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Siegel normality. (English) Zbl 0652.10027

In its original form Siegel’s method implies the algebraic independence of the values of \(E\)-function solutions of a normal system of linear differential equations \((d / dz)y = Ay\), \(A\) an \(n \times n\) matrix with \(a_{ij} \in \mathbb C(z)\). In a fundamental development of the method A.B. Shidlovskij [Izv. Akad. Nauk SSSR, Ser. Mat. 23, 35-66 (1959; Zbl 0085.273)] replaced this normality condition by the algebraic independence (over \(\mathbb C(z)\)) of the functions under consideration. In the measures of algebraic independence Shidlovskij’s development does not yield complete effectivity which can be obtained for normal systems [see W. D. Brownawell, J. Aust. Math. Soc., Ser. A 39, 227-240 (1985; Zbl 0574.10039)]. In the present paper the authors are able to obtain characterizations for Siegel’s normality criterion which give very important results on the normality of families of equations for generalized hypergeometric functions of the type \[ _ pF_{q-1}(z) = \sum_{n=0}^\infty {(u_1)_ n \cdots (u_ p)_ n \over (v_1)_ n \cdots (v_{q-1})_ n} {(-z)^{(q-p)n}\over n!}, \] where the parameters \(u_ i, v_ j\) are rational numbers satisfying certain conditions \(q > p \geq 0\), \(q \geq 2\), and \((u)_0 = 1\), \((u)_ n = u(u + 1) \cdots (u + n - 1)\), \(n \geq 1\). Thus they obtain the algebraic independence (and effective measure of algebraic independence) of the values of these functions and their derivatives at non-zero algebraic points. These results generalize several earlier results, in particular, note the paper of V. K. Salikhov [Dokl. Akad. Nauk. SSSR 254, No. 4, 806-808 (1980; Zbl 0467.34008)].
Reviewer: K.Väänänen

MSC:

11J81 Transcendence (general theory)
34A30 Linear ordinary differential equations and systems
11J85 Algebraic independence; Gel’fond’s method
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