## New backward recurrences for Bessel functions

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- by Henry C. Thacher PDF
- Math. Comp.
**33**(1979), 744-764 Request permission

## 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*.

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## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp.
**33**(1979), 744-764 - MSC: Primary 65D20; Secondary 33A40
- DOI: https://doi.org/10.1090/S0025-5718-1979-0521289-1
- MathSciNet review: 521289