On the measure of linear and algebraic independence for values of entire hypergeometric functions. (English. Russian original) Zbl 0920.11046

Zudilin, V. V.

Math. Notes 61, No. 2, 246-247 (1997); translation from Mat. Zametki 61, No. 2, 302-304 (1997).

The author computes explicit measures of transcendence for values of some generalized hypergeometric functions \(f\) and their derivatives by applying a general theorem from one of his previous papers [V. V. Zudilin, Sb. Math. 187, No. 12, 1791–18181 (1996); translation from Mat. Sb. 187, No. 12, 57–86 (1996; Zbl 0878.11030)]. Let \((E)\) be the linear differential equation of order \(m\) satisfied by \(f\), and denote by \(\Psi_1,\dots, \Psi_m\) a fundamental system of solutions of \((E)\). The problem in applying the above general theorem consists in proving that the functions \(\Psi_j^{(n-1)}\), with \(j\), \(n=1, \dots,m\), are homogeneously algebraically independent over \(\mathbb C(z)\).
In the present paper, the author solves it by using results on Galois groups of linear differential equations [N. M. Katz, Differential Galois groups, Princeton (1990)].

Reviewer: D.Duverney (Lille)

MSC:

11J82	Measures of irrationality and of transcendence
11J91	Transcendence theory of other special functions
12H05	Differential algebra

Keywords:

measures of linear independence; measures of transcendence; generalized hypergeometric functions; Galois groups of linear differential equations

Citations:

Zbl 0878.11030

References:

[1]	V. V. Zudilin,Mat. Sb. [Russian Acad. Sci. Sb. Math.],187, No. 12, 57–86 (1996).
[2]	N. M. Katz,Differential Galois Groups, Univ. Press, Princeton (1990).