Basic analog of Fourier series on a q-linear grid

For 0 < q < 1 define the symmetric q-linear operator acting oil a suitable function f(x) by deltaf(x) = f(q(1/2)x)-f(q (1/2)x). The q-linear initial v alue problem deltaf(x)/deltax = lambdaf(x), f(0) = 1, has two entire functi ons C-q(z) and S-q(z) as linearly independent solutions. The functions C-q( z) and S-q(z) are orthogonal on a discrete set. We consider Fourier expansi ons in these functions and derive analytic bounds on the roots of S-q(Z). ( C) 2001 Academic Press.