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Pfaff--Saalschutz formula

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1: 35.8 Generalized Hypergeometric Functions of Matrix Argument
PfaffSaalschütz Formula
2: 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). …
3: 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. …
4: Bibliography K
  • A. A. Kapaev and A. V. Kitaev (1993) Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.
  • S. Karlin and J. L. McGregor (1961) The Hahn polynomials, formulas and an application. Scripta Math. 26, pp. 33–46.
  • 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.
  • T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
  • 5: 27.20 Methods of Computation: Other Number-Theoretic Functions
    The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
    6: Bibliography S
  • L. Saalschütz (1893) Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen. Springer-Verlag, Berlin (German).
  • 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..
  • 7: Howard S. Cohl
    Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects. In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.
    8: Preface
    Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. …
    9: 5.5 Functional Relations
    §5.5(ii) Reflection
    5.5.3 Γ ( z ) Γ ( 1 - z ) = π / sin ( π z ) , z 0 , ± 1 , ,
    §5.5(iii) Multiplication
    Duplication Formula
    Gauss’s Multiplication Formula
    10: 24.6 Explicit Formulas
    §24.6 Explicit Formulas
    24.6.6 E 2 n = k = 1 2 n ( - 1 ) k 2 k - 1 ( 2 n + 1 k + 1 ) j = 0 1 2 k - 1 2 ( k j ) ( k - 2 j ) 2 n .
    24.6.7 B n ( x ) = k = 0 n 1 k + 1 j = 0 k ( - 1 ) j ( k j ) ( x + j ) n ,
    24.6.12 E 2 n = k = 0 2 n 1 2 k j = 0 k ( - 1 ) j ( k j ) ( 1 + 2 j ) 2 n .