On the measure of linear and algebraic independence for values of entire hypergeometric functions.

*(English. Russian original)*Zbl 0920.11046
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)].

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
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\textit{V. V. Zudilin}, Math. Notes 61, No. 2, 246--247 (1997; Zbl 0920.11046); translation from Mat. Zametki 61, No. 2, 302--304 (1997)

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##### 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). |

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