Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson
Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 381-396.

Let $Q$ be a nonconstant polynomial with integer coefficients and without zeros at the non–negative integers. Essentially with the method of Hermite, a new proof is given on linear independence of values at rational points of the function

$\begin{array}{c}\hfill G\left(x\right)=\sum _{n=0}^{\infty }\phantom{\rule{3.33333pt}{0ex}}\frac{{x}^{n}}{Q\left(1\right)Q\left(2\right)\cdots Q\left(n\right)}.\end{array}$

Soit $Q$ un polynôme non-constant à coefficients entiers, sans racines sur les nombres entiers positifs. Nous donnons ici, essentiellement avec la méthode de Hermite, une nouvelle démonstration de l’indépendence linéaire de certaines valeurs aux points rationnels de la fonction

$\begin{array}{c}\hfill G\left(x\right)=\sum _{n=0}^{\infty }\phantom{\rule{3.33333pt}{0ex}}\frac{{x}^{n}}{Q\left(1\right)Q\left(2\right)\cdots Q\left(n\right)}.\end{array}$

Published online:
DOI: 10.5802/jtnb.496
Rolf Wallisser 1

1 Mathematisches Institut der Universität Freiburg Eckerstr.1 79104 Freiburg, Deutschland
@article{JTNB_2005__17_1_381_0,
author = {Rolf Wallisser},
title = {Linear independence of values of a certain generalisation of the exponential function {\textendash} a new proof of a theorem of {Carlson}},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {381--396},
publisher = {Universit\'e Bordeaux 1},
volume = {17},
number = {1},
year = {2005},
doi = {10.5802/jtnb.496},
zbl = {1096.11024},
mrnumber = {2152230},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.496/}
}
TY  - JOUR
TI  - Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2005
DA  - 2005///
SP  - 381
EP  - 396
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.496/
UR  - https://zbmath.org/?q=an%3A1096.11024
UR  - https://www.ams.org/mathscinet-getitem?mr=2152230
UR  - https://doi.org/10.5802/jtnb.496
DO  - 10.5802/jtnb.496
LA  - en
ID  - JTNB_2005__17_1_381_0
ER  - 
%0 Journal Article
%T Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson
%J Journal de Théorie des Nombres de Bordeaux
%D 2005
%P 381-396
%V 17
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.496
%R 10.5802/jtnb.496
%G en
%F JTNB_2005__17_1_381_0
Rolf Wallisser. Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 381-396. doi : 10.5802/jtnb.496. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.496/

[1] P. Bundschuh, R. Wallisser, Maße für die lineare Unabhängigkeit von Werten ganz transzendenter Lösungen gewisser Funktionalgleichungen I Bd. 69 (1999) II Bd. 73 (2003). Abh. Math Sem. Univ. Hamburg. | Zbl: 1041.11050

[2] F. Carlson, Sur une propriété arithmétique de quelques fonctions entières. Arkiv för Mathematik, Astronomi och Fysik. Bd 25A. N: 07 (1935). | Zbl: 0011.39202

[3] R. Dedekind, Über den Zusammenhang zwischen der Theorie der Ideale und der höheren Kongruenzen, Abh. Akad. Wiss. Göttingen 23 (1878), 3–37.

[4] N.I. Fel’dman, Yu.V. Nesterenko, Transcendental Numbers, Number Theory IV. Encycl. of Math. Sc. 44 (1998), Springer. | MR: 1603608 | Zbl: 0885.11004

[5] I. Gerst, J. Brillhart, On the prime divisors of polynomials. Amer. Math. Monthly 78 (1971). | MR: 279071 | Zbl: 0214.30604

[6] H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, 2nd ed., Physica, Würzburg, 1965.

[7] D.Hilbert, Über die Transcendenz der Zahlen $e$ und $\pi$. Math. Ann. 43 (1893), 216–219.

[8] A. Hurwitz, Beweis der Transzendenz der Zahl $e$. Math. Ann. 43 (1893), 220–221. | MR: 1510809

[9] T. Nagell, Généralisation d’un théorème de Tchebycheff. Journ. de Math. (${8}^{e}$ serie), tome IV (1921).

[10] O. Perron, Irrationalzahlen. Chelsea, New York, (1951). | Zbl: 0029.20303

[11] I. Schur, Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen. S.–B. Berlin. Math. Ges. 11 (1912), 40–50.

[12] Th. Skolem, Some theorems on irrationality and linear independence. Skand. Mat. Kongr. 11, Trondheim 1949, 77–98. | MR: 53981 | Zbl: 0048.03304

Cited by Sources: