A.1. Transformations of the MID
In this Appendix, we list several properties of the MID. A more detailed list together with the proofs can be found in [7].
The MID and the conjugate one are related by
$$ \kern1.8pt\overline{\vphantom{K}\kern7.0pt}\kern-8.8pt K _{n,m}^{(z)}(\bar u|\bar v)=(1-z)^{m-n}K_{m,n}^{(z)}(\bar v|\bar u).$$
(A.1)
The MID has the following property under the shift of one of the parameter sets:
$$K_{n,m}^{(z)}(\bar u-c|\bar v)=\frac{(-z)^{n}(1-z)^{m-n}}{f(\bar v,\bar u)}K_{m,n}^{(1/z)}(\bar v|\bar u),$$
(A.2)
$$\kern1.8pt\overline{\vphantom{K}\kern7.0pt}\kern-8.8pt K _{n,m}^{(z)}(\bar u+c|\bar v)=\frac{(-z)^{n}(1-z)^{m-n}}{f(\bar u,\bar v)} \kern1.8pt\overline{\vphantom{K}\kern7.0pt}\kern-8.8pt K _{m,n}^{(1/z)}(\bar v|\bar u).$$
(A.3)
Some bilinear combinations of the MIDs reduce to a new MID.
Proposition A.1.
Let
\(\bar\xi\)
,
\(\bar u\)
, and
\(\bar v\)
be sets of arbitrary complex numbers such that
\(\#\bar\xi=l\)
,
\(\#\bar u=n\)
and
\(\#\bar v=m\)
. Then
$$ \sum_{\{\bar\xi_{ \scriptscriptstyle{\mathrm{I}} },\bar\xi_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar\xi}z_2^{l_{ \scriptscriptstyle{\mathrm{I}} }} K_{n,l_{ \scriptscriptstyle{\mathrm{I}} }}^{(z_1)}(\bar u|\bar\xi_{ \scriptscriptstyle{\mathrm{I}} })K_{m,l_{ \scriptscriptstyle{\mathrm{II}} }}^{(z_2)}(\bar v|\bar\xi_{ \scriptscriptstyle{\mathrm{II}} })f(\bar\xi_{ \scriptscriptstyle{\mathrm{II}} },\bar\xi_{ \scriptscriptstyle{\mathrm{I}} })f(\bar u,\bar\xi_{ \scriptscriptstyle{\mathrm{II}} }) =K_{n+m,l}^{(z_1z_2)}(\{\bar u,\bar v\}|\bar\xi).$$
(A.4)
Replacing \(K_{n,l_{ \scriptscriptstyle{\mathrm{I}} }}^{(z_1)}(\bar u|\bar\xi_{ \scriptscriptstyle{\mathrm{I}} })\) in (A.4) with the conjugate MID via (A.1) and (A.2), we obtain
$$\sum_{\{\bar\xi_{ \scriptscriptstyle{\mathrm{I}} },\bar\xi_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar\xi}\biggl(-\frac{z_2}{z_1}\biggr)^{\!l_{ \scriptscriptstyle{\mathrm{I}} }} \kern1.8pt\overline{\vphantom{K}\kern7.0pt}\kern-8.8pt K _{n,l_{ \scriptscriptstyle{\mathrm{I}} }}^{(z_1)}(\bar u|\bar\xi_{ \scriptscriptstyle{\mathrm{I}} })K_{m,l_{ \scriptscriptstyle{\mathrm{II}} }}^{(z_2)}(\bar v|\bar\xi_{ \scriptscriptstyle{\mathrm{II}} })f(\bar\xi_{ \scriptscriptstyle{\mathrm{II}} },\bar\xi_{ \scriptscriptstyle{\mathrm{I}} }) =f(\bar\xi,\bar u)K_{n+m,l}^{(z_1z_2)}(\{\bar u-c,\bar v\}|\bar\xi).$$
Setting
\(z_1=z_2=1\) here yields
$$ \sum_{\{\bar\xi_{ \scriptscriptstyle{\mathrm{I}} },\bar\xi_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar\xi}(-1)^{l_{ \scriptscriptstyle{\mathrm{I}} }} \kern1.8pt\overline{\vphantom{K}\kern7.0pt}\kern-8.8pt K _{n,l_{ \scriptscriptstyle{\mathrm{I}} }}^{(1)}(\bar u|\bar\xi_{ \scriptscriptstyle{\mathrm{I}} })K_{m,l_{ \scriptscriptstyle{\mathrm{II}} }}^{(1)}(\bar v|\bar\xi_{ \scriptscriptstyle{\mathrm{II}} })f(\bar\xi_{ \scriptscriptstyle{\mathrm{II}} },\bar\xi_{ \scriptscriptstyle{\mathrm{I}} }) =f(\bar\xi,\bar u)K_{n+m,l}^{(1)}(\{\bar u-c,\bar v\}|\bar\xi).$$
(A.5)
A.2. Other determinants related to the MID
Proposition A.2.
