A generalization of the $ q $-Lidstone series

In this paper, we study the existence of solutions for the general $ q $-Lidstone problem: \begin{document}$ \begin{equation*} (D_{q^{-1}}^{r_n}f)(1) = a_n, \quad(D_{q^{-1}}^{s_n}f)(0) = b_n, \quad (n\in \mathbb{N}) \end{equation*} $\end{document} where $ (r_n)_n $ and $ (s_n)_n $ are two sequences of non-negative integers and $ (a_n)_n $ and $ (b_n)_n $ are two sequences of complex numbers. We define a $ q^{-1} $-standard set of polynomials and then we introduce a generalization of the $ q $-Lidstone expansion theorem.