New backward recurrences for Bessel functions

Abstract:

The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate ${J_v}(x)\;(|x|\;{\text {small}})$, ${e^x}{K_v}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})$, and the modulus and phase of $H_v^{(1)}(x)\;(\operatorname {Re} x > 0,\;|x|\;{\text {large}})$. 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 ${K_{v + 1}}(x)/{K_v}(x)$ the convergence of which increases with $|x|$. The procedures were tested numerically both for integer and fractional values of v, and for real and complex x.