Let
\(\bar u=\{u_1,\ldots,u_N\}\)
,
\(\bar\eta=\{\eta_1,\ldots,\eta_N\}\)
, and
\(z\)
be arbitrary complex numbers. Let
\(F_k^{(1)}(u)\)
and
\(F_k^{(2)}(u)\)
,
\(k=1,\ldots,N\)
, be two sets of functions
$$ \begin{aligned} \, &F_k^{(1)}(u)=\phi_1(u)\biggl(\frac z{g(\bar\eta_k,u)}-h(\bar\eta_k,u)\biggr)+\phi_2(u)\biggl(\frac{1}{g(u,\bar\eta_k)}-z h(u,\bar\eta_k)\biggr), \\ &F_k^{(2)}(u)=\frac{(-1)^{N-1}\phi_1(u)}{g(u,\bar\eta_k)}-\phi_2(u)h(u,\bar\eta_k), \end{aligned}$$
(A.6)
$$ \widehat F_{jk}^{(1)}=F_k^{(1)}(u_j),\qquad\widehat F_{jk}^{(2)}=F_k^{(2)}(u_j).$$
(A.7)
$$ \det_N\widehat F^{(1)}=(z-1)^N\det_N\widehat F^{(2)}.$$
(A.8)
Proof.
Obviously, both determinants can be presented in the form
$$ \det_N\widehat F^{(\ell)}=\sum_{\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u}X^{(\ell)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })\phi_1(\bar u_{ \scriptscriptstyle{\mathrm{I}} })\phi_2(\bar u_{ \scriptscriptstyle{\mathrm{II}} }), \qquad \ell=1,2,$$
(A.9)
where the coefficients
\(X^{(\ell)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })\) are independent of
\(\phi_1\) and
\(\phi_2\). In this equation, we used the shorthand notation for the products of the functions
\(\phi_\ell(u)\),
\(\ell=1,2\). Because
\(\phi_\ell\) are arbitrary functions, Eq. (
A.8) holds if and only if
$$ X^{(1)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })=(z-1)^N X^{(2)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })$$
(A.10)
for an arbitrary partition
\(\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u\). However, it is easy to see that without a loss of generality, it suffices to prove (
A.10) for
\(\bar u_{ \scriptscriptstyle{\mathrm{I}} }=\{u_1,\ldots,u_p\}\) and
\(\bar u_{ \scriptscriptstyle{\mathrm{II}} }=\{u_{p+1},\ldots,u_N\}\). Here,
\(p\) is an arbitrary integer from the set
\(\{0,1,\ldots,N\}\).
