Gauss%E2%80%93Legendre formula
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11: 15.8 Transformations of Variable
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►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1).
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15.8.13
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15.8.14
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15.8.15
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15.8.16
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12: 35.8 Generalized Hypergeometric Functions of Matrix Argument
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§35.8(iii) Case
►Kummer Transformation
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… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions and of matrix argument. …13: 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
►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. … ►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …14: 16.4 Argument Unity
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►The function is well-poised if
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►The function with argument unity and general values of the parameters is discussed in Bühring (1992).
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►For generalizations involving functions see Kim et al. (2013).
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►Balanced series have transformation formulas and three-term relations.
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►Transformations for both balanced and very well-poised are included in Bailey (1964, pp. 56–63).
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15: 16.18 Special Cases
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►The 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
►As 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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16: 31.7 Relations to Other Functions
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§31.7(i) Reductions to the Gauss Hypergeometric Function
… ►Other reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where . … ►
31.7.2
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31.7.3
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►Similar 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).
17: 18.20 Hahn Class: Explicit Representations
18: 16.8 Differential Equations
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►the function satisfies the differential equation
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►We have the connection formula
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►Analytical 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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19: 20.11 Generalizations and Analogs
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