On a multiplicative type sum form functional equation and its role in information theory

• DOI: 10.1007/s10492-006-0018-6
• Publication Date: 2006-10-17T18:34:10Z
• Authors: Prem Nath; Dhiraj Kumar Singh
• Keywords: Gauss sum; Functional equation; Unit interval

In this paper, we obtain all possible general solutions of the sum form functional equations $$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {f(p_i q_j )} } = \sum\limits_{i = 1}^k {g(p_i )} \sum\limits_{j = 1}^l {h(q_j )} $$ and $$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {F(p_i q_j )} } = \sum\limits_{i = 1}^k {G(p_i ) + } \sum\limits_{j = 1}^l {H(q_j ) + \lambda } \sum\limits_{i = 1}^k {G(p_i )} \sum\limits_{j = 1}^l {H(q_j )} $$ valid for all complete probability distributions (p1, ..., pk), (q1, ..., ql), k ≥ 3, l ≥ 3 fixed integers; λ ∈ ℝ, λ ≠ 0 and F, G, H, f, g, h are real valued mappings each having the domain I = [0, 1], the unit closed interval.