\ell>y_{i}\end{smallmatrix}}\frac{1}{b(\ell)-b(y_{i})}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}\frac{1}{b(y_{i})-b(\ell)}\right].$ (319)

We take $b(z)=q^{-z}$ in (319). The second factor becomes,

$\displaystyle\prod_{i<j}(b(y_{i})-b(y_{j}))=\prod_{i<j}(q^{-y_{i}}-q^{-y_{j}})=\prod_{i<j}(1-q^{y_{i}-y_{j}})q^{-y_{i}}=\prod_{i<j}(1-q^{y_{i}-y_{j}})\prod_{i}q^{-(n-i)y_{i}}.$
(320)

For the last product in (319), we have

$\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}\bigl{(}b(\ell)-b(y_{i})\bigr{)}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}\bigl{(}b(y_{i})-b(\ell)\bigr{)}\\\
\propto\left(\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}\frac{1-q^{\ell-
y_{i}}}{1-q}\right)\left(\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}\frac{1-q^{y_{i}-\ell}}{1-q}\right)\left(\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}q^{-\ell}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}q^{-y_{i}}\right).$ (321)

We transform the first factor in the righthand side of (321) as

$\prod_{i=1}^{n}\left(\prod_{\ell=y_{i}+1}^{x_{i}}\frac{1-q^{\ell-
y_{i}}}{1-q}\cdot\prod_{j=1}^{i-1}\left[\prod_{\ell=x_{j+1}+\theta}^{x_{j}}\frac{1-q^{\ell-
y_{i}}}{1-q}\right]\right)\\\
=\prod_{i=1}^{n}\left(\Gamma_{q}(x_{i}-y_{i}+1)\prod_{j=1}^{i-1}\frac{\Gamma_{q}(x_{j}-y_{i}+1)}{\Gamma_{q}(x_{j+1}+\theta-
y_{i})}\right)=\prod_{i=1}^{n}\Gamma_{q}(x_{1}-y_{i}+\theta)\prod_{1\leqslant
i\leqslant j\leqslant
n}\frac{\Gamma_{q}(x_{i}-y_{j}+1)}{\Gamma_{q}(x_{i}-y_{j}+\theta)},$ (322)

where in the last identity we swapped $i$ and $j$. We also transform the
second factor in (321):

$\prod_{i=1}^{n}\left(\prod_{\ell=x_{i+1}+\theta}^{y_{i}-1}\frac{1-q^{y_{i}-\ell}}{1-q}\prod_{j=n}^{i+1}\left[\prod_{\ell=x_{j+1}+\theta}^{x_{j}}\frac{(1-q^{y_{i}-\ell})}{1-q}\right]\right)\\\
=\prod_{i=1}^{n}\left(\Gamma_{q}(y_{i}-x_{i+1}-\theta+1)\prod_{j=i+1}^{n}\frac{\Gamma_{q}(y_{i}-x_{j+1}-\theta+1)}{\Gamma_{q}(y_{i}-x_{j})}\right)\\\
=\prod_{i=1}^{n}\Gamma_{q}(y_{i}-x_{n+1})\prod_{1\leqslant i\leqslant
j\leqslant
n}\frac{\Gamma_{q}(y_{i}-x_{j+1}-\theta+1)}{\Gamma_{q}(y_{i}-x_{j+1})}.$ (323)

Finally, we transform the last factor in the righthand side of (321):

$\prod_{i=1}^{n}\left(\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}q^{-\ell}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}q^{-y_{i}}\right)=q^{-\sum\limits_{i=1}^{n}\left(\sum\limits_{\ell=y_{i}+1}^{x_{i}}\ell+\sum\limits_{j=1}^{i-1}\sum\limits_{\ell=x_{j+1}+\theta}^{x_{j}}\ell\right)-\sum\limits_{i=1}^{n}\left(\sum\limits_{\ell=x_{i+1}+\theta}^{y_{i}-1}y_{i}+\sum\limits_{j=i+1}^{n}\sum\limits_{\ell=x_{j+1}+\theta}^{x_{j}}y_{i}\right)}\\\
\propto
q^{\sum\limits_{i=1}^{n}(y_{i}^{2}+y_{i})/2-\sum\limits_{i=1}^{n}y_{i}(y_{i}-x_{n+1}-(n-i+1)\theta+n-i)}=q^{-\sum\limits_{i=1}^{n}\left(y_{i}^{2}/2+y_{i}(-x_{n+1}-(n-i+1)\theta+n-i-1/2)\right)}.$
(324)

