Irrationality exponents of certain alternating series

Let m be a positive integer, \((w_n)\) $(w_n)$ be a sequence of positive integers, and \((y_n)\) $(y_n)$ be a sequence of nonzero integers with \(y_1\ge 1\) $y_1\ge 1$ . Define \(q_0=1, q_1=w_0, q_{n+1}=q_{n-1}(w_nq_n^m+y_n) \,\,(n\ge 1)\) $q_0=1,q_1=w_0,q_{n+1}=q_{n-1}(w_n{q}_n^m+y_n)(n\ge 1)$ . Under certain assumptions on \((w_n)\) $(w_n)$ and \((y_n)\) $(y_n)$ , we give the exact value of the irrationality exponent of the number \(\begin{aligned} \xi =\sum _{n=1}^{\infty }(-1)^{n-1}\frac{y_1y_2\cdots y_n}{q_nq_{n-1}}. \end{aligned}\) $$\begin{array}{c}\hfill \xi =\sum _{n=1}^{\infty }{(-1)}^{n-1}\frac{y_1{y}_2\cdots y_n}{q_n{q}_{n-1}}.\end{array}$$