To obtain the coefficients \(X^{(\ell)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })\), we should set \(\phi_2(u_j)=0\) for \(j=1,\ldots,p\) and \(\phi_1(u_j)=0\) for \(j=p+1,\ldots,N\) in (A.6) and (A.6). We obtain
$$ X^{(\ell)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })=\det_N\Phi^{(\ell)},\qquad\ell=1,2,$$
(A.11)
where
$$ \begin{alignedat}{3} &\Phi^{(1)}_{jk}=\frac z{g(\bar\eta_k,u_j)}-h(\bar\eta_k,u_j),&\qquad &j=1,\ldots,p, \\ &\Phi^{(1)}_{jk}=\frac{1}{g(u_j,\bar\eta_k)}-z h(u_j,\bar\eta_k),&\qquad &j=p+1,\ldots,N, \end{alignedat}$$
(A.12)
and
$$ \begin{alignedat}{3} &\Phi^{(2)}_{jk}=\frac{(-1)^{N-1}}{g(u_j,\bar\eta_k)},&\qquad &j=1,\ldots,p, \\ &\Phi^{(2)}_{jk}=-h(u_j,\bar\eta_k),&\qquad &j=p+1,\ldots,N. \end{alignedat}$$
(A.13)
Let \(u_j=u'_j+c\) for \(j=1,\ldots,p\), and \(u_j=u'_j\) for \(j=p+1,\ldots,N\). Using \(1/g(u_j,\bar\eta_k)=h(u'_j,\bar\eta_k)\) and \(h(\bar\eta_k,u_j)=(-1)^{N-1}/g(u'_j,\bar\eta_k)\) (see (2.11)) for \(k=1,\ldots,p\), we obtain
$$ X^{(\ell)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })=(-1)^{(\ell+p-1)N}\det_N\widetilde{\Phi}^{(\ell)},\qquad\ell=1,2,$$
(A.14)
where
$$ \widetilde{\Phi}^{(1)}_{jk}=\frac{1}{g(u'_j,\bar\eta_k)}-z h(u'_j,\bar\eta_k),\qquad \widetilde{\Phi}^{(2)}_{jk}=h(u'_j,\bar\eta_k), \qquad j,k=1,\ldots,N.$$
(A.15)
It is easy to see that
\(\det_N\widetilde{\Phi}^{(2)}\) reduces to the Cauchy determinant:
$$ \det_N\widetilde{\Phi}^{(2)} =h(\bar u',\bar\eta)\det_N\biggl(\frac{1}{h(u'_j,\eta_k)}\biggr) =\frac{1}{\Delta(\bar\eta)\Delta'(\bar u')}.$$
(A.16)
The determinant of \(\widetilde{\Phi}^{(1)}\) is computed in Corollary 4.1. Due to (4.12), we have
$$ \det_N\widetilde{\Phi}^{(1)}=\frac{(1-z)^N}{\Delta(\bar\eta)\Delta'(\bar u')}.$$
(A.17)
Comparing this equation with (
A.16) and using (
A.14), we see that
$$ X^{(1)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })=(z-1)^NX^{(2)}(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }).$$
(A.18)
The proposition is proved.
\(\blacksquare\)
Corollary A.1.
Let \(0\leqslant n\leqslant N\), and let \(\bar u=\{u_1,\ldots,u_N\}\), \(\bar\eta=\{\eta_{n+1},\ldots,\eta_N\}\), and \(z\) be arbitrary complex numbers (\(\bar\eta=\varnothing\) for \(n=N\)). We compose two \(N\times N\) matrices \(\widehat F^{(01)}\) and \(\widehat F^{(02)}\) with the entries
$$\widehat F_{jk}^{(01)}=\begin{cases} F_k^{(0)}(u_j), & k=1,\ldots,n,\\ F_k^{(1)}(u_j), & k=n+1,\ldots,N, \end{cases}\qquad \widehat F_{jk}^{(02)}=\begin{cases} F_k^{(0)}(u_j), & k=1,\ldots,n,\\ F_k^{(2)}(u_j), & k=n+1,\ldots,N, \end{cases}$$
(
A.6)
, while \(F_k^{(0)}(u)\), \(k=1,\ldots,n\), are arbitrary functions. Then $$ \det_N\widehat F^{(01)} = (z-1)^{N-n}\det_N\widehat F^{(02)}.$$
(A.19)
Proof.
