functions satisfy the periodic boundary conditions, $$\Upsilon_N(\{u_k\}|\boldsymbol{\mu})\equiv
\det(u_k^{2\mu_l})_{1\leq k,l\leq N}
\label{37}$$ are parametrized by the partitions $\boldsymbol{\mu}$ and by different, up to permutation, sets $\{u_1, \dots, u_N\}$ of solutions of the Bethe equations: $$u_k^{2M}=(-1)^{N-1},\qquad1\leq k\leq N.
\label{38}$$ These solutions have the form: $u_k^2=e^{i2\pi I_k/M}$, where $I_k$ are integers or half-integers (this depends on parity of $N$). Due to the antisymmetry of  with the respect to permutations of the parameters $u_k$, it is sufficient to restrict oneself to the strict partitions $M\geq I_1>I_2>\dots>I_N \geq1$ in order to guarantee the single-valuedness of $u_k$. With the help of  the corresponding normalized average $$\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N\equiv
\frac{\langle\Psi_N|\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}
|\Psi_N\rangle}{\langle\Psi_N\mid\Psi_N\rangle}
\label{39}$$ can be represented as a linear combination of $(N+1)$-point generating functions . Therefore, this average is related to the number of random walks of $N+1$ pedestrians. The initial and the final positions of one of them are fixed at $l_1=1$ and $j_1=m+1$, respectively, while for the rest (*virtual*) pedestrians the choice of their initial and the final positions is arbitrary.

Calculation of equation  is of interest in the thermodynamic limit, when $M$ and $N$ are growing (their ratio remains finite), which means that the number of virtual pedestrians is increasing. In this limit [@26] $$\widetilde F_{m+1;1}({\lambda})\equiv
\langle\sigma_{m+1}^{+}e^{-{\lambda}H_0}\sigma_1^{-}\rangle_N
