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§15.2(i) Gauss Series►The hypergeometric function is defined by the Gauss series … ►On the circle of convergence, , the Gauss series: … ►In 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 .
… ►The Gauss series (15.2.1) converges for . … ►Large values of or , for example, delay convergence of the Gauss series, and may also lead to severe cancellation. …
§16.10 Expansions in Series of Functions… ►Expansions 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).
… ►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).
… ►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, §§8.21, 8.24–8.26).
15.16.1 ,… ►
15.16.2 .… ►
15.16.4… ►For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).
New series expansions of the Gauss hypergeometric function.
Adv. Comput. Math. 39 (2), pp. 349–365.
… ►Every function periodic (mod ) can be expressed as a finite Fourier series of the form … ►is a periodic function of and has the finite Fourier-series expansion … ►Another generalization of Ramanujan’s sum is the Gauss sum associated with a Dirichlet character . …In particular, . ► is separable for some if …
10: Bibliography M
A -analog of the Gauss summation theorem for hypergeometric series in
Adv. in Math. 72 (1), pp. 59–131.