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11: 34.3 Basic Properties: Symbol
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►When any one of is equal to , or , the symbol has a simple algebraic form.
…For these and other results, and also cases in which any one of is or , see Edmonds (1974, pp. 125–127).
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►Even permutations of columns of a symbol leave it unchanged; odd permutations of columns produce a phase factor , for example,
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34.3.15
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►For the polynomials see §18.3, and for the function see §14.30.
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12: 22.7 Landen Transformations
13: 10.51 Recurrence Relations and Derivatives
14: 26.3 Lattice Paths: Binomial Coefficients
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►The number of lattice paths from to , , that stay on or above the line is
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26.3.3
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26.3.4
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26.3.7
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26.3.10
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15: 24.6 Explicit Formulas
16: 34.4 Definition: Symbol
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34.4.1
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►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1.
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34.4.3
►where is defined as in §16.2.
►For alternative expressions for the symbol, written either as a finite sum or as other terminating generalized hypergeometric series of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
17: 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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18: 10.53 Power Series
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10.53.2
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10.53.3
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10.53.4
►For and combine (10.47.10), (10.53.1), and (10.53.2).
For combine (10.47.11), (10.53.3), and (10.53.4).