The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate J v ( x ) ( | x | small ) {J_v}(x)\;(|x|\;{\text {small}}) , e x K v ( x ) ( Re ⁡ x > 0 , | x | large ) {e^x}{K_v}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}}) , and the modulus and phase of H v ( 1 ) ( x ) ( Re ⁡ x > 0 , | x | large ) H_v^{(1)}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}}) . The first converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for K v + 1 ( x ) / K v ( x ) {K_{v + 1}}(x)/{K_v}(x) the convergence of which increases with | x | |x| . The procedures were tested numerically both for integer and fractional values of v, and for real and complex x.