> \dots > l_N$ and $j_1 > j_2 >\dots > j_N$. Let us consider the $N$-point correlation function ($N\leq M$): $$F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})=
\langle\Uparrow\!\!|\sigma_{j_1}^{+}\sigma_{j_2}^{+}\dots
\sigma_{j_N}^{+}e^{-\lambda H_0}\sigma_{l_1}^{-}
\sigma_{l_2}^{-}\dots\sigma_{l_N}^{-}|\!\!\Uparrow\rangle.
\label{20}$$ The present correlator is related to enumeration of the admissible trajectories which are traced by $N$ vicious walkers. Indeed, the application of the operator $\mathcal D^K_{\lambda/2}$ to  results in the average of the type $$\langle\Uparrow\!\!|\sigma_{j_1}^{+} \sigma_{j_2}^{+} \dots
\sigma_{j_N}^{+}(-2H_0)^K \sigma_{l_1}^{-}\sigma_{l_2}^{-} \dots
\sigma_{l_N}^{-} |\!\!\Uparrow\rangle\,.$$ This average provides the numbers $|P_K (l_1, \dots, l_N\rightarrow j_1, \dots, j_N)|$ that can be established with the help of the commutator $$\lbrack H_0,\sigma_{l_1}^{-}\sigma_{l_2}^{-}\dots\sigma_{l_K}^{-}]=
\sum_{k=1}^K\sigma_{l_1}^{-}\dots\sigma_{l_{k-1}}^{-}
[H_0,\sigma_{l_k}^{-}]\sigma_{l_{k+1}}^{-}\dots\sigma_{l_K}^{-}
\label{21}$$ (in this case, differentiation with respect to $\lambda/2$, instead of $\lambda$, allows to take into account the normalization of the hopping matrix ). The condition of non-intersection of trajectories of the walkers is expressed by the vanishing of the correlation function  for any pair of coinciding indices $l_k$ or $j_p$.

Differentiating  with respect to $\lambda$ and applying , we obtain the equation: $$\frac{d}{d\lambda}\,F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)=
\frac12\sum_{k=1}^N\bigl(F_{j_1,
\dots,j_N;l_1,l_2,\dots,l_k+1,\dots,l_N}({\lambda})+
F_{j_1,\dots,j_N;l_1,l_2,\dots,l_k-1,\dots,l_N}({\lambda})\bigr).
\label{22}$$ Equation has been considered in [@16] for the case of periodicity with respect to the lattice argument and with the initial condition: $$F_{j_1, \dots, j_N;l_1, \dots, l_N}(0)\,=\,\prod_{m=1}^N\delta_{j_m,l_m}\,.$$ The function $F_{j_1,j_2, \dots, j_N; l_1,l_2, \dots, l_N}(\lambda)$ can be expressed 