Intuitionistic fuzzy sets with membership and non-membership functions of exponential type

In the paper a special kind of intuitionistic fuzzy sets are considered. For brevity further we will call them exponential intuitionistic fuzzy sets. The name is related to the fact that the membership and non-membership functions of these sets are of exponential form. The mentioned sets are useful for utilizing a novel type of investigations that have been proposed by some authors recently. The entire investigation is based on the equation a^x + b^x = 1, which is considered in detail and solved in the paper and several different explicit representations (some of them using Euler's gamma and beta functions) for its solution are found. At the end of the paper some of our formulas are applied and as a result seven new representations as infinite series for the golden ratio are obtained.