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

• Published in: Boundary value problems Vol. 2017; no. 1; pp. 1 - 18
• Main Authors: Mansour, Zeinab, Al-Towailb, Maryam
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 22.11.2017; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Boundary value problems; Difference and Functional Equations; Difference equations; Fourier series; Green’s function; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Polynomials; q-difference equations; q-Fourier expansions; q-Lidstone polynomials; Green’s function; 05A30; 39A13; 39A05; Fourier expansions; 42A16; 11B68; Lidstone polynomials; 30E25; difference equations
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-017-0908-4

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 ) ) , 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 ) , where n ∈ N and 0 ≤ k ≤ 2 n − 1 . These results are a q -analogue of work by Agarwal and Wong of 1989.