Appendix
The Double Sine Function
The double sine function \(S_2(z):=S_2(z|\varvec{\omega })\), see [Ku] and references therein, is a meromorphic function that satisfies two functional relations
$$\begin{aligned} \frac{S_2(z)}{S_2(z+\omega _1)}=2\sin \frac{\pi z}{\omega _2},\qquad \frac{S_2(z)}{S_2(z+\omega _2)}=2\sin \frac{\pi z}{\omega _1} \end{aligned}$$
(A.1)
and inversion relation
$$\begin{aligned} S_2(z)S_2(-z)=-4\sin \frac{\pi z}{\omega _1}\sin \frac{\pi z}{\omega _2}, \end{aligned}$$
(A.2)
or equivalently
$$\begin{aligned} S_2(z)S_2(\omega _1+\omega _2-z)=1. \end{aligned}$$
(A.3)
The factorization formula
$$\begin{aligned} S_2(z + m \omega _1 + k \omega _2) = (-1)^{mk} \, \frac{S_2(z + m \omega _1) \, S_2(z + k \omega _2)}{S_2(z)} \end{aligned}$$
(A.4)
follows from (A.1). The function \(S_2(z)\) is a meromorphic function of z with poles at
$$\begin{aligned} z_{m,k} = m\omega _1 + k\omega _2, \qquad m,k \ge 1\end{aligned}$$
(A.5)
and zeros at
$$\begin{aligned} z_{-m,-k}=-m\omega _1-k\omega _2,\qquad m,k\ge 0. \end{aligned}$$
(A.6)
For \(\omega _1/\omega _2 \not \in \mathbb {Q}\) all poles and zeros are simple. The residues of \(S_2(z)\) and \(S^{-1}_2(z)\) at these points are
$$\begin{aligned} \underset{z = z_{m,k}}{Res} \, S_2(z)&= \frac{\sqrt{\omega _1\omega _2}}{2\pi }\frac{(-1)^{mk}}{\prod \nolimits _{s=1}^{m - 1}2\sin \dfrac{\pi s\omega _1}{\omega _2}\prod \nolimits _{l=1}^{k - 1}2\sin \dfrac{\pi l\omega _2}{\omega _1}}, \end{aligned}$$
(A.7)
$$\begin{aligned} \underset{z = z_{-m,-k}}{Res} \, S^{-1}_2(z)&= \frac{\sqrt{\omega _1\omega _2}}{2\pi }\frac{(-1)^{mk+m+k}}{\prod \nolimits _{s=1}^m2\sin \dfrac{\pi s\omega _1}{\omega _2}\prod \nolimits _{l=1}^k2\sin \dfrac{\pi l\omega _2}{\omega _1}}. \end{aligned}$$
(A.8)
In the analytic region \( \textrm{Re}\,z \in ( 0, \textrm{Re}\,(\omega _1 + \omega _2) )\) we have the following integral representation for the logarithm of \(S_2(z)\)
$$\begin{aligned} \ln S_2 (z) = \int _0^\infty \frac{dt}{2t} \left( \frac{{\text {sh}}\left[ (2z - \omega _1 - \omega _2)t \right] }{ {\text {sh}}(\omega _1 t) {\text {sh}}(\omega _2 t) } - \frac{ 2z - \omega _1 - \omega _2 }{ \omega _1 \omega _ 2 t } \right) . \end{aligned}$$
(A.9)
It is clear from this representation that the double sine function is homogeneous
$$\begin{aligned} S_2( \gamma z | \gamma \omega _1, \gamma \omega _2 ) = S_2(z|\omega _1, \omega _2), \qquad \gamma \in (0, \infty ) \end{aligned}$$
(A.10)
and invariant under permutation of periods
$$\begin{aligned} S_2(z| \omega _1, \omega _2) = S_2(z | \omega _2, \omega _1). \end{aligned}$$
(A.11)
The double sine function can be expressed through the Barnes double Gamma function \(\Gamma _2(z|\varvec{\omega })\) [B],
$$\begin{aligned} S_2(z|\varvec{\omega })=\Gamma _2(\omega _1+\omega _2-z|\varvec{\omega })\Gamma _2^{-1}(z|\varvec{\omega }), \end{aligned}$$
(A.12)
and its properties follow from the corresponding properties of the double Gamma function. It is also connected to the Ruijsenaars hyperbolic Gamma function \(G(z|\varvec{\omega })\) [R2]
$$\begin{aligned} G(z|\varvec{\omega }) = S_2\Bigl (\imath z + \frac{\omega _1 + \omega _2}{2} \,\Big |\, \varvec{\omega }\Bigr ) \end{aligned}$$
(A.13)
and to the Faddeev quantum dilogarithm \(\gamma (z|\varvec{\omega })\) [F]
$$\begin{aligned} \gamma (z|\varvec{\omega }) = S_2\Bigl (-\imath z + \frac{\omega _1+\omega _2}{2}\, \Big |\, \varvec{\omega }\Bigr ) \exp \Bigl ( \frac{\imath \pi }{2\omega _1 \omega _2} \Bigl [z^2 + \frac{\omega _1^2+\omega _2^2}{12} \Bigr ]\Bigr ). \end{aligned}$$
(A.14)
Both \(G(z|\varvec{\omega })\) and \(\gamma (z|\varvec{\omega })\) were investigated independently.
In the paper we deal only with ratios of double sine functions denoted by \(\mu (x)\) (1.6) and K(x) (1.11)
$$\begin{aligned} \begin{aligned}\mu (x)&=S_2(\imath x)S_2^{-1} (\imath x+g),\\ K(x)&= S_2\left( \imath x+\frac{\omega _1+\omega _2}{2}+\frac{g}{2}\right) S_2^{-1}\left( \imath x+\frac{\omega _1+\omega _2}{2}-\frac{g}{2}\right) . \end{aligned} \end{aligned}$$
(A.15)
Now we will give the key asymptotic formulas and bounds for them, which were derived in [BDKK, Appendices A, B] from the known results for the double Gamma function.
In what follows we assume conditions (1.8), (1.9)
$$\begin{aligned} \textrm{Re}\,\omega _j> 0, \qquad 0< \textrm{Re}\,g < \textrm{Re}\,\omega _1 + \textrm{Re}\,\omega _2, \qquad \nu _g = \textrm{Re}\,\hat{g} > 0, \end{aligned}$$
(A.16)
where we denoted
$$\begin{aligned} \hat{g} = \frac{g}{\omega _1 \omega _2}. \end{aligned}$$
(A.17)
The functions \(\mu (x)\) and K(x) (A.15) with \(x \in \mathbb {R}\) have the following asymptotics
$$\begin{aligned} \mu (x) \sim e^{\pi \hat{g} | x | \pm \imath \frac{\pi \hat{g} g^*}{2} }, \qquad K(x) \sim e^{- \pi \hat{g} |x|}, \qquad x\rightarrow \pm \infty . \end{aligned}$$
(A.18)
and bounds
$$\begin{aligned} |\mu (x)| \le C e^{\pi \nu _g |x|}, \qquad |K(x)| \le C e^{-\pi \nu _g |x|}, \qquad x \in \mathbb {R}\end{aligned}$$
(A.19)
where C is a positive constant uniform for compact subsets of parameters \(\varvec{\omega }, g\) preserving the mentioned conditions, see [BDKK, eq.(B.3)].
Another key result that we need in the paper is the following Fourier transform formula given in [R3, Proposition C.1], which we rewrite in terms of the double sine function using connection formula (A.13). This Fourier transform can be already found in [FKV, PT].
