Continuation of Functions Representable via Infinitely Multiple Exponentials with Alternating Exponents

For infinitely multiple exponentials with two alternating exponents, we construct continuations to unbounded domains, so that these continuations are the limit functions of sequences of finitely multiple exponentials. We prove that, for different exponents, such a continuation is unique in the class of continuously differentiable functions. We extend the domain of existence of infinitely multiple exponentials. For exponents with the same signs, this domain remains bounded, while for exponents with different signs, it coincides with the number axis.