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1: William P. Reinhardt
He has been a National Lecturer for Sigma Xi and Phi Beta Kappa, as well as a Sloan, Dreyfus, and Guggenheim Fellow, and Fulbright Senior Scholar (Australia). …
  • In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
    2: 27.2 Functions
    §27.2 Functions
    Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … Note that σ 0 ( n ) = d ( n ) . …This is Jordan’s function. … Table 27.2.2 tabulates the Euler totient function ϕ ( n ) , the divisor function d ( n ) ( = σ 0 ( n ) ), and the sum of the divisors σ ( n ) ( = σ 1 ( n ) ), for n = 1 ( 1 ) 52 . …
    3: 26.14 Permutations: Order Notation
    The set 𝔖 n 26.13) can be viewed as the collection of all ordered lists of elements of { 1 , 2 , , n } : { σ ( 1 ) σ ( 2 ) σ ( n ) } . …Equivalently, this is the sum over 1 j < n of the number of integers less than σ ( j ) that lie in positions to the right of the j th position: inv ( 35247816 ) = 2 + 3 + 1 + 1 + 2 + 2 + 0 = 11 . The major index is the sum of all positions that mark the first element of a descent: … An excedance in σ 𝔖 n is a position j for which σ ( j ) > j . A weak excedance is a position j for which σ ( j ) j . …
    4: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • N. E. Nørlund (1955) Hypergeometric functions. Acta Math. 94, pp. 289–349.
  • 5: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
  • 6: 26.2 Basic Definitions
    Thus 231 is the permutation σ ( 1 ) = 2 , σ ( 2 ) = 3 , σ ( 3 ) = 1 . … Given a finite set S with permutation σ , a cycle is an ordered equivalence class of elements of S where j is equivalent to k if there exists an = ( j , k ) such that j = σ ( k ) , where σ 1 = σ and σ is the composition of σ with σ 1 . It is ordered so that σ ( j ) follows j . …Here σ ( 1 ) = 2 , σ ( 2 ) = 5 , and σ ( 5 ) = 1 . The function σ also interchanges 3 and 6, and sends 4 to itself. …
    7: 23.9 Laurent and Other Power Series
    §23.9 Laurent and Other Power Series
    23.9.2 ( z ) = 1 z 2 + n = 2 c n z 2 n 2 , 0 < | z | < | z 0 | ,
    23.9.3 ζ ( z ) = 1 z n = 2 c n 2 n 1 z 2 n 1 , 0 < | z | < | z 0 | .
    c 2 = 1 20 g 2 ,
    23.9.7 σ ( z ) = m , n = 0 a m , n ( 10 c 2 ) m ( 56 c 3 ) n z 4 m + 6 n + 1 ( 4 m + 6 n + 1 ) ! ,
    8: 26.12 Plane Partitions
    §26.12(ii) Generating Functions
    26.12.25 pp ( n ) = 1 n j = 1 n pp ( n j ) σ 2 ( j ) ,
    where σ 2 ( j ) is the sum of the squares of the divisors of j . …
    26.12.26 pp ( n ) ( ζ ( 3 ) ) 7 / 36 2 11 / 36 ( 3 π ) 1 / 2 n 25 / 36 exp ( 3 ( ζ ( 3 ) ) 1 / 3 ( 1 2 n ) 2 / 3 + ζ ( 1 ) ) ,
    where ζ is the Riemann ζ -function25.2(i)). …
    9: Bibliography C
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • H. S. Cohl, J. Park, and H. Volkmer (2021) Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.
  • H. S. Cohl (2011) On parameter differentiation for integral representations of associated Legendre functions. SIGMA Symmetry Integrability Geom. Methods Appl. 7, pp. Paper 050, 16.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 10: 26.13 Permutations: Cycle Notation
    σ 𝔖 n is a one-to-one and onto mapping from { 1 , 2 , , n } to itself. An explicit representation of σ can be given by the 2 × n matrix: … In cycle notation, the elements in each cycle are put inside parentheses, ordered so that σ ( j ) immediately follows j or, if j is the last listed element of the cycle, then σ ( j ) is the first element of the cycle. … See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … Given a permutation σ 𝔖 n , the inversion number of σ , denoted inv ( σ ) , is the least number of adjacent transpositions required to represent σ . …