A sequence of the real numbers (Formula Presented) is associated with the some indefinite Stieltjes moment problem and generalized Jacobi matrices. The relation between the α regular indefinite Stieltjes moment problem and shifted Darboux transformation of the generalized Jacobi matrix is studied. The new formulas for the Stieltjes polynomials with the shift are found and one are used to obtain the description of the solutions of the α regular indefinite Stieltjes moment problem.