Combining (320), (321), (322) (323) and (324), we rewrite (319) as

$\prod_{i<j}(1-q^{y_{i}-y_{j}})\prod_{i\leqslant
j}\frac{\Gamma_{q}(x_{i}-y_{j}+\theta)}{\Gamma_{q}(x_{i}-y_{j}+1)}\frac{\Gamma_{q}(y_{i}-x_{j+1})}{\Gamma_{q}(y_{i}-x_{j+1}-\theta+1)}\prod_{i=1}^{n}\frac{q^{-\theta(n-i+1)y_{i}}q^{y_{i}^{2}/2-y_{i}(x_{n+1}+1/2)}w(y_{i})}{\Gamma_{q}(x_{1}-y_{i}+\theta)\Gamma_{q}(y_{i}-x_{n+1})},$

which matches (65) with $w(z)$ as in (66). ∎

###### Proof of Claim 3.17.

We recall the parameters $s^{2}=q^{v}$, and the coefficients (73)

$\displaystyle\begin{split}\psi_{{\bm{\lambda}}\setminus{\bm{\mu}}}^{(i)}(u;q,t,s)&=(u/t)^{|{\bm{\lambda}}|-|{\bm{\mu}}|}t^{n({\bm{\mu}})-n({\bm{\lambda}})}\frac{C_{\bm{\lambda}}^{0}(s^{2}qt/u;q,t)C_{\bm{\mu}}^{0}(u/t;q,t)}{C_{\bm{\lambda}}^{0}(u;q,t)C_{\bm{\mu}}^{0}(s^{2}qt/u;q,t)}\\\
&\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}+j-1}t^{-\lambda_{j}^{\prime}-i+3}s^{2}}{1-q^{\mu_{i}-j+1}t^{\mu_{j}^{\prime}-i}}\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}\neq\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}-j}t^{\lambda_{j}^{\prime}-i+1}}{1-q^{\mu_{i}+j}t^{-\mu_{j}^{\prime}-i+1}s^{2}}\\\
&\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}-j+1}t^{\lambda_{j}^{\prime}-i}}{1-q^{\mu_{i}+j-1}t^{2-\mu_{j}^{\prime}-i}s^{2}}\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}\neq\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}+j}t^{2-\lambda_{j}^{\prime}-i}s^{2}}{1-q^{\mu_{i}-j}t^{\mu_{j}^{\prime}-i+1}},\end{split}$
(325) $\displaystyle C_{\bm{\mu}}^{0}(x;q,t)$
$\displaystyle=\prod_{(i,j)\in{\bm{\mu}}}(1-q^{j-1}t^{1-i}x)=\prod_{i\geqslant
1}(t^{1-i}x;q)_{\mu_{i}}$

We take $u=t^{n+1}$ and $s=t^{n}\sqrt{t_{0}t_{1}t_{2}t_{3}/qt}$, and recall
$t^{n}\sqrt{t_{0}t_{1}t_{2}t_{3}/qt}=q^{v/2}$. We have

$\displaystyle(u/t)^{|{\bm{\lambda}}|-|{\bm{\mu}}|}t^{n({\bm{\mu}})-n({\bm{\lambda}})}=t^{n(|{\bm{\lambda}}|-|{\bm{\mu}}|)+n({\bm{\mu}})-n({\bm{\lambda}})}=q^{-\theta(\sum_{i}(n-(i-1))y_{i}-\sum_{i}(n-(i-1))x_{i})}.$
(326)

and

$\displaystyle\begin{split}&\phantom{{}={}}\frac{C_{\bm{\mu}}^{0}(u/t;q,t)}{C_{\bm{\mu}}^{0}(s^{2}qt/u;q,t)}=\frac{C_{\bm{\mu}}^{0}(t^{n};q,t)}{C_{\bm{\mu}}^{0}(q^{v+1}t^{-n};q,t)}=\frac{C_{\bm{\mu}}^{0}(q^{n\theta};q,t)}{C_{\bm{\mu}}^{0}(q^{v+1-n\theta};q,t)}\\\
&=\prod_{i=1}^{n}\frac{\Gamma_{q}(\mu_{i}+(n-i+1)\theta)}{\Gamma_{q}((n-i+1)\theta)}\frac{\Gamma_{q}(v+1-(n+i-1)\theta)}{\Gamma_{q}(v+1-(n+i-1)\theta+\mu_{i})}\\\
&=\prod_{i=1}^{n}\frac{\Gamma_{q}(y_{i}+n\theta)\Gamma_{q}(v+1-(n+i-1)\theta)}{\Gamma_{q}((n-i+1)\theta)\Gamma_{q}(y_{i}+v+1-n\theta)}.\end{split}$
(327)

