Using properties of regular solid spherical harmonics the one-range addition theorems for first and second derivatives of combined Coulomb (eta = 0) and Yukawa (eta > 0) type potentials f(uvs)(eta, (r) over bar) = r(u-v-1)e(-eta r)T(vs)(x, y, z), where T-vs(x, y, z) = [4 pi/(2 nu + 1)](1/2)r(nu)S(vs)(theta, phi), with integer and noninteger values of indices it are presented. With the help of these addition theorems and complete orthonormal sets of Psi(alpha)-ETOs, where alpha = 1, 0, -1, -2,..., a large number of series expansion formulae for arbitrary one-electron multicenter integrals over integer and noninteger n STOs and their derivatives are established. The final results are expressed in terms of two-center overlap integrals between noninteger n STOs. The series expansion formulae obtained in this work is especially useful for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules, and solids. It should be noted that the results for the noninteger indices of potentials and STOs presented in this paper were not appeared in our past publications. (c) 2005 Elsevier B.V. All rights reserved.