Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Bounds for the distance \c(v,s) - c(v) (+/-1,s')\ between adjacent zeros of ylinder functions are given; s and s' are such that There Exists cv,s'' is an element of ]c(v,s,) c(v+/-1,s') [; c(v,k) stands for the kth positive z ero of the cylinder (Bessel) function C-v(x) = cos alphaJ(v)(x) - sin alpha Y(v)(x), alpha is an element of [0; pi[, v is an element of R. These bounds, together with the application of modified (global) Newton met hods based on the monotonic functions f(v)(x) =x(2v-1) C-v(x)/Cv-1(x) and g (v)(x) = -x(-(2v+1))C(v)(x)/Cv+1(x), give rise to forward (c(v,k) --> c(v,k +1)) and backward (c(v,k+1) --> c(v,k)) iterative relations between consecu tive zeros of cylinder functions. The problem of finding all the positive real zeros of Bessel functions C-v( x) for any real alpha and v inside an interval [x(1,) x(2)], x(1) > 0, is s olved in a simple way.