to one of the nearest sites). Let all numbers $q_k$ be non-negative, and let the inequality $K\geq q_1+q_2+\dots+q_N$ be fulfilled, which means that the steps that can compensate each other are allowed. Let us denote the number of these steps as $2L$, $$L\equiv\frac{K-q_1-q_2-\dots-q_N}{2}\,.
\label{29}$$ Taking into account , the following formula for the number of paths takes place: $$\mathcal T_K(q_1,q_2,\dots,q_N)=\sum_{L_1+L_2+\dots+L_N=L}
P(q_1+L_1,q_2+L_2,\dots,q_N+L_N,L_1,L_2,\dots,L_N),
\label{30}$$ where summation is taken over all non-negative values of $L_1,L_2,\dots,L_N$, provided that their sum is equal to $L$, and the formula  for the number of permutations with repeats is used.

Turning back to the function $F_{j_1,j_2,\dots,j_N;l_1,l_2,\dots,l_N}({\lambda})$ let us define the matrix $(n_{rs})_{1\leq r,s\leq N}$ with the entries $n_{rs}\equiv j_r-l_s$. Then, we arrive to the following

The number of trajectories consisting of $K$ links, which are traced by $N$ vicious walkers on an axis, is expressed through the number of trajectories of the same “length” $K$, which are traced by a single walker travelling over sites of $N$-dimensional lattice of infinite extension: $$\begin{aligned}
|P_K(l_1,\dots,l_N\rightarrow j_1,\dots,j_N)|&\equiv
\mathcal D^K_{\lambda/2}\bigl[F_{j_1,\dots,j_N;l_1,\dots,l_N}(\lambda)\bigr]=
\\
&=
\sum_{S_{a_1,a_2,\dots,a_N}}(-1)^{\mathcal P_S}
\mathcal T_K(n_{a_11},n_{a_22},\dots,n_{a_NN}),
\end{aligned}
\label{31}$$ where summation is taken over all permutations $S_{a_1,a_2,\dots,a_N}\equiv$ $S(\begin{smallmatrix}1,&2,&\dots,&N \\ a_1,&a_2,&\dots,&a_N\end{smallmatrix})$ of the numbers $1,2,\dots,N$, and $\mathcal P_S$ implies a parity of a specific 