On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order

摘要

For the coefficients A_n(#zeta#) and B_n(#zeta#), that occur in the uniform asymptotic expansions of Bessel functions of large order, we give asymptotic expansions as n -> infinity. The coefficients in these asymptotic expansions are again A_m(#zeta#) and B_m(#zeta#), and the asymptotic base consists of functions A~(pq)(n,#zeta#), which can be seen as new generalizations of the Airy function. We describe the asymptotic behaviour of the functions A~(pq)(n,#zeta#), as n -> infinity, and we compute the Taylor-series expansions of A~(pq)(n,#zeta#) at #zeta# = 0. Two numerical examples are included.