On characteristic functions of probability distributions of sums with random permutations of signs

In the paper, the random series
$$ S=\sum_{k=1}^\infty \pm a_k ,\qquad a_k > 0,\qquad \sum_{k=1}^\infty a_k < \infty $$
$S=\sum_{k=1}^\infty \pm a_k$, $a_k > 0$, $\sum_{k=1}^\infty a_k < \infty$ is considered, in which the permutation of signs is subject to the Markov dependence with the matrix of transition probabilities
$$ \begin{pmatrix} p(+1,+1)&p(-1,+1) p(+1,-1)&p(-1,-1) \end{pmatrix}= \begin{pmatrix} 1-\alpha&\alpha \alpha&1-\alpha \end{pmatrix}, \qquad 1<\alpha<1. $$
For the characteristic function $f(z)$ of the sum $S$, the formula
$$ f(z)=\prod^{\infty}_{k=0}\cos(a_kz)+i(1-2\alpha)\sum_{j=0}^{\infty}\psi_j(z)\prod^{\infty}_{k=j+2}\cos(a_kz)\sin(a_{j+1}z), $$
is obtained, where $\psi_j(z)=\mathsf{E}(t_je^{izS_j})$ и $S_j=\sum^j_{k=1}\pm a_k$, $z \in {\mathbf C}^1$.