About the Project
NIST

with respect to summation

AdvancedHelp

(0.002 seconds)

1—10 of 11 matching pages

1: 18.3 Definitions
In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials T n ( x ) , n = 0 , 1 , , N , are orthogonal on the discrete point set comprising the zeros x N + 1 , n , n = 1 , 2 , , N + 1 , of T N + 1 ( x ) : …
2: 3.11 Approximation Techniques
When n > 0 and 0 j n , 0 k n , … Now suppose that X k = 0 when k , that is, the functions ϕ k ( x ) are orthogonal with respect to weighted summation on the discrete set x 1 , x 2 , , x J . … …
3: 25.11 Hurwitz Zeta Function
The Riemann zeta function is a special case: … For other series expansions similar to (25.11.10) see Coffey (2008). … In (25.11.18)–(25.11.24) primes on ζ denote derivatives with respect to s . … When a = 1 , (25.11.35) reduces to (25.2.3). … uniformly with respect to bounded nonnegative values of α . …
4: 1.8 Fourier Series
As n Then the series (1.8.1) converges to the sum …at every point at which f ( x ) has both a left-hand derivative (that is, (1.4.4) applies when h 0 - ) and a right-hand derivative (that is, (1.4.4) applies when h 0 + ). … when f ( x ) and g ( x ) are square-integrable and a n , b n and a n , b n are their respective Fourier coefficients.
Poisson’s Summation Formula
5: 23.18 Modular Transformations
according as the elements [ a b c d ] of 𝒜 in (23.15.3) have the respective forms …Here e and o are generic symbols for even and odd integers, respectively. …
23.18.7 s ( d , c ) = r = 1 c - 1 r c ( d r c - d r c - 1 2 ) , c > 0 .
6: Errata
  • Equations (15.2.3_5), (19.11.6_5)

    These equations, originally added in Other Changes and Other Changes, respectively, have been assigned interpolated numbers.

  • Subsection 19.25(vi)

    The Weierstrass lattice roots e j , were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots e j , and lattice invariants g 2 , g 3 , now link to their respective definitions (see §§23.2(i), 23.3(i)).

    Reported by Felix Ospald.

  • Equations (10.15.1), (10.38.1)

    These equations have been generalized to include the additional cases of J - ν ( z ) / ν , I - ν ( z ) / ν , respectively.

  • Subsections 14.5(ii), 14.5(vi)

    The titles have been changed to μ = 0 , ν = 0 , 1 , and Addendum to §14.5(ii) μ = 0 , ν = 2 , respectively, in order to be more descriptive of their contents.

  • Equation (8.12.18)
    8.12.18 Q ( a , z ) P ( a , z ) } z a - 1 2 e - z Γ ( a ) ( d ( ± χ ) k = 0 A k ( χ ) z k / 2 k = 1 B k ( χ ) z k / 2 )

    The original ± in front of the second summation was replaced by to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

    Reported 2017-01-28 by Richard Paris.

  • 7: 8.12 Uniform Asymptotic Expansions for Large Parameter
    in each case uniformly with respect to λ in the sector | ph λ | 2 π - δ ( < 2 π ). … For the asymptotic behavior of c k ( η ) as k see Dunster et al. (1998) and Olde Daalhuis (1998c). … Higher coefficients A k ( χ ) , B k ( χ ) , up to k = 8 , are given in Paris (2002b). … For asymptotic expansions, as a , of the inverse function x = x ( a , q ) that satisfies the equation …These expansions involve the inverse error function inverfc ( x ) 7.17), and are uniform with respect to q [ 0 , 1 ] . …
    8: Bibliography G
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • G. Gasper (1981) Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.
  • W. Gautschi (1984) Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.
  • J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.
  • R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in U ( n ) . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.
  • 9: 34.7 Basic Properties: 9 j Symbol
    The 9 j symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent 9 j symbols. …
    34.7.4 ( j 13 j 23 j 33 m 13 m 23 m 33 ) { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = m r 1 , m r 2 , r = 1 , 2 , 3 ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) .
    10: Bibliography
  • R. W. Abernathy and R. P. Smith (1993) Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.
  • G. B. Airy (1849) Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.
  • G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.
  • A. Apelblat (1989) Derivatives and integrals with respect to the order of the Struve functions H ν ( x ) and L ν ( x ) . J. Math. Anal. Appl. 137 (1), pp. 17–36.
  • A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.