Further results on a new class of functional equations satisfied by polynomial functions

Abstract

Whenever a numerical method produces accurate results, it creates an interesting functional equation, and because regularities is not assumed, unexpected solutions can emerge. Thus, this paper is mainly devoted to finding solutions to a generalized functional equation constructed in this spirit; namely, we solve the generalized form of the functional equation considered in Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012), then considered in Nadhomi et al. (Aequationes Math 95:1095–1117, 2021) and continued in Okeke and Sablik (Results Math 77:125, https://doi.org/10.1007/s00025-022-01664-x, 2022), that is we find the polynomial functions satisfying the following functional equation, 0.1∑i=1nγiF(aix+biy)=∑j=1m(αjx+βjy)f(cjx+djy),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{i=1}^n \\gamma _i F(a_i x + b_i y)= \\sum _{j=1}^m(\\alpha _j x + \\beta _j y) f(c_j x + d_j y), \\end{aligned}$$\\end{document}for every x,yin mathbb R, gamma _i,alpha _j,beta _j in mathbb R, and a_i,b_i,c_j,d_j in mathbb Q, and its special forms. 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 and those from Okeke and Sablik (Results Math 77:125, https://doi.org/10.1007/s00025-022-01664-x, 2022) where the right hand side of the Fechner–Gselmann equation was studied. It turns out that under some assumptions on the parameters involved, the pair (F, f) solving Eq. (0.1) happens to be a pair of polynomial functions.