Proposition
[R3] For real positive periods \(\omega _1, \omega _2\) we have
$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}} dx \, e^{\frac{2\pi \imath }{\omega _1 \omega _2} y x} S_2\Bigl (\imath x - \imath \nu + \frac{\omega _1 + \omega _2}{2} \Bigr ) S_2^{-1} \Bigl ( \imath x - \imath \rho + \frac{\omega _1 + \omega _2}{2} \Bigr ) \\&\quad = \sqrt{\omega _1 \omega _2} \, e^{ \frac{\pi \imath }{\omega _1 \omega _2} y (\nu + \rho ) } S_2(\imath \rho - \imath \nu ) \, S_2^{-1}\Bigl (\imath y + \frac{\imath (\rho - \nu )}{2} \Bigr ) \, S_2^{-1}\Bigl (-\imath y + \frac{\imath (\rho - \nu )}{2} \Bigr ), \end{aligned} \end{aligned}$$
(A.20)
while the parameters \(\nu , \rho , y\) satisfy the conditions
$$\begin{aligned} -\frac{\omega _1 + \omega _2}{2}< {\textrm{Im}}\, \rho< {\textrm{Im}}\, \nu< \frac{\omega _1 + \omega _2}{2}, \qquad |{\textrm{Im}}\,y | < {\textrm{Im}}\,\frac{\nu - \rho }{2}. \end{aligned}$$
(A.21)
In the special case
$$\begin{aligned} \nu = \frac{\imath g}{2}, \qquad \rho = -\frac{\imath g}{2} \end{aligned}$$
(A.22)
taking \(y = \omega _1 \omega _2 \lambda \) and using homogeneity of the double sine (A.10) (with \(\gamma = \omega _1 \omega _2\)) we arrive at the Fourier transform formula for the function K(x) (A.15)
$$\begin{aligned} \int _{\mathbb {R}} dx \; e^{2 \pi \imath \lambda x } {K}(x) = \sqrt{\omega _1 \omega _2} \, S_2(g) \, \hat{{K}}(\lambda ), \end{aligned}$$
(A.23)
where \(| {\textrm{Im}}\,\lambda | < \textrm{Re}\,\hat{g}/2\) and conditions (A.21) are satisfied due to the inequalities on the coupling constant g (1.8), (1.9). Here we recall the notations
$$\begin{aligned} \hat{K}(\lambda ) = K_{\hat{g}^*}(\lambda |\hat{\varvec{\omega }}), \qquad \hat{g}^* = \frac{g^*}{\omega _1\omega _2}, \qquad \hat{\varvec{\omega }} = \Bigl ( \frac{1}{\omega _2}, \frac{1}{\omega _1} \Bigr ). \end{aligned}$$
(A.24)
Note that the right hand side of (A.23) is analytic function of \(\omega _1, \omega _2\) in the domain \(\textrm{Re}\,\omega _j > 0\). The integral from the left is also analytic with respect to periods. Indeed, due to the bound (A.19) it is absolutely convergent uniformly on compact sets of parameters \(\varvec{\omega }, g\) preserving the conditions (1.8), (1.9). Hence, the formula (A.23) also holds for complex periods under the mentioned conditions.
A Degeneration of Rains Integral Identity
1.1 Hyperbolic \(A_n\rightleftarrows A_m\) Identity
Keep the assumptions \(\textrm{Re}\,\omega _1>0\), \(\textrm{Re}\,\omega _2>0\) and denote
$$\begin{aligned} q=\frac{\omega _1+\omega _2}{2}, \qquad \eta = \textrm{Re}\,\frac{2\pi q}{\omega _1\omega _2} > 0. \end{aligned}$$
(B.1)
In this Appendix it is convenient to use the following notations
$$\begin{aligned} \gamma ^{(2)}(z) =S_2^{-1}(z|\omega _1,\omega _2),\qquad \gamma ^{(2)}( a+u,b-u)= \gamma ^{(2)}( a+u)\gamma ^{(2)}(b-u) \end{aligned}$$
(B.2)
and
$$\begin{aligned} f(\pm z + c) = f(z + c) \, f(-z + c). \end{aligned}$$
(B.3)
Assume that a and b are in the region of analyticity of the double sine function, and
$$\begin{aligned} \alpha =\textrm{Re}\,\frac{a+b}{\omega _1\omega _2}>0,\qquad \beta =\textrm{Re}\,\frac{2q-a-b}{\omega _1\omega _2}>0. \end{aligned}$$
(B.4)
The asymptotical bounds and global analytical properties of the double sine function imply the following lemma, see [BDKK, eq. (A.20), (A.29)] for the details.
Lemma 1
For any \(u \in \mathbb {R}\) we have the uniform bounds
$$\begin{aligned} C_1 e^{-\beta |u|}<| \gamma ^{(2)}( a+\imath u,b-\imath u)|<C_2 e^{-\beta |u|},\qquad C_1,C_2>0. \end{aligned}$$
(B.5)
Here is the hyperbolic limit [Ra2, Theorem 4.6] of \(A_n-A_m\) Rains integral identity [Ra1, Theorem 4.1] (see also [SS])
$$\begin{aligned}{} & {} \frac{1}{(n+1)!}\int _{\mathbb {R}^n} \frac{\prod _{j=1}^{n+1}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell +\imath u_j,f_\ell -\imath u_j) }{\prod _{1\le j<k\le n+1}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^n \frac{du_j}{ \sqrt{\omega _1\omega _2}} = \prod _{j,k=1}^{n+m+2}\gamma ^{(2)}(g_j+f_k)\nonumber \\{} & {} \quad \times \frac{1}{(m+1)!}\int _{\mathbb {R}^m} \frac{\prod _{j=1}^{m+1}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g'_\ell + \imath u_j,f'_\ell -\imath u_j)}{\prod _{1\le j<k\le m+1}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^m \frac{du_j}{ \sqrt{\omega _1\omega _2}} \end{aligned}$$
(B.6)
where integration variables satisfy the relations
$$\begin{aligned} \sum _{j=1}^{n+1}u_j=0, \qquad \quad \sum _{j=1}^{m+1}u_j=0 \end{aligned}$$
(B.7)
in the first and the second integrals correspondingly. External parameters \(g_{\ell }\) and \(f_{\ell }\) obey the following balancing condition:
$$\begin{aligned} G+F=2(m+1)q, \qquad G=\sum _{\ell =1}^{n+m+2}g_\ell ,\qquad F=\sum _{\ell =1}^{n+m+2}f_\ell . \end{aligned}$$
(B.8)
Parameters \(g'_{\ell }\) and \(f'_{\ell }\) are connected with \(g_{\ell }\) and \(f_{\ell }\) by means of the following transformation
$$\begin{aligned} g'_{\ell } =\frac{G}{ m+1}-g_\ell , \qquad f'_{\ell } =\frac{F}{ m+1}-f_\ell , \qquad \ell =1\ldots , n+m+2. \end{aligned}$$
(B.9)
Assume that all the parameters \(f_l\), \(g_l\), \(f'_l\), \(g'_l\), have real positive parts and the sums \(f_l+g_l\) and \(f'_l+g'_l\) are in the region of analyticity of the double sine function. It is achieved, e.g., once these parameters are in a vicinity of the middle point
$$\begin{aligned} f_l=g_l=\frac{m+1}{n+m+2}q. \end{aligned}$$
(B.10)
Then due to Lemma 1 and balancing conditions the integrand of the left-hand side of (B.6) can be bounded by the function
$$\begin{aligned} C\exp \eta \biggl (-(n+1)\sum _{j=1}^{n+1}|u_j|+\sum _{\begin{array}{c} i,j=1 \\ i<j \end{array}}^{n + 1}|u_i-u_j|\biggr ) \le C'\exp \eta \biggl (-\sum _{j=1}^{n+1}|u_j|\biggr ). \end{aligned}$$
(B.11)
Analogous bound we have for the right-hand side of (B.6), so that this identity has a non-empty region of parameters where it is presented by convergent integrals.
