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By using the Laplace transformation, the analytical solutions of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations at the leading-order and next-to-leading-order (NLO) approximations for the gluon, singlet, and nonsinglet quark polarization inside the nucleon are obtained. At NLO approximation, a second Laplace transformation is required. Complete NLO calculations need an iteration method, which we try to employ. As an efficiently mathematical tool, we employ the Jacobi polynomials to extract the polarized nucleon structure, using the analytical solutions for the polarized parton densities.Optimum nucleon structure functions are determined by a 2 analysis of the available experimental data, and their uncertainties are estimated by the Hessian method. Our results for nucleon and deuteron structure functions are in good agreement with all available experimental data and fitting parametrization models
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