Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson

Rolf Wallisser[1]

  • [1] Mathematisches Institut der Universität Freiburg Eckerstr.1 79104 Freiburg, Deutschland

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 381-396
  • ISSN: 1246-7405

Abstract

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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 G ( x ) = ∑ n = 0 ∞ x n Q ( 1 ) Q ( 2 ) ⋯ Q ( n ) .

How to cite

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Wallisser, Rolf. "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 17.1 (2005): 381-396. <http://eudml.org/doc/249419>.

@article{Wallisser2005,
abstract = {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\{align*\} G(x) = \sum \limits \_\{n=0\}^\infty ~ \frac\{x^n\}\{Q(1) Q(2)\cdots Q(n)\}. \end\{align*\}},
affiliation = {Mathematisches Institut der Universität Freiburg Eckerstr.1 79104 Freiburg, Deutschland},
author = {Wallisser, Rolf},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {381-396},
publisher = {Université Bordeaux 1},
title = {Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson},
url = {http://eudml.org/doc/249419},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Wallisser, Rolf
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
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 381
EP - 396
AB - 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{align*} G(x) = \sum \limits _{n=0}^\infty ~ \frac{x^n}{Q(1) Q(2)\cdots Q(n)}. \end{align*}
LA - eng
UR - http://eudml.org/doc/249419
ER -

References

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  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. Zbl1041.11050
  2. F. Carlson, Sur une propriété arithmétique de quelques fonctions entières. Arkiv för Mathematik, Astronomi och Fysik. Bd 25A. N: 07 (1935). Zbl0011.39202
  3. R. Dedekind, Über den Zusammenhang zwischen der Theorie der Ideale und der höheren Kongruenzen, Abh. Akad. Wiss. Göttingen 23 (1878), 3–37. 
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  5. I. Gerst, J. Brillhart, On the prime divisors of polynomials. Amer. Math. Monthly 78 (1971). Zbl0214.30604MR279071
  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 π . Math. Ann. 43 (1893), 216–219. 
  8. A. Hurwitz, Beweis der Transzendenz der Zahl e . Math. Ann. 43 (1893), 220–221. MR1510809
  9. T. Nagell, Généralisation d’un théorème de Tchebycheff. Journ. de Math. ( 8 e serie), tome IV (1921). Zbl48.1173.01
  10. O. Perron, Irrationalzahlen. Chelsea, New York, (1951). Zbl0029.20303
  11. I. Schur, Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen. S.–B. Berlin. Math. Ges. 11 (1912), 40–50. Zbl43.0261.02
  12. Th. Skolem, Some theorems on irrationality and linear independence. Skand. Mat. Kongr. 11, Trondheim 1949, 77–98. Zbl0048.03304MR53981

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