1.2 Removing the Condition \(\textstyle \sum _{j}\,u_j=0\)
To remove conditions (B.7) we shift the external parameters
$$\begin{aligned} g_{\ell } \rightarrow g_{\ell } +\imath L, \qquad f_{\ell } \rightarrow f_{\ell } -\imath L,\qquad L >0, \end{aligned}$$
(B.12)
and then calculate the leading asymptotic of both sides as \(L \rightarrow \infty \).
Denote the domain
$$\begin{aligned} D_j=\{(u_1,\ldots ,u_{n+1})\in \mathbb {R}^{n+1} :{u_j} \ge {u_k}, \, \forall k\not =j\}. \end{aligned}$$
(B.13)
Due to \(S_{n+1}\) symmetry the integrand in the left-hand side, the integral is \(n+1\) times the same integral over the region \(D_{n+1}\). Similarly for the right-hand side, so that we replace the identity (B.6) by the same equality of integrals over the regions \(D_{n+1}\) and \(D_{m+1}\) substituting \(\frac{n+1}{(n+1)!} = \frac{1}{n!}\) in front of the left-hand side and \(\frac{m+1}{(m+1)!} = \frac{1}{m!}\) in front of the right-hand side.
Now consider the left-hand side. Change the integration variables
$$\begin{aligned} u_{j} = v_{j} - L, \qquad j=1,\ldots , n; \qquad u_{n+1} = v_{n+1} +n L. \end{aligned}$$
(B.14)
The integrand transforms as follows
$$\begin{aligned}&\frac{\prod _{j=1}^{n+1}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell + \imath u_j\,,f_\ell -\imath u_j) }{\prod _{1\le j<k\le n+1}\gamma ^{(2)}(\pm \imath (u_j- u_k))} = \frac{\prod _{j=1}^{n}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell + \imath v_j\,,f_\ell -\imath v_j) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (v_j- v_k))} \end{aligned}$$
(B.15)
$$\begin{aligned}&\times \frac{\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell + \imath v_{n+1} +\imath (n+1)L\,,f_\ell -\imath v_{n+1} -\imath (n+1)L) }{\prod _{1\le j\le n}\gamma ^{(2)}(\pm \imath (v_j - v_{n+1} - (n+1)L))}. \end{aligned}$$
(B.16)
Next we recall the asymptotic [BDKK, eq. (A.19)]
$$\begin{aligned} \gamma ^{(2)}(z|\omega _1,\omega _2) = e^{\mp \frac{\imath \pi }{2} B_{2,2}(z|\omega _1,\omega _2)} \Bigl (1+O\bigl (z^{-1}\bigr )\Bigr ) \end{aligned}$$
(B.17)
for \(\pm {\textrm{Im}}(z)>0\) and \(|z| \rightarrow \infty \) along a vertical strip of a fixed width, where \(B_{2,2}(z|\omega _1,\omega _2)\) is a multiple Bernoulli polynomial
$$\begin{aligned} B_{2,2}(z|\omega _1,\omega _2) = \frac{z^2}{\omega _1\omega _2} - \frac{\omega _1+\omega _2}{\omega _1\omega _2}\,z + \frac{\omega ^2_1+3\omega _1\omega _2+\omega ^2_2}{6\omega _1\omega _2} = \frac{(z-q)^2}{\omega _1\omega _2} - \frac{\omega ^2_1+\omega ^2_2}{12\omega _1\omega _2} \end{aligned}$$
(B.18)
and \(2q = \omega _1+\omega _2\). We use the following corollary of (B.17)
$$\begin{aligned} \gamma ^{(2)} (z+a)\gamma ^{(2)}(-z+b)=e^{\pm \frac{ \imath \pi }{\omega _1\omega _2} (2q-a-b)(z+(a-b)/2)} \Bigl (1+O\bigl (z^{-1}\bigr )\Bigr ) \end{aligned}$$
(B.19)
for \(\pm {\textrm{Im}}(z+a)>0\), \({\textrm{Im}}(z-b)>0\) and \(|z| \rightarrow \infty \) along a vertical strip of a fixed width.
The following inequalitites are valid in \(D_{n+1}\) for real positive L
$$\begin{aligned} A=v_{n+1} +(n+1)L> L, \qquad B_j=v_{n+1} - v_{j} +(n+1)L > 0. \end{aligned}$$
(B.20)
Let us take \(g_{\ell },f_{\ell } \in \mathbb {R}\). Then, by (B.19) we have the following asymptotic as \(L \rightarrow \infty \)
$$\begin{aligned} \begin{aligned}&\frac{\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell + \imath v_{n+1} + \imath (n+1)L\,,f_\ell -\imath v_{n+1} -\imath (n+1)L) }{\prod _{1\le j\le n}\gamma ^{(2)}(\pm \imath (v_j - v_{n+1} - (n+1)L))} \\&\quad = \frac{e^{-\frac{\imath \pi }{2\omega _1\omega _2} \sum _{\ell =1}^{n+m+2} \left( g_\ell + f_\ell -2q\right) \left( g_\ell - f_\ell +2\imath v_{n+1}+\imath 2(n+1)L\right) } }{e^{\frac{\imath \pi }{2\omega _1\omega _2} \sum _{j=1}^{n}(2q) \left( -2\imath v_j + 2\imath v_{n+1} + 2(n+1)\imath L\right) }}. \biggl (1+O\bigl (A^{-1}\bigr )+\sum _{j=1}^nO\bigl (B_j^{-1}\bigr )\biggr ) \end{aligned} \end{aligned}$$
(B.21)
Using the balancing conditions (B.7) and (B.8) we can simplify the exponent in the right-hand side of (B.21) and rewrite it as
$$\begin{aligned} \exp \biggl (-\frac{2q(n+1)L}{\omega _1\omega _2} \biggr ) \, \exp \frac{\imath \pi }{2\omega _1\omega _2}\biggl (2q\left( G-F\right) +\sum _{\ell =1}^{n+m+2}\left( - g^2_\ell + f^2_\ell \right) \biggr ). \end{aligned}$$
(B.22)
The error term O(1/A) can be replaced by O(1/L) due to (B.20), errow terms \(O(1/B_j)\) are not small only in a vicinity of the zero point of the measure function, which can be droped in the total integral or replaced by o(1). Thus, we have finally the estimate for \(L\rightarrow \infty \)
$$\begin{aligned} \begin{aligned}&\frac{\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell + \imath v_{n+1} + \imath (n+1)L\,,f_\ell -\imath v_{n+1} -\imath (n+1)L) }{\prod _{1\le j\le n}\gamma ^{(2)}(\pm \imath (v_j - v_{n+1} - (n+1)L))} \\&\quad =\exp \biggl (-\frac{2q(n+1)L}{\omega _1\omega _2} \biggr ) \, \exp \frac{\imath \pi }{2\omega _1\omega _2}\biggl (2q\left( G-F\right) +\sum _{\ell =1}^{n+m+2}\bigl (- g^2_\ell + f^2_\ell \bigr ) \biggr )\bigl (1+o(1)\bigr ). \end{aligned} \end{aligned}$$
(B.23)
In the same manner we can write down a uniform bound for the integrands in the right-hand side of (B.15) and (B.16). According to Lemma 1 and balancing conditions, the product in the right-hand side of (B.15) is restricted by
$$\begin{aligned} C\exp \eta \biggl (-(n+1)\sum _{j=1}^{n}|v_j|+\sum _{\begin{array}{c} i,j = 1 \\ i < j \end{array}}^n|v_i-v_j|\biggr ) \le C'\exp \eta \biggl (-\sum _{j=1}^{n}|v_j|\biggr ) \end{aligned}$$
(B.24)
while the product (B.16) is restricted by
$$\begin{aligned} C''\frac{\exp \bigl [-(n+1)\eta (v_{n+1}+(n+1)L)\bigr ]}{\exp \bigl [-\eta \sum _{j=1}^n(v_{n+1}-v_j+(n+1)L) \bigr ]}= C''\exp \bigl [-(n+1)L\eta \bigr ]. \end{aligned}$$
