Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals

Abstract

Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for the Macdonald function . The results can be applied, for instance, to study the summability of the divergent Kontorovich-Lebedev integrals in the sense of Jones. Namely, we answer affirmatively a question (cf. [Ehrenmark U. Summability experiments with a class of divergent inverse Kontorovich-Lebedev transforms. Comput Math Appl. 2018;76(1):141–154.]) whether these integrals converge for even entire functions of the exponential type in a weak sense.