Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Taylor-Dirichlet series and algebraic differential-difference equations


Author: Frank Wadleigh
Journal: Proc. Amer. Math. Soc. 80 (1980), 83-89
MSC: Primary 30B50
DOI: https://doi.org/10.1090/S0002-9939-1980-0574513-3
MathSciNet review: 574513
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that if a convergent Taylor-Dirichlet series

$\displaystyle \sum\limits_{k = 0}^\infty {{P_k}(s){e^{ - {\lambda _k}s}},\quad ... ...{C}},{P_k}} (s) \in {\mathbf{C}}[s],\mathcal{R}({\lambda _k}) \uparrow \infty ,$

satisfies an algebraic differential-difference equation then the set of its exponents $ \{ {\lambda _k}\} _{k = 0}^\infty $ has a finite, linear, integral basis. This generalizes a theorem of A. Ostrowski.

An application of the theorem to a problem of oscillation in neuro-muscular systems is indicated.


References [Enhancements On Off] (What's this?)

  • [1] M. Blambert and M. Berland, Sur la convergence des éléments LC-dirichlétiens, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), A963-A966. MR 53#3278. MR 0399434 (53:3278)
  • [2] R. E. Langer, On the zeros of exponential sums and integrals, Bull. Amer. Math. Soc. 37 (1931), 213-239. MR 1562129
  • [3] B. Lepson, On hyperdirichlet series and on related questions of the general theory of functions, Trans. Amer. Math. Soc. 172 (1952), 18-45. MR 13#636. MR 0045801 (13:636e)
  • [4] G. Lunc, On series of Taylor-Dirichlet type, Izv. Akad. Nauk Armjan. SSR Ser. Fiz.-Mat. Nauk 14 (1961), no. 2, 7-16. MR 24#A212. (Russian) MR 0130349 (24:A212)
  • [5] A. Miškelevičius, On the convergence of Dirichlet series, Litovsk. Mat. Sb. 3 (1963), no. 2, 105-113. MR 34#4466. (Russian) MR 0204627 (34:4466)
  • [6] M. N. Oguztöreli and R. B. Stein, An analysis of oscillations in neuro-muscular systems, J. Math. Biol. 2 (1975), 87-105. MR 0389254 (52:10085)
  • [7] A. Ostrowski, Über Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1921), 241-298. MR 1544442
  • [8] L. S. Pontryagin, On the zeros of some elementary transcendental functions, Izv. Akad. Nauk SSSR Ser. Mat. 6 (1942), 115-134. MR 0007918 (4:214a)
  • [9] K. Väisälä, Verallgemeinerung des Begriffes der Dirichletschen Reihen, Acta Univ. Dorp. AI 2 (1921), 3-32.
  • [10] G. Valiron, Sur les solutions des équations différentielles linéaires d'ordre infini et à coéfficients constants, Ann. École Norm. 46 (1929), 25-53. MR 1509291

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30B50

Retrieve articles in all journals with MSC: 30B50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0574513-3
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society