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Transcendence of special values of Pochhammer functions. (English) Zbl 1197.11086

The exceptional set of a single-variable transcendental function is the set of algebraic numbers at which it assumes algebraic values. The paper under review characterizes the exceptional set of \(m\)th order Pochhammer functions, \(m>3\), which are solutions of \(m\)th order Picard–Fuchs differential equations with \(m+1\) regular singular points, in terms of certain members of a family of abelian varieties. In particular, criteria for the exceptional set to be finite and for it to be infinite are established.

MSC:

11J91 Transcendence theory of other special functions
11G18 Arithmetic aspects of modular and Shimura varieties
33C65 Appell, Horn and Lauricella functions
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