Arithmetical aspects of certain functional equations
Lutz G. Lucht
Acta Arithmetica, Tome 80 (1997), p. 257-277 / Harvested from The Polish Digital Mathematics Library
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The classical system of functional equations      1/n∑ν=0n-1F((x+ν)/n)=n-sF(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=0n-1F((x+ν)/n)=∑d=1∞λn(d)F(dx) (n ∈ ℕ) with complex valued sequences λn. 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.

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:207091 
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Lutz G. Lucht. Arithmetical aspects of certain functional equations. Acta Arithmetica, Tome 80 (1997) pp. 257-277. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav82i3p257bwm/

[00000] [1] E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964. | Zbl 0144.06802

[00001] [2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, London, 1960. | Zbl 0086.25803

[00002] [3] D. Kubert, The universal ordinary distribution, Bull. Soc. Math. France 107 (1979), 179-202. | Zbl 0409.12021

[00003] [4] M. Kuczma, Functional Equations in a Single Variable, PWN-Polish Scientific Publishers, Warszawa, 1968.

[00004] [5] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, London, 1986. | Zbl 0629.12016

[00005] [6] L. G. Lucht, Arithmetical sequences and systems of functional equations, Aequationes Math. 53 (1997), 73-90. | Zbl 0881.39021

[00006] [7] C. Methfessel, Multiplicative and additive recurrent sequences, Arch. Math. (Basel) 63 (1994), 321-328. | Zbl 0812.11014

[00007] [8] J. Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. 29 (1983), 281-322. | Zbl 0557.10031

[00008] [9] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.

[00009] [10] T. Popoviciu, Remarques sur la définition fonctionnelle d'un polynôme d'une variable réelle, Mathematica (Cluj) 12 (1936), 5-12. | Zbl 0015.34403

[00010] [11] W. Rudin, Real and Complex Analysis, McGraw-Hill, London, 1970. | Zbl 0954.26001

[00011] [12] M. F. Yoder, Continuous replicative functions, Aequationes Math. 13 (1975), 251-261. | Zbl 0333.39007