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Functional Equation Characterizing Polynomial Functions and an Algorithm

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Abstract

In the present paper we solve the generalized form of the functional equation considered in Fechner and Gselmann (Publ Math Debr 80(1–2):143–154, 2012), that is we find general solutions of the following functional equation.

$$\begin{aligned} F(x + y) - F(x) - F(y) = \sum \limits _{i=1}^m (a_i x + b_i y)f(\alpha _i x + \beta _i y) \end{aligned}$$
(0.1)

for all \(x,y,a_i,b_i \in \mathbb {R}\) and \(\alpha _i,\beta _i \in \mathbb {Q}.\) Thus we continue investigations presented in Nadhomi et al. (Aequationes Math, 95:1095–1117, 2021) where we generalized the left hand side of Fechner–Gselmann equation. It turns out that under some mild assumption, the pair (F, f) solving (0.1) happens to be a pair of polynomial functions, and in some important cases just the usual polynomials (despite the fact that we assume no regularity of solutions a priori). In the second part of the present paper we formulate an algorithm written in the computer algebra system Maple which determines the polynomial solutions of the functional equations belonging to the class (0.1) (cf. also Borus and Gilányi in 2013 IEEE 4th International Conference on Cognitive Infocommunications (CogInfoCom), pp 559–562, 2013; Aequationes Math 94(4):723–736, 2020; Gilányi in Math Pannon 9(1):55–70, 1998).

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Acknowledgements

1. We are also grateful to Dr. Attila Gilányi, who provided expertise that greatly assisted us during this research in particular, in the area that involves computer algorithms. 2. We are grateful to the referee for the valuable remarks which enabled us to improve the final layout.

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  1. Institute of Mathematics, University of Silesia, Bankowa 14, 40 007, Katowice, Poland

    Chisom Prince Okeke & Maciej Sablik

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  1. Chisom Prince Okeke
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Okeke, C.P., Sablik, M. Functional Equation Characterizing Polynomial Functions and an Algorithm. Results Math 77, 125 (2022). https://doi.org/10.1007/s00025-022-01664-x

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