In this study, a new method is proposed for evaluating electric multipole transition (radial) matrix elements of the generalized type H-nl,nt(k) in hydrogenic atom and ions using the Slater type orbitals (STOs). The formula obtained allows the determination of all multipole transition matrix elements between two different nonrelativistic radial wave functions R-nl and R-nl. A comparative study carried out between the results of analytical computations and other numerical simulations shows that the methods agree well and emphasizing thus the effectiveness and accuracy of the proposed analytical expressions. The simple equation thus obtained has been found to be remarkable accurate and has shown a wide range of applicability. (C) 2011 Elsevier B.V. All rights reserved.