In the following, we simplify the products in
$\psi_{{\bm{\lambda}}/{\bm{\mu}}}^{(i)}(u;q,t,s)$. For the first product term
in (325)

$\displaystyle\begin{split}\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}+j-1}t^{-\lambda_{j}^{\prime}-i+3}s^{2}}{1-q^{\mu_{i}-j+1}t^{\mu_{j}^{\prime}-i}}&=\prod_{k}\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}=k}\frac{1-q^{\lambda_{i}+j-1}t^{-k-i+3}s^{2}}{1-q^{\mu_{i}-j+1}t^{k-i}}\\\
&=\prod_{k}\prod_{1\leqslant i\leqslant k\atop\lambda_{k+1}+1\leqslant
j\leqslant\mu_{k}}\frac{1-q^{\lambda_{i}+j-1}t^{-k-i+3}s^{2}}{1-q^{\mu_{i}-j+1}t^{k-i}}\\\
&=\prod_{i\leqslant
k}\frac{\Gamma_{q}(x_{i}+y_{k}+v+\theta)\Gamma_{q}(y_{i}-y_{k}+1)}{\Gamma_{q}(x_{i}+x_{k+1}+v+2\theta)\Gamma_{q}(y_{i}-x_{k+1}-\theta+1)}.\end{split}$
(328)

For the second product term in (325)

$\displaystyle\begin{split}\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}\neq\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}-j}t^{\lambda_{j}^{\prime}-i+1}}{1-q^{\mu_{i}+j}t^{-\mu_{j}^{\prime}-i+1}s^{2}}&=\prod_{k}\prod_{(i,j)\in{\bm{\lambda}}\atop\lambda_{j}^{\prime}=k,\mu_{j}^{\prime}=k-1}\frac{1-q^{\lambda_{i}-j}t^{k-i+1}}{1-q^{\mu_{i}+j}t^{-k-i+2}s^{2}}\\\
&=\prod_{k}\prod_{1\leqslant i\leqslant k\atop\mu_{k}+1\leqslant
j\leqslant\lambda_{k}}\frac{1-q^{\lambda_{i}-j}t^{k-i+1}}{1-q^{\mu_{i}+j}t^{-k-i+2}s^{2}}\\\
&=\prod_{i\leqslant
k}\frac{\Gamma_{q}(x_{i}-y_{k}+\theta)\Gamma_{q}(y_{i}+y_{k}+1+v)}{\Gamma_{q}(x_{i}-x_{k}+\theta)\Gamma_{q}(y_{i}+x_{k}+1+v)}.\end{split}$
(329)

For the third product term in (325), we have

$\displaystyle\begin{split}\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}-j+1}t^{\lambda_{j}^{\prime}-i}}{1-q^{\mu_{i}+j-1}t^{2-\mu_{j}^{\prime}-i}s^{2}}&=\prod_{k}\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}=\mu_{j}^{\prime}=k}\frac{1-q^{\lambda_{i}-j+1}t^{k-i}}{1-q^{\mu_{i}+j-1}t^{2-k-i}s^{2}}\\\
&=\prod_{k}\prod_{1\leqslant i\leqslant k,\atop\lambda_{k+1}+1\leqslant
j\leqslant\mu_{k}}\frac{1-q^{\lambda_{i}-j+1}t^{k-i}}{1-q^{\mu_{i}+j-1}t^{2-k-i}s^{2}}\\\
&=\prod_{i\leqslant
k}\frac{\Gamma_{q}(x_{i}-x_{k+1}-\theta+1)\Gamma_{q}(y_{i}+x_{k+1}+v+\theta)}{\Gamma_{q}(x_{i}-y_{k}+1)\Gamma_{q}(y_{i}+y_{k}+v)}.\end{split}$
(330)

