Irrationality exponents of certain alternating series

Let m be a positive integer, \((w_n)\) be a sequence of positive integers, and \((y_n)\) be a sequence of nonzero integers with \(y_1\ge 1\). Define \(q_0=1, q_1=w_0, q_{n+1}=q_{n-1}(w_nq_n^m+y_n) \,\,(n\ge 1)\). Under certain assumptions on \((w_n)\) and \((y_n)\), we give the exact value of the irrationality exponent of the number

$$\begin{aligned} \xi =\sum _{n=1}^{\infty }(-1)^{n-1}\frac{y_1y_2\cdots y_n}{q_nq_{n-1}}. \end{aligned}$$