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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.

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

This paper introduces a q-analog of the class of completely convex functions. We prove specific properties, including that q-completely convex functions have convergent q-Lidstone series expansions. We also provide a sufficient and necessary condition for a real function to have an absolutely convergent q-Lidstone series expansion.

This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f(z) has such an expansion, in which case convergence is uniform on compact sets.

In this paper, we present the q-Lidstone polynomials which are q-Bernoulli polynomials generated by the third Jackson q-Bessel function, based on the Green’s function of a certain q-difference equation. Also, we provide the q-Fourier series expansions of these polynomials and derive some results related to them.

Abstract In this paper, we study some properties of q-Lidstone polynomials by using Green’s function of certain q-differential systems. The q-Fourier series expansions of these polynomials are given. As an application, we prove the existence of solutions for the linear q-difference equations ( − 1 ) n D q − 1 2 n y ( x ) = ϕ ( x , y ( x ) , D q − 1 y ( x ) , D q − 1 2 y ( x ) , … , D q − 1 k y ( x ) ) , $$ (-1)^{n} D_{q^{-1}}^{2n} y(x)= \phi\bigl(x,y(x), D_{q^{-1}}y(x), D_{q^{-1}}^{2}y(x), \ldots , D_{q^{-1}}^{k}y(x)\bigr), $$ subject to the boundary conditions D q − 1 2 j y ( 0 ) = β j , D q − 1 2 j y ( 1 ) = γ j ( β j , γ j ∈ C , j = 0 , 1 , … , n − 1 ) , $$ D_{q^{-1}}^{2j}y(0)= \beta_{j},\qquad D_{q^{-1}}^{2j}y(1)= \gamma_{j} \quad(\beta _{j},\gamma_{j} \in\mathbb{C}, j=0,1,\ldots,n-1), $$ where n ∈ N $n\in\mathbb{N}$ and 0 ≤ k ≤ 2 n − 1 $0\leq k\leq2n-1$ . These results are a q-analogue of work by Agarwal and Wong of 1989.

In this paper, we introduce the complementary q-Lidstone interpolating polynomial of degree 2 n , which involves interpolating data at the odd-order q-derivatives. For this polynomial, we will provide a q-Peano representation of the error function. Next, we use these results to prove the existence of solutions of the complementary q-Lidstone boundary value problems. Some examples are included.