Finally for the last product term in (325), we have

$\displaystyle\begin{split}\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}\neq\mu_{j}^{\prime}}\frac{1-q^{\lambda_{i}+j}t^{2-\lambda_{j}^{\prime}-i}s^{2}}{1-q^{\mu_{i}-j}t^{\mu_{j}^{\prime}-i+1}}&=\prod_{k}\prod_{(i,j)\in{\bm{\mu}}\atop\lambda_{j}^{\prime}=k+1,\mu_{j}^{\prime}=k}\frac{1-q^{\lambda_{i}+j}t^{1-k-i}s^{2}}{1-q^{\mu_{i}-j}t^{k-i+1}}\\\
&=\prod_{k}\prod_{1\leqslant i\leqslant k\atop\mu_{k+1}+1\leqslant
j\leqslant\lambda_{k+1}}\frac{1-q^{\lambda_{i}+j}t^{1-k-i}s^{2}}{1-q^{\mu_{i}-j}t^{k-i+1}}\\\
&=\prod_{1\leqslant i\leqslant
k}\frac{\Gamma_{q}(y_{i}-x_{k+1})\Gamma_{q}(x_{i}+x_{k+1}+1+v)}{\Gamma_{q}(y_{i}-y_{k+1})\Gamma_{q}(x_{i}+y_{k+1}+1+v)}.\end{split}$
(331)

Putting (326), (LABEL:e:ct1), (328), (329), (330), and (331) all together, we
get the following expression for the transition probability (74):

$\displaystyle\begin{split}&\mathbb{P}({\bm{y}}|{\bm{x}})=\prod_{i=1}^{n}\left[q^{-\theta(\sum_{i}(n-(i-1))y_{i}-\sum_{i}(n-(i-1))x_{i})}\frac{\Gamma_{q}(y_{i}+n\theta)\Gamma_{q}(v+1-(n+i-1)\theta)}{\Gamma_{q}((n-i+1)\theta)\Gamma_{q}(y_{i}+v+1-n\theta)}\right]\\\
&\prod_{i\leqslant k\leqslant
n}\frac{\Gamma_{q}(x_{i}+y_{k}+v+\theta)\Gamma_{q}(y_{i}-y_{k}+1)}{\Gamma_{q}(x_{i}+x_{k+1}+2\theta+v)\Gamma_{q}(y_{i}-x_{k+1}-\theta+1)}\prod_{i\leqslant
k\leqslant
n+1}\frac{\Gamma_{q}(x_{i}-y_{k}+\theta)\Gamma_{q}(y_{i}+y_{k}+1+v)}{\Gamma_{q}(x_{i}-x_{k}+\theta)\Gamma_{q}(y_{i}+x_{k}+1+v)}\\\
&\prod_{i\leqslant k\leqslant
n}\frac{\Gamma_{q}(x_{i}-x_{k+1}-\theta+1)\Gamma_{q}(y_{i}+x_{k+1}+v+\theta)}{\Gamma_{q}(x_{i}-y_{k}+1)\Gamma_{q}(y_{i}+y_{k}+v)}\prod_{i\leqslant
k\leqslant
n}\frac{\Gamma_{q}(y_{i}-x_{k+1})\Gamma_{q}(x_{i}+x_{k+1}+1+v)}{\Gamma_{q}(y_{i}-y_{k+1})\Gamma_{q}(x_{i}+y_{k+1}+1+v)}\\\
&\propto\prod_{1\leqslant i<k\leqslant n}(1-q^{y_{i}-y_{k}})\prod_{1\leqslant
i\leqslant k\leqslant
n}\frac{\Gamma_{q}(y_{i}-x_{k+1})\Gamma_{q}(x_{i}-y_{k}+\theta)}{\Gamma_{q}(x_{i}-y_{k}+1)\Gamma_{q}(y_{i}-x_{k+1}-\theta+1)}\\\
&\times\prod_{1\leqslant i\leqslant k\leqslant
n}(1-q^{y_{i}+y_{k}+v})\left[\prod_{i=1}^{n}\prod_{k=1}^{n+1}\frac{\Gamma_{q}(y_{i}+x_{k}+v+\theta)}{\Gamma_{q}(y_{i}+x_{k}+1+v)}\right]\prod_{i=1}^{n}q^{-\theta\sum_{i=1}^{n}(n-(i-1))y_{i}}.\end{split}$
(332)

This finishes the proof of (75). In the following we check that (75) is a
special case of (27), with $b(z)=q^{-z}+q^{z+v}$. We notice that the
difference factorizes:

