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.

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.

In this paper, we employ the fractional q-calculus in solving a triple system of q-Integral equations, where the kernel is the third Jackson q-Bessel functions. The solution is reduced to two simultaneous Fredholm qintegral equation of the second kind. Examples are included. We also apply a result in [6] for solutions of dual q 2 -integral equations to solve certain triple integral equations.2000 Mathematics Subject Classification. primary 45F10; secondary 31B10, 26A3, 33D45. Key words and phrases. The third Jackson q-Bessel functions, fractional q-integral operators, triple q-integral equations.

No abstract

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.