(B.25)
Here we used the condition
$$\begin{aligned} \sum _{j=1}^{n+1}v_j=0. \end{aligned}$$
The estimates (B.24) and (B.25) show that the transformed integrals in the left-hand side of (B.6), multiplied by
$$\begin{aligned} \exp \frac{2\pi q(n+1)L}{\omega _1\omega _2} \, \exp \frac{-\imath \pi }{2\omega _1\omega _2}\biggl (2q(G-F)+\sum _{\ell =1}^{n+m+2} \left( - g^2_\ell + f^2_\ell \right) \biggr ) \end{aligned}$$
(B.26)
have a convergend majorant and thus tend to the limit equal to the convergent integral
$$\begin{aligned} \frac{1}{n!}\int _{\mathbb {R}^n} \frac{\prod _{j=1}^{n}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell +\imath v_j\,,f_\ell -\imath v_j) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (v_j- v_k))} \prod _{j=1}^n \frac{dv_j}{ \sqrt{\omega _1\omega _2}} \end{aligned}$$
(B.27)
Next perform the same calculation in the m-integral in the right-hand side of (B.6). The shifts of external variables
$$\begin{aligned} g_{\ell } \rightarrow g_{\ell } +\imath L, \qquad f_{\ell } \rightarrow f_{\ell } -\imath L \end{aligned}$$
(B.28)
induce different shifts of \(g'_{\ell }\) and \(f'_{\ell }\). Due to the relations (B.9) we have
$$\begin{aligned} g'_{\ell } \rightarrow g'_{\ell } + \frac{n+1}{m+1} \, \imath L, \qquad f'_{\ell } \rightarrow f'_{\ell } - \frac{n+1}{m+1}\imath L. \end{aligned}$$
(B.29)
Repeating the same steps we conclude that the integral in the right-hand side of (B.6), multiplied by (B.26) tends to the convergent integral
$$\begin{aligned} \frac{1}{m!}\int _{\mathbb {R}^m} \frac{\prod _{j=1}^{m}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g'_\ell +\imath u_j\,,f'_\ell -\imath u_j) }{\prod _{1\le j<k\le m}\gamma ^{(2)}(\pm \imath (u_j- u_k))} \prod _{j=1}^m \frac{du_j}{ \sqrt{\omega _1\omega _2}}. \end{aligned}$$
(B.30)
After all we obtain the relation
$$\begin{aligned}{} & {} \frac{1}{n!}\int _{\mathbb {R}^n} \frac{\prod _{j=1}^{n}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g_\ell +\imath u_j,f_\ell -\imath u_j) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^n \frac{du_j}{ \sqrt{\omega _1\omega _2}} = \prod _{j,k=1}^{n+m+2}\gamma ^{(2)}(g_j+f_k)\nonumber \\{} & {} \quad \times \frac{1}{m!}\int _{\mathbb {R}^m} \frac{\prod _{j=1}^{m}\prod _{\ell =1}^{n+m+2}\gamma ^{(2)}(g'_\ell +\imath u_j,f'_\ell -\imath u_j)}{\prod _{1\le j<k\le m}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^m \frac{du_j}{ \sqrt{\omega _1\omega _2}} \end{aligned}$$
(B.31)
valid under balancing conditions (B.8), (B.9).
1.3 First reduction
In what follows we perform some reductions of the relation (B.31), where the external parameters obey the balancing condition (B.8) and are connected by the relation (B.9).
Let us perform the shifts
$$\begin{aligned} g_{n+m+2} \rightarrow g_{n+m+2} -\imath L, \qquad f_{n+m+2} \rightarrow f_{n+m+2} +\imath L\end{aligned}$$
(B.32)
which are compatible with balancing condition. Due to the relations (B.9) we have
$$\begin{aligned} \begin{aligned}&g'_{n+m+2} \rightarrow g'_{n+m+2} - \frac{\imath L}{ m+1}+\imath L, \qquad f'_{n+m+2} \rightarrow f'_{n+m+2} + \frac{\imath L}{ m+1}-\imath L\\&g'_{\ell } \rightarrow g'_{\ell } - \frac{\imath L}{ m+1},\qquad f'_{\ell } \rightarrow f'_{\ell } +\frac{\imath L}{ m+1}, \qquad \ell =1,\ldots , n+m+1. \end{aligned}\end{aligned}$$
(B.33)
Next we calculate calculate the leading asymptotics of both sides of the identity (B.31) as \(L \rightarrow \infty \). Denote for simplicity
$$\begin{aligned} g_{n+m+2} = a, \qquad f_{n+m+2} = b, \qquad g'_{n+m+2} = a', \qquad f'_{n+m+2} = b'. \end{aligned}$$
(B.34)
We have the following pointwise limit in left-hand side as \(L \rightarrow \infty \)
$$\begin{aligned} \begin{aligned}&\prod _{j=1}^{n}\gamma ^{(2)}(a + \imath u_{j} -\imath L\,,b-\imath u_{j}+\imath L) \rightarrow \\&\prod _{j=1}^{n} e^{\frac{\imath \pi }{2\omega _1\omega _2} \left( (a + \imath u_{j} -\imath L-q)^2 -(b-\imath u_{j}+\imath L-q)^2\right) } = e^{\frac{\imath \pi }{2\omega _1\omega _2} I_1} \end{aligned}\end{aligned}$$
(B.35)
where
$$\begin{aligned} \begin{aligned} I_1 =&\sum _{j=1}^{n} \left( a+b-2q\right) \left( a-b+2\imath u_j -2\imath L\right) \\&= \left( a+b-2q\right) \left( a-b-2\imath L\right) n + 2\left( a+b-2q\right) \sum _{j=1}^{n} \imath u_j \end{aligned} \end{aligned}$$
(B.36)
In right-hand side of the identity (B.31) we have to shift all the integration variables
$$\begin{aligned} u_j \rightarrow u_j +\frac{L}{m+1}\end{aligned}$$
(B.37)
to remove L-dependence in almost all functions except containing \(g'_{n+m+2} = a'\) and \(f'_{n+m+2} = b'\), so that
$$\begin{aligned} \begin{aligned}&\prod _{j=1}^{m}\gamma ^{(2)}(a' + u_{j} +\imath L\,,b'-u_{j}-\imath L) \rightarrow \\&\prod _{j=1}^{n} e^{-\frac{\imath \pi }{2\omega _1\omega _2} \left( (a' + \imath u_{j} +\imath L-q)^2 -(b'-\imath u_{j}-\imath L-q)^2\right) } = e^{\frac{\imath \pi }{2\omega _1\omega _2} I_2} \end{aligned} \end{aligned}$$
(B.38)
where
$$\begin{aligned} \begin{aligned} I_2&= \sum _{j=1}^{m} \left( 2q-a'-b'\right) \left( a'-b'+2\imath u_j +2\imath L\right) \\&=\left( 2q-a'-b'\right) \left( a'-b'+2\imath L\right) m + 2\left( a'+b'-2q\right) \sum _{j=1}^{m} \imath u_j\\&= (a+b){\frac{m}{m+1}}(G-F) +2\imath L(a+b)m - \left( a^2-b^2\right) m +2(a+b)\sum _{j=1}^{m} \imath u_j. \end{aligned} \end{aligned}$$
(B.39)
We also have L-dependence in prefactor in right-hand side of (B.31) and
$$\begin{aligned} \begin{aligned}&\prod _{\ell =1}^{n+m+1}\gamma ^{(2)}(a -\imath L+f_{\ell }\,,b +\imath L+g_{\ell }) \rightarrow \\&\prod _{\ell =1}^{n+m+1} e^{\frac{\imath \pi }{2\omega _1\omega _2} \left( (a -\imath L+f_{\ell } -q)^2 -(b +\imath L+g_{\ell } -q)^2\right) } = e^{\frac{\imath \pi }{2\omega _1\omega _2} I_3} \end{aligned}\end{aligned}$$
(B.40)
where
