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1: 18.5 Explicit Representations
§18.5 Explicit Representations
Chebyshev
2: 26.13 Permutations: Cycle Notation
An explicit representation of σ can be given by the 2 × n matrix: …
3: 18.20 Hahn Class: Explicit Representations
§18.20 Hahn Class: Explicit Representations
4: Bibliography T
  • P. G. Todorov (1978) Une nouvelle représentation explicite des nombres d’Euler. C. R. Acad. Sci. Paris Sér. A-B 286 (19), pp. A807–A809.
  • 5: 16.11 Asymptotic Expansions
    Explicit representations for the coefficients c k are given in Volkmer and Wood (2014). …
    6: 18.37 Classical OP’s in Two or More Variables
    Explicit Representation
    7: 14.30 Spherical and Spheroidal Harmonics
    Explicit Representation
    8: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2004c) Integral representations for computing real parabolic cylinder functions. Numer. Math. 98 (1), pp. 105–134.
  • H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • K. I. Gross and R. A. Kunze (1976) Bessel functions and representation theory. I. J. Functional Analysis 22 (2), pp. 73–105.
  • E. Grosswald (1985) Representations of Integers as Sums of Squares. Springer-Verlag, New York.
  • 9: 27.13 Functions
    Each representation of n as a sum of elements of S is called a partition of n , and the number S ( n ) of such partitions is often of great interest. …
    §27.13(iv) Representation by Squares
    In fact, there are four representations, given by 5 = 2 2 + 1 2 = 2 2 + ( - 1 ) 2 = ( - 2 ) 2 + 1 2 = ( - 2 ) 2 + ( - 1 ) 2 , and four more with the order of summands reversed. … 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. …
    10: 10.74 Methods of Computation
    It should be noted, however, that there is a difficulty in evaluating the coefficients A k ( ζ ) , B k ( ζ ) , C k ( ζ ) , and D k ( ζ ) , from the explicit expressions (10.20.10)–(10.20.13) when z is close to 1 owing to severe cancellation. … 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. …
    §10.74(iii) Integral Representations
    For evaluation of the Hankel functions H ν ( 1 ) ( z ) and H ν ( 2 ) ( z ) for complex values of ν and z based on the integral representations (10.9.18) see Remenets (1973). … The integral representation used is based on (10.32.8). …