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 , , and the modulus and phase of . 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 the convergence of which increases with . 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

DOI:
https://doi.org/10.1090/S0025-5718-1979-0521289-1

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