$$\begin{aligned} \begin{aligned} I_3&= \sum _{\ell =1}^{n+m+1} \left( a+b+f_{\ell }+g_{\ell }-2q\right) \left( a-b+f_{\ell }-g_{\ell } -2\imath L\right) \\&= \left( (a+b)(n+m)-2q n\right) \left( a-b-2L\right) +\left( a+b-2q\right) \left( a-b\right) \\&\quad + \left( a+b-2q\right) \left( F-G\right) + \sum _{\ell =1}^{n+m+1} \left( f_{\ell }+g_{\ell }\right) \left( f_{\ell }-g_{\ell }\right) . \end{aligned} \end{aligned}$$
(B.41)
We have
$$\begin{aligned} \begin{aligned} I_2+I_3&= -2\imath L\left( a+b-2q\right) n + \left( a+b-2q\right) \left( a-b\right) (n+1)\\&\quad +\left( a+b-2q\right) \left( F-G\right) +(a+b){\frac{m}{m+1}}(G-F) \\&\quad + \sum _{\ell =1}^{n+m+1} \left( f_{\ell }+g_{\ell }\right) \left( f_{\ell }-g_{\ell }\right) +2(a+b)\sum _{j=1}^{m} \imath u_j. \end{aligned} \end{aligned}$$
(B.42)
The L-dependence in \(I_1\) and \(I_2+I_3\) is the same so that asymptotic behaviour of both sides of the identity is the same and in the limit we arrive at
$$\begin{aligned} \begin{aligned}&\frac{1}{n!}\int _{\mathbb {R}^n}\, e^{\frac{\pi }{\omega _1\omega _2}\left( 2q-a-b\right) \sum _{j=1}^{n} u_j}\, \frac{\prod _{j=1}^{n}\prod _{\ell =1}^{n+m+1}\gamma ^{(2)}(g_\ell + \imath u_j\,,f_\ell -\imath u_j) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^n \frac{ du_j}{\sqrt{\omega _1\omega _2}} \\&\quad = e^{\frac{\imath \pi }{2\omega _1\omega _2}\,\varphi (a,b,f,g)}\,\gamma ^{(2)}(a+b)\, \prod _{j,k=1}^{n+m+1}\gamma ^{(2)}(g_j+f_k) \\&\qquad \times \frac{1}{m!}\int _{\mathbb {R}^m}\, e^{\frac{\pi }{\omega _1\omega _2}\left( a+b\right) \sum _{j=1}^{m} u_j}\, \frac{\prod _{j=1}^{m}\prod _{\ell =1}^{n+m+1}\gamma ^{(2)}(g'_\ell +\imath u_j,f'_\ell -\imath u_j)}{\prod _{1\le j<k\le m}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^m\frac{ du_j}{\sqrt{\omega _1\omega _2}} \end{aligned}\end{aligned}$$
(B.43)
provided we are able to obtain the uniform bounds for the corresponding integrands. Here we introduced the function
$$\begin{aligned} \begin{aligned} \varphi (a,b,f,g)&= \left( a+b-2q\right) \left( F-G+a-b\right) + (a+b){\frac{m}{m+1}}(G-F)\\&\quad +\sum _{\ell =1}^{n+m+1} \left( f^2_{\ell }-g^2_{\ell }\right) \\&= (a^2-b^2){\frac{m}{m+1}} + \left( {\frac{a+b}{m+1}}-2q\right) \sum _{\ell =1}^{n+m+1} \left( f_{\ell }-g_{\ell }\right) + \sum _{\ell =1}^{n+m+1} \left( f^2_{\ell }-g^2_{\ell }\right) . \end{aligned} \end{aligned}$$
(B.44)
Estimate first the nominator of the integrand in the left-hand side of (B.31). Collect its factors containing the variable \(u_j\). They are equal to
$$\begin{aligned} G_j= \gamma ^{(2)}(a+\imath u_j-\imath L, b-\imath u_j+\imath L)\prod _{l=1}^{n+m+1}\gamma ^{(2)}(g_l+\imath u_j,f_l-\imath u_j).\end{aligned}$$
(B.45)
Due to Lemma 1
$$\begin{aligned} \begin{aligned} |G_j|&<C_j\exp \,\textrm{Re}\,\frac{\pi }{\omega _1\omega _2}\biggl (-\sum _{l=1}^{n+m+1}(2q-g_l-f_l)|u_j|-(2q-a-b)|u_j-L|\biggr )\\&= \exp \,\textrm{Re}\,\frac{\pi }{\omega _1\omega _2}\bigl ( -(2qn+a+b)|u_j|-(2q-a-b)|u_j-L|\big ). \end{aligned}\end{aligned}$$
(B.46)
Assuming the condition (B.4) for parameters a and b we see that the last line of (B.46) is represented by exponent of the piecewise linear function \(-\delta (u_j)\), where
$$\begin{aligned} \delta (u_j)=\alpha |u_j|+\beta |u_j-L| \end{aligned}$$
with positive coefficients \(\alpha \) and \(\beta \), \(\alpha >\beta \). Elementary analysis of its graph shows that
$$\begin{aligned} \delta (u_j)>\beta L+(\alpha -\beta )|u_j|. \end{aligned}$$
(B.47)
Thus, we have the bound
$$\begin{aligned} |G_j|<C_j \exp \left( -\pi \textrm{Re}\,\frac{2q-a-b}{\omega _1\omega _2}L -2\pi \textrm{Re}\,\frac{q(n-1)+a+b}{\omega _1\omega _2}|u_j|\right) . \end{aligned}$$
(B.48)
Multiplying over all j we arrive to the desired asymptotics
$$\begin{aligned} \exp \biggl ( -\pi n\, \textrm{Re}\,\frac{2q - a - b}{\omega _1\omega _2} L \biggr ) \end{aligned}$$
(B.49)
multiplied by the integral with integrand uniformly bounded by
$$\begin{aligned}{} & {} C \exp \, \textrm{Re}\,\frac{2\pi }{\omega _1\omega _1}\biggl (- \sum _{j=1}^{n}\bigl (q(n-1)+(a+b)\bigr )|u_j|+q \sum _{\begin{array}{c} i,j=1 \\ i<j \end{array}}^n |u_i-u_j|\biggr )\nonumber \\{} & {} \quad \le C'\exp \biggl ( -\textrm{Re}\,\frac{2\pi (a+b)}{\omega _1\omega _2}\sum _{j = 1}^n|u_j| \biggr ). \end{aligned}$$
(B.50)
The latter absolutely converges once
$$\begin{aligned} \textrm{Re}\,\frac{a+b}{\omega _1\omega _2}>0.\end{aligned}$$
(B.51)
In the same manner we estimate the integral in the right-hand side of (B.31). Collect all factors of the nominator containing the shifted variable \(u_j\) into the product \(G'_j\)
$$\begin{aligned} G'_j= \gamma ^{(2)}(a'+\imath u_j-\imath L, b'-\imath u_j+\imath L)\prod _{l=1}^{n+m+1}\gamma ^{(2)}(g'_l+\imath u_j,f'_l-\imath u_j).\end{aligned}$$
(B.52)
Following the same lines as before we see that the integrand is a product of its asymptotics
$$\begin{aligned} \exp \biggl (-\frac{\pi m}{\omega _1\omega _2}(2q-a'-b')L\biggr )=\exp \biggl (-\frac{\pi m}{\omega _1\omega _2}(a+b)L\biggr ) \end{aligned}$$
(B.53)
multiplied by the function which can be estimated by a uniform absolutely integrable function
$$\begin{aligned}{} & {} \exp \, \textrm{Re}\,\frac{2\pi }{\omega _1\omega _1}\biggl ( - \sum _{j=1}^{m}\bigl (q(n-1)+(a'+b')\bigr )|u_j|+q \sum _{\begin{array}{c} i,j=1 \\ i<j \end{array}}^m|u_i-u_j|\biggr )\nonumber \\{} & {} \quad \le C'\exp \biggl ( -\textrm{Re}\,\frac{2\pi (a'+b')}{\omega _1\omega _2}\sum _{j = 1}^m|u_j| \biggr )=C'\exp \biggl (-\textrm{Re}\,\frac{2\pi (2q-a-b)}{\omega _1\omega _2}\sum _{j = 1}^m|u_j| \biggr ).\nonumber \\ \end{aligned}$$
(B.54)
Combining this bound with the limit (B.40) we also see that the right-hand side of (B.31) divided by its asymptotics is given by uniformly bounded integral. This finishes the proof of the relation (B.43).
