*(English)*Zbl 0871.65012

Two new variants of the $\overline{D}$-transformation due to the author [J. Inst. Math. Appl. 26, 1-20 (1980; Zbl 0464.65002)] are designed for computing oscillatory integrals ${\int}_{a}^{\infty}g\left(t\right)\xb7{\mathcal{C}}_{\nu}\left(t\right)dt$, where $g\left(t\right)$ is a non-oscillatory function and ${\mathcal{C}}_{\nu}\left(x\right)$ stands for an arbitrary linear combination of the Bessel functions of the first and second kinds ${J}_{\nu}\left(x\right)$ and ${Y}_{\nu}\left(x\right)$, of arbitrary real order $\nu $.

The author also points out that the application to such integrals of an additional approach involving the so-called $mW$-transformation as well introduced by himself [see Math. Comput. 51, No. 183, 249-266 (1988; Zbl 0694.40004); see also *D. Levin* and *A. Sidi*, Appl. Math. Comput. 9, 175-215 (1981; Zbl 0487.65003)] produces similarly good results. Finally, convergence and stability results of both the $\overline{D}$-transformation and the $mW$-transformation in all of their forms are discussed and a numerical example is added.