q-Lidstone polynomials and existence results for q-boundary value problems

Abstract

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.

Keywords

• q-difference equations
• Green’s function
• q-Lidstone polynomials
• q-Fourier expansions