Pfaff--Saalschutz%20formula
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1: 24.1 Special Notation
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►Unless otherwise noted, the formulas in this chapter hold for all values of the variables and , and for all nonnegative integers .
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►It was used in Saalschütz (1893), Nielsen (1923), Schwatt (1962), and Whittaker and Watson (1927).
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2: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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On formulas involving both the Bernoulli and Fibonacci numbers.
Scripta Math. 23, pp. 27–35.
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An extension of Saalschütz’s summation theorem for the series
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Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The addition formula for Laguerre polynomials.
SIAM J. Math. Anal. 8 (3), pp. 535–540.
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3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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Pfaff–Saalschütz Formula
…4: Gergő Nemes
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►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.
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5: Wolter Groenevelt
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►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.
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6: 16.4 Argument Unity
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►When the function is said to be balanced or Saalschützian.
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Pfaff–Saalschütz Balanced Sum
… ►Balanced 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
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Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen.
Springer-Verlag, Berlin (German).
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Calculation of the gamma function by Stirling’s formula.
Math. Comp. 25 (114), pp. 317–322.
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Gaussian Quadrature Formulas.
Prentice-Hall Inc., Englewood Cliffs, N.J..
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