Developing the determinant \(\det\widehat F^{(01)}\) with respect to the first \(n\) columns, we obtain
$$ \det_N\widehat F^{(01)}=\sum_{\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u} (-1)^{P_{ \scriptscriptstyle{\mathrm{I}} , \scriptscriptstyle{\mathrm{II}} }} \det_n\bigl(F_k^{(0)}(u^{ \scriptscriptstyle{\mathrm{I}} }_j)\bigr) \det_{N-n}\bigl(F_k^{(1)}(u^{ \scriptscriptstyle{\mathrm{II}} }_j)\bigr),$$
(A.20)
where the sum is taken over partitions
\(\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u\) such that
\(\#\bar u_{ \scriptscriptstyle{\mathrm{I}} }=n\) and
\(\#\bar u_{ \scriptscriptstyle{\mathrm{II}} }\,{=}\,N-n\). The notation
\(u^{ \scriptscriptstyle{\mathrm{I}} }_j\) (resp.
\(u^{ \scriptscriptstyle{\mathrm{II}} }_j\)) means the
\(j\)th element of the subset
\(\bar u_{ \scriptscriptstyle{\mathrm{I}} }\) (resp.
\(\bar u_{ \scriptscriptstyle{\mathrm{II}} }\)). Finally,
\(P_{ \scriptscriptstyle{\mathrm{I}} , \scriptscriptstyle{\mathrm{II}} }\) is the parity of the partition
\(\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u\).
Due to Proposition A.2,
$$ \det_{N-n}\bigl(F_k^{(1)}(u^{ \scriptscriptstyle{\mathrm{II}} }_j)\bigr)=(z-1)^{N-n}\det_{N-n}\bigl(F_k^{(2)}(u^{ \scriptscriptstyle{\mathrm{II}} }_j)\bigr)$$
(A.21)
for an arbitrary subset
\(\bar u_{ \scriptscriptstyle{\mathrm{II}} }\). Hence,
$$ \det_N\widehat F^{(01)}=(z-1)^{N-n}\sum_{\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u} (-1)^{P_{ \scriptscriptstyle{\mathrm{I}} , \scriptscriptstyle{\mathrm{II}} }}\det_n\bigl(F_k^{(0)}(u^{ \scriptscriptstyle{\mathrm{I}} }_j)\bigr)\det_{N-n}\bigl(F_k^{(2)}(u^{ \scriptscriptstyle{\mathrm{II}} }_j)\bigr).$$
(A.22)
Taking the sum over partitions, we arrive at (
A.19).
\(\blacksquare\) In particular, setting
$$ \phi_1(u)=(-1)^{N}\hat\lambda_1(u)h(\bar v,u),\qquad \phi_2(u)=\hat\lambda_2(u)h(u,\bar v),$$
(A.23)
we obtain
$$ F_k^{(1)}(u_j)=\mathcal N^{(2)}_{jk},\qquad k=n+1,\ldots,N,$$
(A.24)
where
\(\mathcal N^{(2)}_{jk}\) is given by (
5.16). By Corollary
A.1, we obtain
$$ \det_N\widehat F^{(01)} = (z-1)^{N-n}\det_N\widehat F^{(02)},$$
(A.25)
where
$$ \begin{aligned} \, &\widehat F_{jk}^{(01)}=\begin{cases} F_k^{(0)}(u_j),& k=1,\ldots,n,\\ \mathcal N^{(2)}_{jk}, & k=n+1,\ldots,N, \end{cases} \\ &\widehat F_{jk}^{(02)}=\begin{cases} F_k^{(0)}(u_j), & k=1,\ldots,n, \\ (-1)^{n+1}\hat\lambda_1(u_j)\dfrac{h(\bar v,u_j)}{g(u_j,\bar\eta_k)}-{} & \\ \qquad -\hat\lambda_2(u_j)h(u_j,\bar v)h(u_j,\bar\eta_k), &k=n+1,\ldots,N, \end{cases} \end{aligned}$$
(A.26)
and
\(F_k^{(0)}(u)\),
\(k=1,\ldots,n\), are arbitrary functions.
