Gauss series
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1: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
โบThe hypergeometric function is defined by the Gauss series … โบOn the circle of convergence, , the Gauss series: … โบIn that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that is equal to the first terms of the Maclaurin series for .2: 15.19 Methods of Computation
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โบThe Gauss series (15.2.1) converges for .
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โบLarge values of or , for example, delay convergence of the Gauss series, and may also lead to severe cancellation.
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3: 16.10 Expansions in Series of Functions
§16.10 Expansions in Series of Functions
… โบExpansions of the form are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).4: 34.4 Definition: Symbol
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โบFor 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).
5: 34.2 Definition: Symbol
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โบFor 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, §§8.21, 8.24–8.26).
6: 19.19 Taylor and Related Series
7: 15.16 Products
§15.16 Products
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15.16.1
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15.16.2
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15.16.4
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โบFor further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).
8: Bibliography L
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New series expansions of the Gauss hypergeometric function.
Adv. Comput. Math. 39 (2), pp. 349–365.
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9: 27.10 Periodic Number-Theoretic Functions
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โบEvery function periodic (mod ) can be expressed as a finite Fourier
series of the form
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โบis a periodic function of and has the finite Fourier-series expansion
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โบAnother generalization of Ramanujan’s sum is the Gauss sum
associated with a Dirichlet character .
…In particular, .
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is separable for some if
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10: Bibliography M
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A -analog of the Gauss summation theorem for hypergeometric series in
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Adv. in Math. 72 (1), pp. 59–131.
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