the regularity of the varied problem (\[Var\_disc\_mom\_syst\]), we get the formula $$\label{f_reg_expansion} f(z)=H_n(\{\mu_k\},\{\lambda_k\},h;z)- p\,
h_{n-1}z^{n-1}-q\,h_{2n-1}z^{2n-1}+O(z^{2n}).$$

In order to obtain the main result of this section, we now show that the above-mentioned problem is indeed regular for a certain choice of the parameters $p$ and $q$. Below we give a possible way of such a choice. The generating polynomial of the system (\[Var\_disc\_mom\_syst\]) for $\alpha_k$ and $\beta_k$ from (\[alpha-beta\]) has the form $$\label{modif_equ} {G}_n(\lambda)={G}_n(p,q;\lambda):= \left|
  \begin{array}{ccccc}
  1 & \lambda & \cdots & \lambda^{n-1} & \lambda^n \\
    s_0 & s_1 & \cdots & s_{n-1}+p & s_{n} \\
    s_1 & s_2 & \cdots & s_n & s_{n+1} \\
 \cdots & \cdots & \cdots & \cdots & \cdots \\
   s_{n-1}+p & s_{n} & \cdots & s_{2n-2} & s_{2n-1}+q \\
  \end{array}
\right|.$$ Obviously, for the parameters $p$ and $q$, being sufficiently large in modulus (comparing with the moments $s_k$ and independently of each other), the roots of this polynomial are arbitrarily close to those of the polynomial of the form (\[modif\_equ\]) with $s_k=0$, $k=\overline{0,2n-1}$. The latter polynomial, as it can be easily checked by expanding the determinant along the first row, has the form $$(-1)^{n(n+1)/2}p^{n}(\lambda^n-q/p),$$ and all its $n$ roots are pairwise distinct. From here, 