Arithmetical aspects of certain functional equations

• Autorzy: Lutz G. Lucht
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 39A10: Difference equations, additive
• 11A25: Arithmetic functions; related numbers; inversion formulas
• 39B62: Functional inequalities, including subadditivity, convexity, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1997
• Tom: 82
• Numer: 3
• Strony: 257-277

Daty

wydano

1997

otrzymano

1996-10-01

poprawiono

1996-12-04

Twórcy

autor

Lutz G. Lucht

• Institut für Mathematik, Technische Universität Clausthal, Erzstrasse 1, 38678 Clausthal-Zellerfeld, Germany

Bibliografia

• [1] E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964.
• [2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, London, 1960.
• [3] D. Kubert, The universal ordinary distribution, Bull. Soc. Math. France 107 (1979), 179-202.
• [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.
• [6] L. G. Lucht, Arithmetical sequences and systems of functional equations, Aequationes Math. 53 (1997), 73-90.
• [7] C. Methfessel, Multiplicative and additive recurrent sequences, Arch. Math. (Basel) 63 (1994), 321-328.
• [8] J. Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities, Enseign. Math. 29 (1983), 281-322.
• [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.
• [11] W. Rudin, Real and Complex Analysis, McGraw-Hill, London, 1970.
• [12] M. F. Yoder, Continuous replicative functions, Aequationes Math. 13 (1975), 251-261.