$\displaystyle
b(z_{1})-b(z_{2})=q^{-z_{1}}+q^{z_{1}+v}-q^{-z_{2}}-q^{z_{2}+v}=(q^{-z_{1}}-q^{-z_{2}})(1-q^{z_{1}+z_{2}+v}).$

We recall the transition probability (27)

$\mathbb{P}({\bm{y}}|{\bm{x}})=\frac{1}{Z({\bm{x}})}\prod_{1\leqslant
i<j\leqslant
n}\bigl{(}b(y_{i})-b(y_{j})\bigr{)}\,\prod_{i=1}^{n}\left[w(y_{i})\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}\frac{1}{b(\ell)-b(y_{i})}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}\frac{1}{b(y_{i})-b(\ell)}\right].$ (333)

With $b(z)=q^{-z}+q^{z+v}$, the first product becomes

$\displaystyle\prod_{i<j}(b(y_{i})-b(y_{j}))=\prod_{i<j}(q^{-y_{i}}-q^{-y_{j}})(1-q^{y_{i}+y_{j}+v}).$
(334)

For the denominators in (333) we have

$\displaystyle\begin{split}&\phantom{{}={}}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}\bigl{(}b(\ell)-b(y_{i})\bigr{)}\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}\bigl{(}b(y_{i})-b(\ell)\bigr{)}\\\
&=\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}(q^{-\ell}-q^{-y_{i}})(1-q^{y_{i}+\ell+v})\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}(q^{-y_{i}}-q^{-\ell})(1-q^{y_{i}+\ell+v})\\\
&\propto\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell>y_{i}\end{smallmatrix}}(q^{-\ell}-q^{-y_{i}})\prod\limits_{\begin{smallmatrix}\ell\in\mathbb{L}({\bm{x}})\\\
\ell<y_{i}\end{smallmatrix}}(q^{-y_{i}}-q^{-\ell})\prod_{\ell\in\mathbb{L}\setminus\\{y_{i}\\}}(1-q^{\ell+y_{i}+v}).\end{split}$
(335)

The first two products on the righthand side of (LABEL:e:1fp) are computed in
(321), (322), (323) and (324). For the last product term on the righthand side
of (LABEL:e:1fp), we have

$\displaystyle\begin{split}\prod_{i=1}^{n}\,\prod_{\ell\in\mathbb{L}\setminus\\{y_{i}\\}}(1-q^{\ell+y_{i}+v})&=\prod_{i=1}^{n}\frac{\prod_{\ell\in\mathbb{L}}(1-q^{\ell+y_{i}+v})}{1-q^{2y_{i}+v}}=\prod_{i=1}^{n}\frac{\prod_{j=1}^{n}\prod_{\ell=x_{j+1}+\theta}^{x_{j}}(1-q^{\ell+y_{i}+v})}{1-q^{2y_{i}+v}}\\\
&=\prod_{i=1}^{n}\left[\frac{1}{1-q^{2y_{i}+v}}\prod_{j=1}^{n}\frac{\Gamma_{q}(y_{i}+x_{j}+v+1)}{\Gamma_{q}(y_{i}+x_{j+1}+v+\theta)}\right]\\\
&=\prod_{i=1}^{n}\frac{1}{1-q^{2y_{i}+v}}\cdot\prod_{i=1}^{n}\prod_{j=1}^{n+1}\frac{\Gamma_{q}(y_{i}+x_{j}+v+1)}{\Gamma_{q}(y_{i}+x_{j}+v+\theta)}\cdot\prod_{i=1}^{n}\frac{\Gamma_{q}(y_{i}+x_{1}+v+\theta)}{\Gamma_{q}(y_{i}+x_{n+1}+v+1)}.\end{split}$
(336)

Combining (336) with transformations of (LABEL:e:1fp) from the previous proof,
we rewrite (333) as