A.3. Summation formula
Let a function \(H(\bar u)\) of \(N\) variables \(\bar u=\{u_1,\ldots,u_N\}\) be defined as
$$ H(\bar u)=\Delta(\bar u)\det_N\Phi_k(u_j),$$
(A.27)
where
\(\Phi_k(u)\) is a set of functions of one variable.
Proposition A.3.
Let
\(\phi_1(u)\)
and
\(\phi_2(u)\)
be one-variable functions. Then
$$\begin{aligned} \, \sum_{\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u}\phi_1(\bar u_{ \scriptscriptstyle{\mathrm{I}} })\phi_2(\bar u_{ \scriptscriptstyle{\mathrm{II}} }) f(\bar u_{ \scriptscriptstyle{\mathrm{II}} },\bar u_{ \scriptscriptstyle{\mathrm{I}} })H(\{\bar u_{ \scriptscriptstyle{\mathrm{I}} }-c,\bar u_{ \scriptscriptstyle{\mathrm{II}} }\})= \Delta(\bar u)\det_N\bigl(\phi_1(u_j)\Phi_k(u_j-c)+\phi_2(u_j)\Phi_k(u_j)\bigr). \end{aligned}$$
(A.28)
Proof is similar to the one of Proposition A.2.
The right-hand side of (A.28) can be presented as
$$\Delta(\bar u)\det_N\bigl(\phi_1(u_j)\Phi_k(u_j-c)+\phi_2(u_j)\Phi_k(u_j)\bigr)= \sum_{\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u}X(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })\phi_1(\bar u_{ \scriptscriptstyle{\mathrm{I}} })\phi_2(\bar u_{ \scriptscriptstyle{\mathrm{II}} }),$$
where the coefficients
\(X(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })\) are independent of
\(\phi_1\) and
\(\phi_2\). Because
\(\phi_\ell\) are arbitrary functions, Eq. (
A.28) holds if and only if
$$ X(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })=f(\bar u_{ \scriptscriptstyle{\mathrm{II}} },\bar u_{ \scriptscriptstyle{\mathrm{I}} })H(\{\bar u_{ \scriptscriptstyle{\mathrm{I}} }-c,\bar u_{ \scriptscriptstyle{\mathrm{II}} }\})$$
(A.29)
for an arbitrary partition
\(\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u\). Due to the symmetry of (
A.28) over
\(\bar u\), it suffices to prove (
A.29) for
\(\bar u_{ \scriptscriptstyle{\mathrm{I}} }=\{u_1,\ldots,u_p\}\) and
\(\bar u_{ \scriptscriptstyle{\mathrm{II}} }=\{u_{p+1},\ldots,u_N\}\), where
\(p\) is an arbitrary integer from the set
\(\{0,1,\ldots,N\}\).
To obtain the coefficients \(X(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })\), we should set \(\phi_2(u_j)=0\) for \(j=1,\ldots,p\) and \(\phi_1(u_j)=0\) for \(j=p+1,\ldots,N\) in (A.28). We obtain
$$ X(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })=\Delta(\bar u_{ \scriptscriptstyle{\mathrm{I}} })\Delta(\bar u_{ \scriptscriptstyle{\mathrm{II}} })g(\bar u_{ \scriptscriptstyle{\mathrm{II}} },\bar u_{ \scriptscriptstyle{\mathrm{I}} })\det_N\widetilde{\Phi},$$
(A.30)
where
$$ \begin{alignedat}{3} &\widetilde{\Phi}_{jk}=\Phi_k(u_j-c),&\qquad &j=1,\ldots,p, \\ &\widetilde{\Phi}_{jk}=\Phi_k(u_j),&\qquad &j=p+1,\ldots,N. \end{alignedat}$$
