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

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 2n q-1 y(x) = φ(x, y(x), D q-1 y(x), D 2 q-1 y(x),. .. , D k q-1 y(x)), subject to the boundary conditions D 2j q-1 y(0) = β j , D 2j q-1 y(1) = γ j (β j , γ j ∈ C, j = 0, 1,. .. , n-1), where n ∈ N and 0 ≤ k ≤ 2n-1. These results are a q-analogue of work by Agarwal and Wong of 1989.

The $q$-Lidstone series involving $q$-Bernoulli and $q$-Euler polynomials generated by the third Jackson $q$-Bessel function

“…The analogous problem for the classical case was studied in [17] by Whittaker. Secondly, we are interested in constructing the q-Fourier series for the q-Lidstone polynomials A n (z) and B n (z), and applying such expansions to a solution of certain q-boundary value problems as in [12] and [13].…”

“…The other form expand the function in q-Lidstone polynomials based on q-Euler polynomials and the coefficients contain the even and odd powers of the q-derivative δqf (z) δqz . The publications [12,13] are the most affiliated with this work. This article is organized as follows: in Section 2, we state some definitions and present some background on q-analysis which we need in our investigations.…”

Odd and even $q$-type Lidstone polynomial sequences

“…In [21] the authors construct the complementary q-Lidstone polynomials ν n (z) and τ n (z) of degree 2n satisfying (1.6)    ν 0 (z) = 1 = τ 0 (z), D q −1 ν n (0) = D q −1 τ n (1) = 0, D 2 q −1 τ n (z) = τ n−1 (z) and D 2 q −1 ν n (z) = ν n−1 (z). Some applications related to q-Lidstone expansion theorem are studied in [20,21].…”

The Complementary q-Lidstone Interpolating Polynomials and Applications

“…In [6], we studied the boundary value problems, which consist of an even order q-differential equation and the q-Lidstone boundary conditions. This paper extends this technique to solve the following problem:…”