New analytical formulas for construction of the gamma-ray shielding formulas for a plane, a disk, a cylindrical or a spherical surface (volume) sources are proposed depending on the functional of the binomial coefficients. Numerical stability and optimization of such analytical relations become more important with increasing degree and order. Some numerical results are presented for significant mapping examples and they are briefly discussed. The computation results are compared with those of previously published studies. Using the derived expressions the gamma-ray shielding formulas can be evaluated most efficiently and accurately for mu t = 0, 1, 2. (C) 2006 Elsevier Ltd. All rights reserved.