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New backward recurrences for Bessel functions

Author: Henry C. Thacher
Journal: Math. Comp. 33 (1979), 744-764
MSC: Primary 65D20; Secondary 33A40
MathSciNet review: 521289
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Abstract: The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate $ {J_v}(x)\;(\vert x\vert\;{\text{small}})$, $ {e^x}{K_v}(x)\;(\operatorname{Re} x > 0,\;\vert x\vert\;{\text{large}})$, and the modulus and phase of $ H_v^{(1)}(x)\;(\operatorname{Re} x > 0,\;\vert x\vert\;{\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 $ \vert x\vert$. The procedures were tested numerically both for integer and fractional values of v, and for real and complex x.

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Keywords: Bessel functions, modified Bessel functions, Airy function, Hankel function, Kelvin function, power series, continued fraction, differential equation, recurrence, Miller algorithm
Article copyright: © Copyright 1979 American Mathematical Society

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