Ratios of Bessel Functions and Roots of aJ x. qxJX x. s0

Derivatives with respect to order n and argument x of the ratio J x.rJ x. of n nq1 Bessel functions are studied for all real values of n and x. Our results generalize and sharpen previously known results, and allow the deduction of a more complete description, including monotonicity and multiplicity, of the positive roots of the equation aJn x.qxJnX x.s0 for all real n and a.