\bigr|_{M,N\gg1}\stackrel{\defa}{=}\Tr^{\prime}(\sigma_{m+1}^{+}
e^{-{\lambda}H_0}\sigma_1^{-}),
\label{40}$$ where the notation $\Tr^{\prime}$ points out that the procedure presented in Section \[sec2\] is used for the calculation of the normalized average. The difference-differential relation, analogous to the equation  is valid for ${\widetilde F}_{m+1;1}({\lambda})$ : $$\begin{aligned}
&\frac d{d{\lambda}}\,{\widetilde F}_{m+1;1}({\lambda})=
\frac12\,({\widetilde F}_{m;1}({\lambda})+
{\widetilde F}_{m+2;1}({\lambda}))-
\Tr^{{\prime}}(H_0\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-})-{}\notag
\\
&\qquad{}-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr)-
\Tr^{\prime}\biggl(\biggl(\frac{1-\sigma_{m+1}^{z}}{2}\biggr)
\sigma_{m+2}^{+}e^{-{\lambda}H_0}\sigma_{1}^{-}\biggr).
\label{41}\end{aligned}$$ The form of the present equation makes it possible to suppose that the average ${\widetilde F}_{m+1;1}({\lambda})$ can also be of interest as a generating function of the random walks.

Let us turn to the calculation of ${\widetilde F}_{m+1;1}({\lambda})$ in the fermionic representation . It is convenient to reduce the problem to calculation of the generating function of the form: $$\widetilde{\mathcal G}\equiv\Tr^{\prime}
(e^Sc_{m+1}e^{-\lambda H_{0}}c^{\dagger}_1e^{-\nu H_{0}}),
\label{42}$$ where the operator $S$ is defined just like in  (i.e., by means of the matrix $\widehat S = \break \diag\{S_1,S_2,\dots,S_M\}$). Indeed, the functional ${\widetilde F}_{m+1;1}({\lambda})$ corresponds to the choice of $\nu=0$ and $S_k=-i\pi\theta(m-k)$, where $\theta(m-k)$ is the Heavyside function, $\theta(0)=1$. The second term in the right-hand side of  corresponds to the differentiation by $\nu$ at the point $\nu=0$, in the third term we put $\nu=0$ and differentiate with respect to $S_{m+1}$. In both cases we put $S_k=-i\pi\theta(m-k)$. Taking into account the 