In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind with argument + expressed as an integral of a product of two 's, one with argument and another with argument . We take advantage of recently obtained asymptotics for with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of , we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.

We provide an explicit formula for a Sonine-Gegenbauer integral, which seems to be unknown in the literature so far. For another type of these integrals, we show a dependence relation over the rational functions, including the explicit calculation of the coefficient functions.