Arithmetical aspects of certain functional equations

Lutz G. Lucht

Acta Arithmetica (1997)

  • Volume: 82, Issue: 3, page 257-277
  • ISSN: 0065-1036

Abstract

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The classical system of functional equations       1 / n ∑ ν = 0 n - 1 F ( ( x + ν ) / n ) = n - s F ( x ) (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to       1 / n ∑ ν = 0 n - 1 F ( ( x + ν ) / n ) = ∑ d = 1 ∞ λ n ( d ) F ( d x ) (n ∈ ℕ) with complex valued sequences λ n . This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.

How to cite

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Lutz G. Lucht. "Arithmetical aspects of certain functional equations." Acta Arithmetica 82.3 (1997): 257-277. <http://eudml.org/doc/207091>.

@article{LutzG1997,
abstract = {The classical system of functional equations      $ 1/n ∑_\{ν=0\}^\{n-1\} F((x+ν)/n) = n^\{-s\} F(x)$ (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to      $ 1/n ∑_\{ν=0\}^\{n-1\} F((x+ν)/n) = ∑_\{d=1\}^∞ λ_n(d)F(dx)$ (n ∈ ℕ) with complex valued sequences $λ_n$. This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.},
author = {Lutz G. Lucht},
journal = {Acta Arithmetica},
keywords = {extended replicativity equations; functional equations; arithmetic functions; Fourier coefficients; recurrent sequences; multiplicative generating functions; aperiodic solutions; difference equations},
language = {eng},
number = {3},
pages = {257-277},
title = {Arithmetical aspects of certain functional equations},
url = {http://eudml.org/doc/207091},
volume = {82},
year = {1997},
}

TY - JOUR
AU - Lutz G. Lucht
TI - Arithmetical aspects of certain functional equations
JO - Acta Arithmetica
PY - 1997
VL - 82
IS - 3
SP - 257
EP - 277
AB - The classical system of functional equations      $ 1/n ∑_{ν=0}^{n-1} F((x+ν)/n) = n^{-s} F(x)$ (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to      $ 1/n ∑_{ν=0}^{n-1} F((x+ν)/n) = ∑_{d=1}^∞ λ_n(d)F(dx)$ (n ∈ ℕ) with complex valued sequences $λ_n$. This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.
LA - eng
KW - extended replicativity equations; functional equations; arithmetic functions; Fourier coefficients; recurrent sequences; multiplicative generating functions; aperiodic solutions; difference equations
UR - http://eudml.org/doc/207091
ER -

References

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  1. [1] E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964. Zbl0144.06802
  2. [2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, London, 1960. Zbl0086.25803
  3. [3] D. Kubert, The universal ordinary distribution, Bull. Soc. Math. France 107 (1979), 179-202. Zbl0409.12021
  4. [4] M. Kuczma, Functional Equations in a Single Variable, PWN-Polish Scientific Publishers, Warszawa, 1968. 
  5. [5] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, London, 1986. Zbl0629.12016
  6. [6] L. G. Lucht, Arithmetical sequences and systems of functional equations, Aequationes Math. 53 (1997), 73-90. Zbl0881.39021
  7. [7] C. Methfessel, Multiplicative and additive recurrent sequences, Arch. Math. (Basel) 63 (1994), 321-328. Zbl0812.11014
  8. [8] J. Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. 29 (1983), 281-322. Zbl0557.10031
  9. [9] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924. 
  10. [10] T. Popoviciu, Remarques sur la définition fonctionnelle d'un polynôme d'une variable réelle, Mathematica (Cluj) 12 (1936), 5-12. Zbl0015.34403
  11. [11] W. Rudin, Real and Complex Analysis, McGraw-Hill, London, 1970. Zbl0954.26001
  12. [12] M. F. Yoder, Continuous replicative functions, Aequationes Math. 13 (1975), 251-261. Zbl0333.39007

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