We can write down the relation (B.43) in a slightly different form by separating f and g-dependence in the function \(\varphi (a,b,f,g)\). Namely, denote
$$\begin{aligned} \varphi (g) = \left( {\frac{a+b}{m+1}}-2q\right) \sum _{\ell =1}^{n+m+1} g_{\ell } + \sum _{\ell =1}^{n+m+1} g^2_{\ell }. \end{aligned}$$
(B.55)
Then
$$\begin{aligned} \varphi (a,b,f,g) = (a^2-b^2){\frac{m}{m+1}}-\varphi (g)+\varphi (f). \end{aligned}$$
(B.56)
The external parameters \(g_{\ell }\) and \(f_{\ell }\) obey the following balancing condition
$$\begin{aligned} g+f+(a+b) = 2q(m+1), \qquad 2q = \omega _1+\omega _2 \end{aligned}$$
(B.57)
where
$$\begin{aligned} g=\sum _{\ell =1}^{n+m+1}g_\ell , \qquad f=\sum _{\ell =1}^{n+m+1}f_\ell . \end{aligned}$$
(B.58)
Parameters \(g'_{\ell }\) and \(f'_{\ell }\) are connected with \(g_{\ell }\) and \(f_{\ell }\) by simple transformation
$$\begin{aligned} g'_{\ell } =\frac{g+a}{ m+1}-g_\ell , \qquad f'_{\ell } =\frac{f+b}{ m+1}-f_\ell , \qquad \ell =1,\ldots , n+m+1 \end{aligned}$$
(B.59)
and in the same way
$$\begin{aligned} a' =\frac{g+a}{ m+1}-a, \qquad b' =\frac{f+b }{m+1}-b, \qquad a'+b' = 2q-a-b. \end{aligned}$$
(B.60)
Finally, using the notation (B.56) we rewrite the relation (B.43) as
$$\begin{aligned} \begin{aligned}&e^{\varphi (g)}\,\frac{1}{n!}\int _{\mathbb {R}^n}\, e^{\frac{\pi }{\omega _1\omega _2}\left( 2q-a-b\right) \sum _{j=1}^{n} u_j}\, \frac{\prod _{j=1}^{n}\prod _{\ell =1}^{n+m+1}\gamma ^{(2)}(g_\ell + \imath u_j\,,f_\ell -\imath u_j) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (u_j- u_k)) }\prod _{j=1}^n \frac{du_j}{\sqrt{\omega _1\omega _2}} \\&\quad = \gamma ^{(2)}(a+b)\,e^{\frac{\imath \pi }{2\omega _1\omega _2}\,(a^2-b^2){\frac{m}{m+1}}}\, \prod _{j,k=1}^{n+m+1}\gamma ^{(2)}(g_j+f_k)\\&\qquad \times e^{\varphi (f)}\,\frac{1}{m!}\int _{\mathbb {R}^m}\, e^{\frac{\pi }{\omega _1\omega _2}\left( a+b\right) \sum _{j=1}^{m} u_j}\, \frac{\prod _{j=1}^{m}\prod _{\ell =1}^{n+m+1}\gamma ^{(2)}(g'_\ell + \imath u_j,f'_\ell -\imath u_j)}{\prod _{1\le j<k\le m}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \\&\qquad \quad \prod _{j=1}^m \frac{du_j}{\sqrt{\omega _1\omega _2}}. \end{aligned}\end{aligned}$$
(B.61)
This relation is valid under the conditions (B.4) and (B.51).
We should note that a special case of obtained formula is presented in the forthcoming paper [SS].
1.4 Second reduction
Now we set \(m=n\) and use the following parametrization
$$\begin{aligned} g_{k} = -\imath x_k+\frac{g^*}{2}, \qquad g_{n+k} = - \imath z_k+\frac{g}{2}, \qquad f_{k} = \imath x_k+\frac{g^*}{2}, \qquad f_{n+k} = \imath z_k+\frac{g}{2} \end{aligned}$$
(B.62)
for \(k = 1, \ldots , n\) and
$$\begin{aligned} g_{2n+1} = q-a + \imath \sum _{k=1}^n (x_k+z_k), \qquad f_{2n+1} = q-b - \imath \sum _{k=1}^n (x_k+z_k). \end{aligned}$$
(B.63)
Then since \(g + g^* = 2q\) from the relations (B.59) we also have
$$\begin{aligned} g'_{k} = \imath x_k+\frac{g}{2}, \qquad g'_{n+k} = iz_k+\frac{g^*}{2}, \qquad f'_{k} = -ix_k+\frac{g}{2}, \qquad f'_{n+k} = -iz_k+\frac{g^*}{2} \end{aligned}$$
(B.64)
for \(k = 1, \ldots , n\) and
$$\begin{aligned} g'_{2n+1} = a - \imath \sum _{k=1}^n (x_k+z_k), \qquad f'_{2n+1} = b + \imath \sum _{k=1}^n (x_k+z_k). \end{aligned}$$
(B.65)
The product behind the integral in the right-hand side of (B.61)
$$\begin{aligned} \prod _{j,k=1}^{2n+1}\gamma ^{(2)}(g_j+f_k)&= \gamma ^{(2)}(g_{2n+1}+f_{2n+1})\, \prod _{k=1}^{2n}\gamma ^{(2)}(g_{2n+1}+f_k)\, \\&\quad \times \prod _{k=1}^{2n}\gamma ^{(2)}(f_{2n+1}+g_k)\, \prod _{j,k=1}^{2n}\gamma ^{(2)}(g_j+f_k). \end{aligned}$$
Then the relation (B.61) takes the form
$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^n}\!\! e^{\frac{\pi (2q-a-b)}{\omega _1\omega _2}\sum \limits _{j=1}^{n} u_j}\, \!\prod _{j=1}^{n}\textstyle \gamma ^{(2)}\Bigl (q-a + \imath \sum \limits _{k=1}^n (x_k+z_k)+ \imath u_j, q-b - \imath \sum \limits _{k=1}^n (x_k+z_k)-\imath u_j \Bigr ) \\&\quad \times \frac{\prod _{j=1}^{n}\prod _{k=1}^{n} \gamma ^{(2)}\left( \pm \imath (x_k-u_j)+\frac{g^*}{2}\,,\pm \imath (z_k-u_j)+\frac{g}{2}\right) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \\&\quad \prod _{j=1}^n \frac{du_j}{\sqrt{\omega _1\omega _2}} = H(x,z,a,b) \\&\quad \times \int _{\mathbb {R}^n} e^{\frac{\pi (a+b)}{\omega _1\omega _2}\sum \limits _{j=1}^{n} u_j}\, \prod _{j=1}^{n}\textstyle \gamma ^{(2)}\Bigl (a - \imath \sum \limits _{k=1}^n (x_k+z_k)- \imath u_j\,, b + \imath \sum \limits _{k=1}^n (x_k+z_k)+\imath u_j\Bigr ) \\&\quad \times \frac{\prod _{j=1}^{n}\prod _{k=1}^{n} \gamma ^{(2)}\left( \pm \imath (x_k-u_j)+\frac{g}{2}\,,\pm \imath (z_k-u_j)+\frac{g^*}{2}\right) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^n\frac{du_j}{\sqrt{\omega _1\omega _2}} \end{aligned} \end{aligned}$$
