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1: 24.1 Special Notation
Unless otherwise noted, the formulas in this chapter hold for all values of the variables x and t , and for all nonnegative integers n . … It was used in Saalschütz (1893), Nielsen (1923), Schwatt (1962), and Whittaker and Watson (1927). …
2: 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.
  • R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.
  • Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series F r + 2 r + 3 . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
  • 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 (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
  • 3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
    PfaffSaalschütz Formula
    4: Gergő Nemes
    As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …
    5: Wolter Groenevelt
    As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …
    6: 16.4 Argument Unity
    When k = 1 the function is said to be balanced or Saalschützian. …
    PfaffSaalschütz Balanced Sum
    Balanced F 3 4 ( 1 ) series have transformation formulas and three-term relations. … See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …
    7: Bibliography S
  • L. Saalschütz (1893) Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen. Springer-Verlag, Berlin (German).
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • R. Spira (1971) Calculation of the gamma function by Stirling’s formula. Math. Comp. 25 (114), pp. 317–322.
  • A. H. Stroud and D. Secrest (1966) Gaussian Quadrature Formulas. Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 8: 17.7 Special Cases of Higher ϕ s r Functions
    q -PfaffSaalschütz Sum
    Nonterminating Form of the q -Saalschütz Sum
    17.7.11 ϕ 3 4 ( q n , q n + 1 , c , c e , c 2 q / e , q ; q , q ) = q ( n + 1 2 ) ( e q n , e q n + 1 , c 2 q 1 n / e , c 2 q n + 2 / e ; q 2 ) ( e , c 2 q / e ; q ) .
    9: 20 Theta Functions
    Chapter 20 Theta Functions
    10: Foreword
    In 1964 the National Institute of Standards and Technology11 1 Then known as the National Bureau of Standards. published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … November 20, 2009 …