Two theorems on meromorphic functions used as a principle for proofs on irrationality

Thomas Nopper; Rolf Wallisser

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 253-261
  • ISSN: 1246-7405

Abstract

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In this paper we discuss two theorems on meromorphic functions of Nikishin and Chudnovsky. Our purpose is to show, how to derive some well-known but not obvious results on irrationality in a systematic and simple way from properties of meromorphic functions with arithmetic conditions. As far as it stands, we have no new results on irrationality, to the contrary some results on numbers of the corollaries are known already since a long time to be transcendental (cf. [4], [9] and [10]). Our main intention lies in theorems on meromorphic functions whose Taylor coefficients are arithmetically characterized. Like Niven [6] used Hermite's method to give all known results on irrationality of trigonometric functions, we use methods going back to Nikishin [5] and Chudnovsky (cf. [2] and [8]), to give results on irrationality of values of non-elementary functions.

How to cite

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Nopper, Thomas, and Wallisser, Rolf. "Two theorems on meromorphic functions used as a principle for proofs on irrationality." Journal de théorie des nombres de Bordeaux 13.1 (2001): 253-261. <http://eudml.org/doc/248724>.

@article{Nopper2001,
abstract = {In this paper we discuss two theorems on meromorphic functions of Nikishin and Chudnovsky. Our purpose is to show, how to derive some well-known but not obvious results on irrationality in a systematic and simple way from properties of meromorphic functions with arithmetic conditions. As far as it stands, we have no new results on irrationality, to the contrary some results on numbers of the corollaries are known already since a long time to be transcendental (cf. [4], [9] and [10]). Our main intention lies in theorems on meromorphic functions whose Taylor coefficients are arithmetically characterized. Like Niven [6] used Hermite's method to give all known results on irrationality of trigonometric functions, we use methods going back to Nikishin [5] and Chudnovsky (cf. [2] and [8]), to give results on irrationality of values of non-elementary functions.},
author = {Nopper, Thomas, Wallisser, Rolf},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {253-261},
publisher = {Université Bordeaux I},
title = {Two theorems on meromorphic functions used as a principle for proofs on irrationality},
url = {http://eudml.org/doc/248724},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Nopper, Thomas
AU - Wallisser, Rolf
TI - Two theorems on meromorphic functions used as a principle for proofs on irrationality
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 253
EP - 261
AB - In this paper we discuss two theorems on meromorphic functions of Nikishin and Chudnovsky. Our purpose is to show, how to derive some well-known but not obvious results on irrationality in a systematic and simple way from properties of meromorphic functions with arithmetic conditions. As far as it stands, we have no new results on irrationality, to the contrary some results on numbers of the corollaries are known already since a long time to be transcendental (cf. [4], [9] and [10]). Our main intention lies in theorems on meromorphic functions whose Taylor coefficients are arithmetically characterized. Like Niven [6] used Hermite's method to give all known results on irrationality of trigonometric functions, we use methods going back to Nikishin [5] and Chudnovsky (cf. [2] and [8]), to give results on irrationality of values of non-elementary functions.
LA - eng
UR - http://eudml.org/doc/248724
ER -

References

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  1. [1] D. Bertrand, A Transcendence Criterion for Meromorphic Functions. In: Transcendence Theory: Advances and Applications, Chapter 12. Academic PressLondon, 1977. Zbl0365.10024MR498419
  2. [2] G.V. Chudnovsky, Contributions to the Theory of Transcendental Numbers. Chapter 9. AMS, 1984. Zbl0594.10024MR772027
  3. [3] M. Erdélyi, T. Oberhettinger, Higher Transcendental Functions, Vol. II. Mc Graw-HillNew York, 1953. Zbl0052.29502
  4. [4] W. Maier, Potenzreihen irrationalen Grenzwertes. J. Reine Angew. Math.156 (1927), 93-148. Zbl53.0340.02JFM53.0340.02
  5. [5] E.M. Nikishin, A Property of the Taylor Coefficients. Math. Notes 29 (1981), 525-527. Zbl0481.10035MR615500
  6. [6] I. Niven, Irrational Numbers. The Mathematical Association of America. Quinn and Boden Company, 1956. Zbl0070.27101MR80123
  7. [7] A.N. Parshin, A.N. Shafarevich, Number Theory IV. Encyclopaedia of Math. Sciences, Vol. 44. Springer, 1998. Zbl1155.00324MR1603604
  8. [8] E. Reyssat, Travaux récents de G. V. Chudnovsky. Séminaire Delange-Pisot-Poitou n° 29 (1977). Zbl0369.10020
  9. [9] A. Shidlovsky, Über die Transzendenz und algebraische Unabhängigkeit von Werten einiger Klassen ganzer Funktionen. Doklady Akad. Nauk SSSR96 (1954), 697-700. Zbl0055.27801
  10. [10] C.L. Siegel, Über einige Anwendungen Diophantischer Approximationen. Abh. Preuss. Akad. Wiss.1929, Nr.1, 70S. JFM56.0180.05
  11. [11] C.L. Siegel, Transcendental Numbers. Princeton University Press, 1949. Zbl0039.04402MR32684

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