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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
2: Bibliography O
  • K. Okamoto (1987b) Studies on the Painlevé equations. II. Fifth Painlevé equation P V . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.
  • A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.
  • F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
  • F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
  • M. Onoe (1955) Formulae and Tables, The Modified Quotients of Cylinder Functions. Technical report Technical Report UDC 517.564.3:518.25, Vol. 4, Report of the Institute of Industrial Science, University of Tokyo, Institute of Industrial Science, Chiba City, Japan.
  • 3: Bibliography Y
  • T. Yoshida (1995) Computation of Kummer functions U ( a , b , x ) for large argument x by using the τ -method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).
  • 4: Bibliography N
  • A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
  • W. Narkiewicz (2000) The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.
  • G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
  • I. Niven, H. S. Zuckerman, and H. L. Montgomery (1991) An Introduction to the Theory of Numbers. 5th edition, John Wiley & Sons Inc., New York.
  • Number Theory Web (website)
  • 5: Guide to Searching the DLMF
  • term:

    a textual word, a number, or a math symbol.

  • phrase:

    any double-quoted sequence of textual words and numbers.

  • proximity operator:

    adj, prec/n, and near/n, where n is any positive natural number.

  • $ stands for any number of alphanumeric characters
    (the more conventional * is reserved for the multiplication operator)
    6: 26.6 Other Lattice Path Numbers
    §26.6 Other Lattice Path Numbers
    Delannoy Number D ( m , n )
    Motzkin Number M ( n )
    Narayana Number N ( n , k )
    §26.6(iv) Identities
    7: 26.5 Lattice Paths: Catalan Numbers
    §26.5 Lattice Paths: Catalan Numbers
    §26.5(i) Definitions
    C ( n ) is the Catalan number. …
    §26.5(ii) Generating Function
    §26.5(iii) Recurrence Relations
    8: 24.15 Related Sequences of Numbers
    §24.15 Related Sequences of Numbers
    §24.15(i) Genocchi Numbers
    §24.15(ii) Tangent Numbers
    §24.15(iii) Stirling Numbers
    §24.15(iv) Fibonacci and Lucas Numbers
    9: 27.2 Functions
    §27.2(i) Definitions
    where p 1 , p 2 , , p ν ( n ) are the distinct prime factors of n , each exponent a r is positive, and ν ( n ) is the number of distinct primes dividing n . … (See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) …
    §27.2(ii) Tables
    10: 26.14 Permutations: Order Notation
    As an example, 35247816 is an element of 𝔖 8 . The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … For example, maj ( 35247816 ) = 2 + 6 = 8 . … The Eulerian number, denoted n k , is the number of permutations in 𝔖 n with exactly k descents. …
    §26.14(iii) Identities