Differential-difference Equations in Entire Functions

  • Genrich Belitskii
  • Vadim Tkachenko
Part of the Operator Theory: Advances and Applications book series (OT, volume 197)


For a linear differential-difference equation with real shifts in the complex plane we prove a theorem of existence of entire solutions for an arbitrary entire function in the r.h.s. and, using it, show that the space of entire solutions of the corresponding homogeneous equation is infinite dimensional.

Mathematics Subject Classification (2000)

Primary:30D05 Secondary:34K06 


Linear functional equations entire functions 


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  1. [1]
    Nörlund, N.E. Differenzenrechnung, Springer Verlag, Berlin, 1924.Google Scholar
  2. [2]
    Gel’fond, A.O. Calculus of Finite Differences, Third corrected edition, Nauka, Moscow, 1967; English translation in: International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, 1971.Google Scholar
  3. [3]
    Naftalevich, A. On a differential-difference equation, Michigan Math. J. 22, no. 3, 205–223, 1975.MathSciNetGoogle Scholar
  4. [4]
    Naftalevich, A. Application of an iteration method for the solution of a difference equation, Matematicheskii Sbornik, 57 (99), 151–178, 1962.Google Scholar
  5. [5]
    Hurwitz, A. Mathematische Werke, Birkhäuser Verlag, Basel, Bd. II, S. 752, 1933.zbMATHGoogle Scholar
  6. [6]
    Belitskii G., and Tkachenko V. One-dimensional Functional Equations, Birkhäuser Verlag, Basel-Boston-Berlin, 2003.zbMATHGoogle Scholar
  7. [7]
    Levin, B.Ya. Lectures on Entire Functions, AMS, Translations of Mathematical Monographs, 150, 1996.Google Scholar
  8. [8]
    Markuschevich, A.I. Theory of Analytic Functions, v. 2, Nauka, Moscow, 1968.Google Scholar
  9. [9]
    Naimark M.A. Linear Differential Operators, Ungar, New York, 1968.zbMATHGoogle Scholar
  10. [10]
    Hartman, F. Ordinary Differential Equations, John Wiley & Sons, New York-London-Sydney, 1964.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Genrich Belitskii
    • 1
  • Vadim Tkachenko
    • 1
  1. 1.Dept. of MathematicsBen-Gurion UniversityBeer—ShevaIsrael

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