Expansion of the exponential function into an infinite product (Q1866884)

The main result of this paper is as follows. Let an infinite product \[ P(z) = \prod_{_{\substack{ n=1\\2\nmid n}}}^\infty \frac{1-a_nz^n}{1+a_nz^n}\tag{1} \] be given, where \(a_1 = -\frac12\), and \(a_3\), \(a_5\), \(a_7\), \(\dots\) are determined by the relation \[ \sum\limits_{k\mid n} \frac{k}{n}\,a_k^{\frac{n}{k}} = 0,\qquad n=3,5,7,\dots \] where \(\frac{k}{n}\) means summation with respect to all positive integral divisors \(n\). Series (1) converges to the functions \(e^z\) absolutely in the circle \(\,K = \{z\in\mathbb C\: | z| <2\}\,\) and uniformly in the circle \(\,K_d = \{z\in\mathbb C\: | z| \leq d<2\}\).