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Wallisser, Rolf

Linear independence of values of a certain generalization of the exponential function. II. (English) Zbl 1364.11134

Funct. Approximatio, Comment. Math. 49, No. 1, 79-90 (2013).
Summary: The arithmetical nature of values at rational points of the hypergeometric series \[ _QG_R(x) := \sum_{n=0}^{\infty} \frac{R(1)R(2) \cdots R(n)}{Q(1)Q(2)\cdots Q(n)} x^n \] is studied. \(R\) and \(Q\) are polynomials with integer coefficients. Using deep results on higher congruences going essentially back to Frobenius, Dedekind, Nagell and Schinzel a measure of \(\mathbb{Q}\)-linear independence of such values is given.
In contrast to former investigations no preseribed factorisations of the polynomials \(Q\) and \(R\) are necessary. Here, however, congruences to those primes \(p\) are used for which \(Q\) mod \(p\) splits completely into a product of linear factors. To get the measure the fact is used that these primes have a Dirichlet-density.
In six applications results, proven in some particular cases by different techniques by F. Carlson [Ark. Mat. Astron. Fys. 25, No. 7, 1–13 (1935; Zbl 0011.39202)], K. Inkeri [Nieuw Arch. Wiskd., III. Ser. 24, 226–230 (1976; Zbl 0338.10026)], P. L. Ivankov [Math. USSR, Sb. 72, No. 1, (1992; Zbl 0776.11039); translation from Mat. Sb. 182, No. 2, 282–302 (1991); Russ. Math. 51, No. 7, 45–49 (2007; Zbl 1154.33003); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2007, No. 7, 48–52 (2007)], J. Popken [Über arithmetische Eigenschaften analytischer Funktionen. Groningen: Diss (1935; Zbl 0013.27004)] and P. Bundschuh and the author [Analysis, München 32, No. 1, 67–83 (2012; Zbl 1305.11062)] are derived.
For Part I, see [the author, ibid. 35, 339–357 (2006; Zbl 1192.11043)].

Cited in 1 Document

MSC:

11J72 Irrationality; linear independence over a field
11J82 Measures of irrationality and of transcendence

Keywords:

irrationality; \(Q\)-linear independence

Citations:

Zbl 1096.11024; Zbl 0011.39202; Zbl 0338.10026; Zbl 0776.11039; Zbl 1154.33003; Zbl 0013.27004; Zbl 1305.11062; Zbl 1192.11043
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Full Text: DOI Euclid

References:

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