Functiones et Approximatio Commentarii Mathematici

Linear independence of values of a certain generalization of the exponential function II

Rolf Wallisser

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Abstract

The arithmetical nature of values at rational points of the hypergeometric series \[ _Q G_R(x) := \sum_{n=0}^\infty \frac{R(1) R(2) \cdots R(n)}{Q(1) Q(2) \cdots Q(n)} x^n \] is studied. $R$ and $Q$ are polynomials with integer coefficients. Using deep results on higher con\-gruences 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 Carlson [2], Inkeri [15], Ivankov [17,18], Popken [26] and Bundschuh--Wallisser [1] are derived.

Article information

Source
Funct. Approx. Comment. Math., Volume 49, Number 1 (2013), 79-90.

Dates
First available in Project Euclid: 20 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.facm/1379686435

Digital Object Identifier
doi:10.7169/facm/2013.49.1.5

Mathematical Reviews number (MathSciNet)
MR3127900

Zentralblatt MATH identifier
1364.11134

Subjects
Primary: 11J72: Irrationality; linear independence over a field 11J82: Measures of irrationality and of transcendence
Secondary: 11K60: Diophantine approximation [See also 11Jxx] 11J25: Diophantine inequalities [See also 11D75] 11A07: Congruences; primitive roots; residue systems

Keywords
irrationality $\mathbb{Q}$-linear independence

Citation

Wallisser, Rolf. Linear independence of values of a certain generalization of the exponential function II. Funct. Approx. Comment. Math. 49 (2013), no. 1, 79--90. doi:10.7169/facm/2013.49.1.5. https://projecteuclid.org/euclid.facm/1379686435


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