Gauss%E2%80%93Legendre%20formula
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11: 16.4 Argument Unity
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โบThe function is well-poised if
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โบThe function with argument unity and general values of the parameters is discussed in Bühring (1992).
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โบFor generalizations involving functions see Kim et al. (2013).
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โบBalanced series have transformation formulas and three-term relations.
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โบTransformations for both balanced and very well-poised are included in Bailey (1964, pp. 56–63).
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12: 15.9 Relations to Other Functions
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Legendre
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15.9.7
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§15.9(iv) Associated Legendre Functions; Ferrers Functions
โบAny hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. … โบThe following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …13: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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§35.8(iii) Case
โบKummer Transformation
… โบPfaff–Saalschütz Formula
… โบThomae Transformation
… โบMultidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions and of matrix argument. …14: 3.5 Quadrature
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§3.5(v) Gauss Quadrature
… โบGauss–Legendre Formula
… โบGauss–Chebyshev Formula
… โบGauss–Jacobi Formula
… โบGauss–Laguerre Formula
…15: 27.2 Functions
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โบEuclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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โบGauss and Legendre conjectured that is asymptotic to as :
…(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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16: 16.18 Special Cases
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โบThe and functions introduced in Chapters 13 and 15, as well as the more general functions introduced in the present chapter, are all special cases of the Meijer -function.
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16.18.1
โบAs a corollary, special cases of the and functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer -function.
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17: 31.7 Relations to Other Functions
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§31.7(i) Reductions to the Gauss Hypergeometric Function
… โบOther reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where . … โบ
31.7.2
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31.7.3
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โบSimilar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities , , and , where and are related to as in §19.2(ii).
18: 20.11 Generalizations and Analogs
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§20.11(i) Gauss Sum
โบFor relatively prime integers with and even, the Gauss sum is defined by … … โบ … โบSimilar identities can be constructed for , , and . …19: 16.8 Differential Equations
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โบthe function satisfies the differential equation
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โบWe have the connection formula
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โบAnalytical continuation formulas for near are given in Bühring (1987b) for the case , and in Bühring (1992) for the general case.
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16.8.10
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