code: def second_order_score(y, mean, scale, shape, skewness):
          
          return ((shape+1)/shape)*(y-mean)/(np.power(scale,2) + (np.power(y-mean,2)/shape))/((shape+1)*((np.power(scale,2)*shape) - np.power(y-mean,2))/np.power((np.power(scale,2)*shape) + np.power(y-mean,2),2))
docs: GAS t Update term potentially using second-order information - native Python function
  
          Parameters
          ----------
          y : float
              datapoint for the time series
  
          mean : float
              location parameter for the t distribution
  
          scale : float
              scale parameter for the t distribution
  
          shape : float
              tail thickness parameter for the t distribution
  
          skewness : float
              skewness parameter for the t distribution
  
          Returns
          ----------
          - Adjusted score of the t family