(A.31)
Let \(u_j=u'_j+c\) for \(j=1,\ldots,p\), and \(u_j=u'_j\) for \(j=p+1,\ldots,N\). Using \(g(x,y+c)=-1/h(y,x)\) for any \(x\) and \(y\), we obtain
$$ \Delta(\bar u_{ \scriptscriptstyle{\mathrm{I}} })\Delta(\bar u_{ \scriptscriptstyle{\mathrm{II}} })g(\bar u_{ \scriptscriptstyle{\mathrm{II}} },\bar u_{ \scriptscriptstyle{\mathrm{I}} })= (-1)^{p(N-p)}\frac{\Delta(\bar u'_{ \scriptscriptstyle{\mathrm{I}} })\Delta(\bar u'_{ \scriptscriptstyle{\mathrm{II}} })}{h(\bar u'_{ \scriptscriptstyle{\mathrm{I}} },\bar u'_{ \scriptscriptstyle{\mathrm{II}} })}= \frac{\Delta(\bar u')}{f(\bar u'_{ \scriptscriptstyle{\mathrm{I}} },\bar u'_{ \scriptscriptstyle{\mathrm{II}} })}.$$
(A.32)
Then (
A.30) takes the form
$$ X(\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} })=\frac{\Delta(\bar u')}{f(\bar u'_{ \scriptscriptstyle{\mathrm{I}} },\bar u'_{ \scriptscriptstyle{\mathrm{II}} })}\det_N\Phi_k(u'_j)= \frac{H(\bar u')}{f(\bar u'_{ \scriptscriptstyle{\mathrm{I}} },\bar u'_{ \scriptscriptstyle{\mathrm{II}} })}.$$
(A.33)
Returning to the original variables and using
\(f(x-c,y)=1/f(y,x)\) for any
\(x\) and
\(y\), we arrive at (
A.29).
\(\blacksquare\) We deal with a particular case of the sum in (A.28) in Eq. (5.8). Indeed, let
$$\phi_1(u)=\lambda_1(u)f(\bar v,u),\qquad\phi_2(u)=\frac{\rho_2}{\rho_1}\lambda_2(u),\qquad \Phi_k(u)=\frac{f(u,\bar v)}{g(u,\bar\eta_k)}-z h(u,\bar\eta_k),$$
where
\(\bar\eta=\{\eta_1,\ldots,\eta_N\}\) are arbitrary complex numbers. Then
$$ \Delta'(\bar\eta) H(\bar u)=(1-z)^{N-n}K^{(z)}_{N,n}(\bar u|\bar v).$$
(A.34)
Hence, due to Proposition
A.3, we have
$$\begin{aligned} \, \sum_{\{\bar u_{ \scriptscriptstyle{\mathrm{I}} },\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}\vdash\bar u} \biggl(\frac{\rho_2}{\rho_1}\biggr)^{\!N_{ \scriptscriptstyle{\mathrm{II}} }} &\lambda_2(\bar u_{ \scriptscriptstyle{\mathrm{II}} })\lambda_1(\bar u_{ \scriptscriptstyle{\mathrm{I}} })f(\bar u_{ \scriptscriptstyle{\mathrm{II}} },\bar u_{ \scriptscriptstyle{\mathrm{I}} })f(\bar v,\bar u_{ \scriptscriptstyle{\mathrm{I}} }) K^{(1)}_{N,n}(\{\bar u_{ \scriptscriptstyle{\mathrm{I}} }-c,\bar u_{ \scriptscriptstyle{\mathrm{II}} }\}|\bar v)= \nonumber\\ &=(1-z)^{N-n}\Delta'(\bar\eta)\Delta(\bar u)\det_N\mathcal{M}_{jk}, \end{aligned}$$
(A.35)
where
$$\begin{aligned} \, \mathcal{M}_{jk}={}&\lambda_1(u_j)f(\bar v,u_j)\biggl(\frac{f(u_j-c,\bar v)}{g(u_j-c,\bar\eta_k)}-z h(u_j-c,\bar\eta_k)\biggr)+{} \nonumber\\ &+\frac{\rho_2}{\rho_1}\lambda_2(u_j)\biggl(\frac{f(u_j,\bar v)}{g(u_j,\bar\eta_k)}-z h(u_j,\bar\eta_k)\biggr). \end{aligned}$$
(A.36)
Using Eqs. (
2.11), we easily transform this result to representation (
5.10).
