  We present new asymptotic series for the Legendre and Jacobi functions of the
first and second kinds in terms of Bessel functions with appropriate arguments.
The results are useful in the context of scattering problems, improve on known
limiting results, and allow the calculation of corrections to the leading
Bessel-function approximations for these functions. Our derivations of these
series are based on Barnes-type representations of the Legendre, Jacobi, and
Bessel functions; our method appears to be new. We use the results, finally, to
obtain asymptotic Bessel function expansions for the rotation functions needed
to describe the scattering of particles with spin.
