The statistics of continued fractions for polynomials over a finite field

Given a finite held F of order q and polynomials a, b is an element of F[X] of degrees m < n respectively, there is the continued fraction representation b/a = a(1) + 1/(a(2) + 1/(a(2) + 1/a(3) + ... + 1/a(r))). Let CF(n, k, q) denote the number of such pairs for which deg b = n, deg a < n, and for 1 less than or equal to j less than or equal to r, deg a(j) less than or equal to k. We give both an exact recurrence relation, and an asymptotic analysis, for CF(n, k, q). The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of r. Averaged over all a and b as above, this presents no difficulties. The average value of r is n(1 - 1/q), and there is full information about the distribution. When b is fixed and only a is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of r that differs from this average by more than O(root n/q).