In this paper, for one-and two-center one-electron integrals between irregular solid harmonics (ISHs) and Slater-type orbitals (STOs) with same screening parameters distinct analytical formulas are obtained. First, an expression is derived in terms of ISHs and STOs using a Fourier transform method and Taylor expansion. The second expression is written by taking the two-center overlap integrals over STOs with the different screening parameters to a limit. Numerical calculations of these two formulae have been performed for chosen quantum numbers. Finally, the two-center basic nuclear attraction integrals have been calculated as a special case of these integrals between ISHs and STOs. Results obtained have been compared with the literature. This comparison shows a perfect match.