A multi-dimensional functional equation having cubic forms as solutions


Won-Gil Park - Department of Mathematics Education, College of Education, Mokwon University, Daejeon 35349, Republic of Korea.
Jae-Hyeong Bae - Humanitas College, Kyung Hee University, Yongin 17104, Republic of Korea.


In this paper, we obtain some results on the m-variable cubic functional equation \[f(2x_1 + y_1,..., 2x_m + y_m) + f(2x_1 - y_1,..., 2x_m - y_m)\\ = 2f(x_1 + y_1,..., x_m + y_m) + 2f(x_1 - y_1,..., x_m - y_m) + 12f(x_1,..., x_m).\] The cubic form \(f(x_1,..., x_m) = \sum_{1\leq i\leq j\leq k\leq m} a_{ijk}x_ix_jx_k\) is a solution of the above functional equation.



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