  The polarized nucleon structure function in the nonsinglet case is
investigated here by a new insight rather than conventional perturbative QCD
(pQCD). For this purpose we note that the solution of the evolution equations
in moment space involves noninteger powers of the coupling constant. Therefore
it is possible to employ a new approach which is called fractional analytical
perturbation theory Consequently, it is possible to remove the Landau
singularities of the renormalized coupling, i.e., at the scales $Q \sim
\Lambda$, using this approach. This provides an opportunity to continue the
desired calculations toward small values of energy scales even less than the
$\Lambda$ scale. To modify the analytical perturbation theory, a newer approach
is introduced, called 2$\delta$anQCD, in which the spectral function of the
holomorphic coupling is parameterized in the low-energy region by two delta
functions. This model gives us more reliable results for the considered QCD
observables, even in the deep infrared region. We calculate the nonsinglet part
of the polarized nucleon structure function, using the 2$\delta$anQCD model,
and compare it with the result from the underlying pQCD where both are in a new
defined scheme, called the Lambert scheme. For this purpose we employ the anQCD
package in the \textit{Mathematica} environment to establish the analytic
(holomorphic) coupling constant. The results at various energy scales are also
compared with the available experimental data, and it turns out that there is a
good consistency between them. The results show that the obtained nucleon
structure function at small energy scales has smoother behavior when using the
2$\delta$anQCD model than the underlying pQCD. In fact the coupling constant in
analytic QCD behaves moderately and it makes the result approach the available
data in a better way.
