We prove that a linear q -difference equation of order n has a fundamental set of n -linearly independent solutions. A q -type Wronskian is derived for the n th order case extending the results of Swarttouw–Meijer (1994) in the regular case. Fundamental systems of solutions are constructed for the n -th order linear q -difference equation with constant coefficients. A basic analog of the method of variation of parameters is established.

This paper is devoted to studying a q-analogue of Sturm-Liouville eigenvalue problems. We formulate a self-adjoint q-difference operator in a Hilbert space. Some of the properties of the eigenvalues and the eigenfunctions are discussed. Green's function is constructed and the problem in question is inverted into a q-type Fredholm integral operator with a symmetric kernel. The set of eigenfunctions is shown to be a complete orthogonal set in the Hilbert space. Examples involving basic trigonometric functions are involved.

We investigate the zeros of q-Bessel functions of the second and third types as well as those of the associated finite q-Hankel transforms. We derive asymptotic relations of the zeros of the q-Bessel functions by comparison with zeros of the theta function. The asymptotics of q-Bessel functions are also given. Zeros of finite q-Hankel transforms of q-summable functions are shown to be real and simple except for a finite number of possible non real zeros. Sufficient conditions are given to guarantee that all zeros are real. We give some applications concerning zeros of combinations of q-Bessel functions.