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Keywords:
sum form functional equation; additive function; multiplicative function
sum form functional equation; additive function; multiplicative function
Summary:
In this paper, we obtain all possible general solutions of the sum form functional equations \[ \align \sum_{i=1}^{k}\sum_{j=1}^{\ell}f(p_iq_j)=&\sum_{i=1}^{k}g(p_i) \sum_{j=1}^{\ell}h(q_j)\\ \text{and} \sum_{i=1}^{k}\sum_{j=1}^{\ell}F(p_iq_j)=&\sum_{i=1}^{k} G(p_i)+\sum_{j=1}^{\ell}H(q_j)+ \lambda\sum_{i=1}^{k}G(p_i)\sum_{j=1}^{\ell}H(q_j) \endalign
\] valid for all complete probability distributions $(p_1,\ldots ,p_k)$, $(q_1,\ldots ,q_\ell )$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in \mathbb{R}$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.
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