6 Explicit Formulas
24.6.6 E 2 n = k = 1 2 n ( - 1 ) k 2 k - 1 ( 2 n + 1 k + 1 ) j = 0 1 2 k - 1 2 ( k j ) ( k - 2 j ) 2 n .
24.6.7 B n ( x ) = k = 0 n 1 k + 1 j = 0 k ( - 1 ) j ( k j ) ( x + j ) n ,
24.6.12 E 2 n = k = 0 2 n 1 2 k j = 0 k ( - 1 ) j ( k j ) ( 1 + 2 j ) 2 n .
2: 10.49 Explicit Formulas
§10.49 Explicit Formulas
§10.49(i) Unmodified Functions
10.49.7 h n ( 2 ) ( z ) = e - i z k = 0 n ( - i ) k - n - 1 a k ( n + 1 2 ) z k + 1 .
§10.49(ii) Modified Functions
§10.49(iv) Sums or Differences of Squares
3: 27.13 Functions
Explicit formulas for r k ( n ) have been obtained by similar methods for k = 6 , 8 , 10 , and 12 , but they are more complicated. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …
4: 3.8 Nonlinear Equations
§3.8(iv) Zeros of Polynomials
Explicit formulas for the zeros are available if n 4 ; see §§1.11(iii) and 4.43. No explicit general formulas exist when n 5 . …
5: Bibliography H
  • K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
  • F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.
  • 6: 18.11 Relations to Other Functions
    §18.11(i) Explicit Formulas
    7: Bibliography N
  • G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
  • 8: 10.74 Methods of Computation
    In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. …
    9: Bibliography T
  • P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
  • 10: 8.11 Asymptotic Approximations and Expansions
    This reference also contains explicit formulas for b k ( λ ) in terms of Stirling numbers and for the case λ > 1 an asymptotic expansion for b k ( λ ) as k . … This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …