In this paper, we study the existence of solutions for the general $q$-Lidstone problem:

$$(D_{q^{-1}}^{r_n}f)(1)=a_n,(D_{q^{-1}}^{s_n}f)(0)=b_n,(n\in \mathbb{N})$$

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.

In this paper, we consider the $q$-type $k$-Lidstone series. The series follows from expanding certain classes of entire functions in terms of Jackson $q^{-1}$- derivatives at integers congruent to r modulo k, where $k$ is a positive integer. We study the main properties of the fundamental polynomials that appear in the series expansion. We include a detailed study for the case $k=3$ with some examples.