# Arithmetical aspects of certain functional equations

Acta Arithmetica (1997)

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

## Access Full Article

top## Abstract

top## How to cite

topLutz 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

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.