Computation of modified Bessel functions and their ratios

Source abstract

An efficient algorithm for calculating ratios r v ( x ) = I v + 1 ( x ) / I v ( x ) , v ≧ 0 , x ≧ 0 {r_v}(x) = {I_{v + 1}}(x)/{I_v}(x),v \geqq 0,x \geqq 0 , is presented. This algorithm in conjunction with the recursion relation for r v ( x ) {r_v}(x) gives an alternative to other recursive methods for I v ( x ) {I_v}(x) when approximations for low-order Bessel functions are available. Sharp bounds on r v ( x ) {r_v}(x) and I v ( x ) {I_v}(x) are also established in addition to some monotonicity properties of r v ( x ) {r_v}(x) and r v ′ ( x ) r’_{v}(x) .