(B.66)
where
$$\begin{aligned} \begin{aligned}&H(x,z,a,b)= e^{\frac{\pi }{\omega _1\omega _2}\, \left( (2q-a-b)\sum \limits _{k=1}^n (x_k+z_k) - g^{*}\sum \limits _{k=1}^n x_k- g\sum \limits _{k=1}^n z_k\right) } \\&\quad \times \prod _{\begin{array}{c} j,k = 1 \\ j\ne k \end{array}}^{n}\gamma ^{(2)}(\imath (x_k-x_j)+g^*)\, \gamma ^{(2)}(\imath (z_k-z_j)+g)\\&\quad \times \prod _{k=1}^{n}\textstyle \gamma ^{(2)}\Bigl (q-a + \imath \sum \limits _{k=1}^n (x_k+z_k)+\imath x_k+\frac{g^*}{2}\Bigr )\, \gamma ^{(2)}\Bigl (q-a + \imath \sum \limits _{k=1}^n (x_k+z_k)+\imath z_k+\frac{g}{2}\Bigr )\\&\quad \times \prod _{k=1}^{n}\textstyle \gamma ^{(2)}\Bigl (q-b - \imath \sum \limits _{k=1}^n (x_k+z_k)-\imath x_k+\frac{g^*}{2}\Bigr )\, \gamma ^{(2)}\Bigl (q-b - \imath \sum \limits _{k=1}^n (x_k+z_k)-\imath z_k+\frac{g}{2}\Bigr ). \end{aligned} \end{aligned}$$
(B.67)
Now we shift \(a \rightarrow a-\imath L\) and \(b \rightarrow b+\imath L\) and calculate asymptotics as \(L \rightarrow \infty \) using the relation (B.35).
In the left-hand side of (B.66) we have
$$\begin{aligned} \begin{aligned} \prod _{j=1}^{n}\textstyle \gamma ^{(2)} \Bigl (q-a + \imath \sum \limits _{k=1}^n (x_k+z_k)+ \imath u_j+\imath L, q-b - \imath \sum _{k=1}^n (x_k+z_k)-\imath u_j-\imath L \Bigr ) \rightarrow \\ \rightarrow e^{\frac{\imath \pi }{2\omega _1\omega _2} I_1},\qquad I_1 = 2\imath (a+b)\Big (nL+\sum _{j=1}^n u_j\Big ) +n(a+b)\Big (2 \imath \sum _{k=1}^n (x_k+z_k) -(a-b)\Big ). \end{aligned} \end{aligned}$$
(B.68)
In right-hand side of (B.66) in the integrand we have
$$\begin{aligned} \begin{aligned}&\prod _{j=1}^{n}\textstyle \gamma ^{(2)} \Bigl (a - \imath \sum \limits _{k=1}^n (x_k+z_k)- \imath u_j-\imath L, b + \imath \sum \limits _{k=1}^n (x_k+z_k)+\imath u_j+\imath L \Bigr ) \rightarrow e^{\frac{\imath \pi }{2\omega _1\omega _2} I_2},\\&I_2 = 2\imath (2q-a-b)\Big (nL+\sum _{j=1}^n u_j\Big )- n(a+b-2q) \Big (2 \imath \sum _{k=1}^n (x_k+z_k) -(a-b)\Big ) \end{aligned} \end{aligned}$$
(B.69)
and for the two factors outside of the integral we have
$$\begin{aligned} \begin{aligned}&\prod _{k=1}^{n}\textstyle \gamma ^{(2)}\Bigl (q - a + \imath \sum \limits _{k=1}^n (x_k+z_k)+ \imath x_k+\frac{g^*}{2}+\imath L, \\&\quad \textstyle q-b -\imath \sum \limits _{k=1}^n (x_k + z_k)- \imath x_k+\frac{g^*}{2} -\imath L\Bigr ) \rightarrow e^{\frac{\imath \pi }{2\omega _1\omega _2} I_3},\\&I_3 = (a+b-g^*)\Big (2\imath n L+2\imath \sum _{k=1}^n\big ((n+1)x_k+nz_k\big )-n(a-b)\Big ) , \end{aligned} \end{aligned}$$
(B.70)
and
$$\begin{aligned}&\prod _{k=1}^{n}\textstyle \gamma ^{(2)}\Big (q - a + \imath \sum \limits _{k=1}^n (x_k + z_k) + \imath z_k+ \frac{g}{2}+\imath L, \\&\quad \textstyle q-b - \imath \sum \limits _{k=1}^n (x_k + z_k) - \imath z_k+\frac{g}{2}-\imath L\Bigr ) \rightarrow e^{\frac{\imath \pi }{2\omega _1\omega _2} I_4},\\&I_4 =(a+b-g)\Big (2\imath n L+2\imath \sum _{k=1}^n\big (nx_k+(n+1)z_k\big )-n(a-b)\Big ). \end{aligned}$$
Collecting all these calculations we see that integrands from both sides have equal asymptotics
$$\begin{aligned} \exp \Big (-\frac{\pi nL}{\omega _1\omega _2}(a+b)\Big )\end{aligned}$$
(B.71)
while the rest has a poinwise limit, so that the initial relation is reduced to the following equality
$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^n} e^{\frac{2\pi \lambda }{\omega _1\omega _2}\sum \limits _{j=1}^{n} u_j}\, \frac{\prod _{j=1}^{n}\prod _{k=1}^{n} \gamma ^{(2)}\left( \pm \imath (x_k-u_j)+\frac{g^*}{2}\,,\pm \imath (z_k-u_j)+\frac{g}{2}\right) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^n du_j \\&\quad = e^{\frac{\pi \lambda }{\omega _1\omega _2}\sum \limits _{k=1}^n (x_k+z_k)}\, \prod _{\begin{array}{c} j,k = 1 \\ j\ne k \end{array}}^{n}\gamma ^{(2)}(\imath (x_k-x_j)+g^*)\, \gamma ^{(2)}(\imath (z_k-z_j)+g) \\&\quad \times \int _{\mathbb {R}^n} e^{-\frac{2\pi \lambda }{\omega _1\omega _2}\sum \limits _{j=1}^{n} u_j}\, \frac{\prod _{j=1}^{n}\prod _{k=1}^{n} \gamma ^{(2)}\left( \pm \imath (x_k-u_j)+\frac{g}{2}\,,\pm \imath (z_k-u_j)+\frac{g^*}{2}\right) }{\prod _{1\le j<k\le n}\gamma ^{(2)}(\pm \imath (u_j- u_k)) } \prod _{j=1}^n du_j \end{aligned} \end{aligned}$$
(B.72)
where
$$\begin{aligned} \lambda =q-a-b,\qquad \Big |\textrm{Re}\,\frac{\lambda }{\omega _1\omega _2}\Big |<\frac{\eta }{2}=\frac{1}{2} \textrm{Re}\,\big (\omega _1^{-1}+\omega _2^{-1}\big ) \end{aligned}$$
(B.73)
provided we can obtain the uniform integrable bounds for the integrands divided by asymptotics (B.71).
