\exp(-s_g\ln [R(x+y)R(x-y)]),
\\
\label{sg}
s_g\equiv \frac{1}{2} \sum_{t=0}^3g_t=\frac{1}{2}\sum_{t=0}^3g'_t.
\\
\label{w}
w(g;x)\equiv 1/c(g;x)c(g;-x),
\\
  c(g;x)\equiv R(x+i\alpha/2)^{-g_0}R(x+i\alpha/2-\pi/2r)^{-g_1}R(x)^{-g_2}R(x-\pi/2r)^{-g_3},\nonumber
\\
R(z)\equiv \prod_{l=0}^{\infty}(1-\exp[-(2l+1)r\alpha+2irz])(1-\exp[-(2l+1)r\alpha-2irz]),\qquad z\in{{\mathbb C}}.\nonumber\end{gathered}$$

At this point we would like to mention that a close relative of the kernel ${{\cal S}}(g;x,y)$ has appeared in the literature before, a fact we only recently became aware of. Indeed, it can be tied in with the kernel that can be found on p. 124 of a monograph by Slavyanov and Lay [@slla]. (It seems this kernel was first introduced in a paper by Kazakov and Slavyanov [@kasl].) More precisely, when a $\wp$-function substitution is made in the latter kernel, it can be factorized in terms of the Weierstrass $\sigma$-function; the nontrivial part is then an analytic continuation of ${{\cal S}}(g;x,y)$. Both in [@slla] and in Takemura’s paper [@take2], this kernel is used to relate Heun functions defined on different intervals.

The entire function $R(z)$ plays a crucial role in this paper. It satisfies $$\begin{gathered}
\label{R2}
R(z)=\exp\left( -\sum_{n=1}^{\infty}\frac{\cos(2nrz)}{n\sinh(nr\alpha)}\right),\qquad |\Im z|<\alpha/2,
\\
\label{dupl}
R(2z)=\prod_{{\delta}=+,-}R(z+i{\delta}\alpha/4)R(z+\pi/2r+i{\delta}\alpha/4),
\\
\label{Rs}
R(z+i\alpha/2)=ipe^{-irz}s(z),\qquad p=2r\prod_{k=1}^{\infty}(1-e^{-2kr\alpha})^2.\end{gathered}$$ Here, $s(z)$ is related to the $\sigma$-function by $$\begin{gathered}
\label{ssig}
s(z)=\exp\left(-\eta z^2r/\pi\right)\sigma(z;\pi/2r,i\alpha/2),\end{gathered}$$ so that $$\begin{gathered}
f(z+i\alpha)=-\exp(r\alpha -2irz)f(z),\qquad f=s,R.\end{gathered}$$ (For more details, see [@fo], where the functions $s$ and $R$ were introduced.)

From 