Let us estimate the integrand of the left-hand side of (B.66). Due to Lemma 1 its nominator is bounded by
$$\begin{aligned} C\prod _{j=1}^n \exp \textrm{Re}\,\gamma A_j(u_j),\qquad \gamma = \frac{\pi }{\omega _1\omega _2} \end{aligned}$$
(B.74)
with some constant C and
$$\begin{aligned} A_j(u_j)= (2q-a-b)u_j-g\sum _{k=1}^n|u_j-x_k|-g^*\sum _{k = 1}^n|u_j-z_k|-(a+b) \Bigl |u_j+L-\sum _{k = 1}^n(x_k+z_k) \Bigr | \end{aligned}$$
(B.75)
Using triangle inequalities
$$\begin{aligned} - |u_j-x_k|\le -|u_j|+|x_k|,\qquad - |u_j-z_k| \le -|u_j|+|z_k| \end{aligned}$$
we can replace \(A_j(u_j)\) by \(\tilde{A}_j(u_j)+C'_j\), where \(C'_j\) is some constant and
$$\begin{aligned} \tilde{A_j}(u_j)= (2q-a-b)u_j-2qn|u_j|-(a+b)|u_j+L| \end{aligned}$$
(B.76)
The function \(\xi (x)=-\textrm{Re}\,\gamma \tilde{A_j}(x)\) is a piecewise linear function of the form
$$\begin{aligned} \xi (x)= n(\alpha +\beta )|x|+\beta |x+L|-\alpha x \end{aligned}$$
(B.77)
where \(\alpha ,\beta >0\) are given by (B.4). Analysing the graph of this function, we get inequality
$$\begin{aligned} \xi (x)\ge \beta L+\big ((n-1)(\alpha +\beta )+2\min (\alpha ,\beta )|x|\big ).\end{aligned}$$
(B.78)
This inequality implies the following bound for the integrand in the left-hand side of (B.66) divided by its asymptotics (B.71)
$$\begin{aligned} \begin{aligned} C\exp \eta&\Big (-(n-1)-\min (\alpha ,\beta )\sum _{j=1}^n|u_j|+\sum _{\begin{array}{c} i,j = 1 \\ i<j \end{array}}^n|u_i-u_j|\Big )\\&\le C' \exp \biggl (-\eta \min (\alpha ,\beta )\sum _{j=1}^n|u_j|\biggr ). \end{aligned} \end{aligned}$$
(B.79)
The latter is absolutely integrable function. The right-hand side is analysed in a similar manner.
Some Inequalities
In our previous paper we proved the following little lemma [BDKK2, Lemma 1].
Lemma
For any \(\varepsilon \in [0, 2]\), \(y_1, y_2, y \in \mathbb {R}\) we have
$$\begin{aligned} |y_1 - y_2| - |y_1 - y| - |y_2 - y| \le \varepsilon \left( |y_1| + |y_2| - |y| \right) . \end{aligned}$$
(C.1)
Now with its help we prove one inequality used in the main text. Define
$$\begin{aligned} L_n(\varvec{y}_{n - 1}, \varvec{x}_n) = \sum _{ \begin{array}{c} i, j = 1 \\ i< j \end{array} }^n | x_i - x_j | + \sum _{ \begin{array}{c} i, j = 1 \\ i < j \end{array} }^{n - 1} | y_ i - y_j| - \sum _{i = 1}^{n}\sum _{j = 1}^{n - 1} |x_i - y_j|. \end{aligned}$$
(C.2)
As before, by \(\Vert \varvec{x}_n \Vert \) denote \(L^1\)-norm. The following statement implicitly appeared during the proof of Lemma 2 in [BDKK2].
Lemma 2
For any \(\varepsilon \in [0, 2]\) we have
$$\begin{aligned} L_n \le (n - 1) \varepsilon \Vert \varvec{x}_n \Vert - \varepsilon \Vert \varvec{y}_{n - 1} \Vert . \end{aligned}$$
(C.3)
Proof
Both sides of the stated inequality are symmetric with respect to components of \(\varvec{x}_n, \varvec{y}_{n - 1}\). Therefore, without loss of generality we assume the ordering
$$\begin{aligned} x_1 \ge \ldots \ge x_n, \qquad y_1 \ge \ldots \ge y_{n - 1}. \end{aligned}$$
(C.4)
For the vector \(\varvec{x}_n\) with ordered components we write
$$\begin{aligned} \sum _{\begin{array}{c} i, j = 1 \\ i < j \end{array}}^n | x_i - x_j | = \sum _{m = 1}^{\lfloor n/2 \rfloor } (n - 2m + 1) | x_m - x_{n - m + 1} |. \end{aligned}$$
(C.5)
Similarly for \(\varvec{y}_{n - 1}\). Consequently,
$$\begin{aligned} \begin{aligned} L_n&= \sum _{m = 1}^{\lfloor n/2 \rfloor } (n - 2m + 1) | x_m - x_{n - m + 1} | + \sum _{m = 1}^{\lfloor (n - 1)/2 \rfloor } (n - 2m) | y_m - y_{n - m} | \\&\quad - \sum _{i = 1}^{n}\sum _{j = 1}^{n - 1} |x_i - y_j|. \end{aligned} \end{aligned}$$
(C.6)
Next step it to regroup terms. Consider term with \(m = 1\) from the first sum and terms with \(i = 1, n\) from the third double sum and write the estimate
$$\begin{aligned} \begin{aligned}&(n - 1) | x_1 - x_n | - \sum _{j = 1}^{n - 1} \left( | x_1 - y_j | + | x_n - y_j | \right) \\&\quad \le (n - 1) \, \varepsilon \left( | x_1 | + |x_n | \right) - \varepsilon \, \Vert \varvec{y}_{n - 1} \Vert , \end{aligned} \end{aligned}$$
(C.7)
where we used inequality (C.1) multiple times. Similarly let us estimate the term with \(m > 1\) from the first sum together with the corresponding terms from the third double sum
$$\begin{aligned} (n - 2m + 1) | x_m - x_{n - m + 1} | - \sum _{j = m}^{n - m} \left( | x_m - y_j | + | x_{n - m + 1} - y_j | \right) \le 0, \end{aligned}$$
(C.8)
where we used triangle inequality multiple times. Remaining from the third double sum terms can be grouped with terms from the second sum
$$\begin{aligned} (n - 2m) | y_m - y_{n - m} | - \sum _{i = m + 1}^{n - m} \left( | x_i - y_m | + | x_i - y_{n - m} | \right) \le 0, \end{aligned}$$
(C.9)
where we again used triangle inequalities. Collecting everything together we have
$$\begin{aligned} L_n \le (n - 1) \, \varepsilon \left( | x_1 | + |x_n | \right) - \varepsilon \, \Vert \varvec{y}_{n - 1} \Vert \le (n - 1) \, \varepsilon \Vert \varvec{x}_n \Vert - \varepsilon \, \Vert \varvec{y}_{n - 1} \Vert . \end{aligned}$$
(C.10)
\(\square \)