$\displaystyle\prod_{1\leqslant i<j\leqslant
n}(1-q^{y_{i}-y_{j}})\prod_{1\leqslant i\leqslant j\leqslant
n}(1-q^{y_{i}+y_{j}+v})\prod_{1\leqslant i\leqslant j\leqslant
n}\frac{\Gamma_{q}(x_{i}-y_{j}+\theta)}{\Gamma_{q}(x_{i}-y_{j}+1)}\frac{\Gamma_{q}(y_{i}-x_{j+1})}{\Gamma_{q}(y_{i}-x_{j+1}-\theta+1)}$
$\displaystyle\times\prod_{i=1}^{n}\prod_{j=1}^{n+1}\frac{\Gamma_{q}(y_{i}+x_{j}+v+\theta)}{\Gamma_{q}(y_{i}+x_{j}+v+1)}\prod_{i=1}^{n}\left[\frac{\Gamma_{q}(y_{i}+x_{n+1}+v+1)}{\Gamma_{q}(y_{i}+x_{1}+v+\theta)}\frac{q^{-\sum_{i}(n-i+1)\theta
y_{i}}q^{y_{i}^{2}/2-y_{i}(x_{n+1}+1/2)}w(y_{i})}{\Gamma_{q}(x_{1}-y_{i}+\theta)\Gamma_{q}(y_{i}-x_{n+1})}\right],$

which matches (LABEL:e:KMdensityhicopy) if

$\displaystyle
w(z)=q^{-z^{2}/2+z(x_{n+1}+1/2)}{\Gamma_{q}(x_{1}-z+\theta)}{\Gamma_{q}(z-x_{n+1})}\frac{\Gamma_{q}(z+x_{1}+v+\theta)}{\Gamma_{q}(z+x_{n+1}+v+1)}.$

This finishes the proof of Claim 3.17. ∎

###### Proof of Theorem 3.19.

Let us do some simplifications in $\psi^{(i)}_{\bm{\lambda}/\bm{\mu}}$ of
Theorem 3.16 in the $q=t$ case. We additionally set
$t^{2n}t_{0}t_{1}t_{2}t_{3}=p$. Note that if
$\lambda^{\prime}_{j}=\mu^{\prime}_{j}$, then $(i,j)\in{{\bm{\lambda}}}$ is
equivalent to $(i,j)\in{{\bm{\mu}}}$. On the other hand, if
$(i,j)\in{{\bm{\lambda}}}/{{\bm{\mu}}}$, then necessarily
$\lambda^{\prime}_{j}\neq\mu^{\prime}_{j}$. Hence,

$\prod_{\begin{subarray}{c}(i,j)\in{{\bm{\lambda}}}\\\
\lambda^{\prime}_{j}=\mu^{\prime}_{j}\end{subarray}}\frac{1-q^{\lambda_{i}+j}q^{-\lambda^{\prime}_{j}-i}p}{1-q^{\mu_{i}-j+1}q^{\mu^{\prime}_{j}-i}}\prod_{\begin{subarray}{c}(i,j)\in{{\bm{\lambda}}}\\\
\lambda^{\prime}_{j}\neq\mu^{\prime}_{j}\end{subarray}}\frac{1-q^{\lambda_{i}-j}q^{\lambda^{\prime}_{j}-i+1}}{1-q^{\mu_{i}+j-1}q^{-\mu^{\prime}_{j}-i}p}\prod_{\begin{subarray}{c}(i,j)\in{{\bm{\mu}}}\\\
\lambda^{\prime}_{j}=\mu^{\prime}_{j}\end{subarray}}\frac{1-q^{\lambda_{i}-j+1}q^{\lambda^{\prime}_{j}-i}}{1-q^{\mu_{i}+j-1}q^{-\mu^{\prime}_{j}-i}p}\prod_{\begin{subarray}{c}(i,j)\in{{\bm{\mu}}}\\\
\lambda^{\prime}_{j}\neq\mu^{\prime}_{j}\end{subarray}}\frac{1-q^{\lambda_{i}+j}q^{-\lambda^{\prime}_{j}-i}p}{1-q^{\mu_{i}-j}q^{\mu^{\prime}_{j}-i+1}}\\\
=\prod_{(i,j)\in{{\bm{\lambda}}}}(1-q^{\lambda_{i}-j}q^{\lambda^{\prime}_{j}-i+1})(1-q^{\lambda_{i}+j}q^{-\lambda^{\prime}_{j}-i}p)\prod_{(i,j)\in{{\bm{\mu}}}}\frac{1}{(1-q^{\mu_{i}-j+1}q^{\mu^{\prime}_{j}-i})(1-q^{\mu_{i}+j-1}q^{-\mu^{\prime}_{j}-i}p)}\\\
\times\prod_{(i,j)\in{{\bm{\lambda}}}/{{\bm{\mu}}}}\frac{1}{1-q^{\mu_{i}+j-1}q^{-\mu^{\prime}_{j}-i}p}\prod_{(i,j)\in{{\bm{\lambda}}}/{{\bm{\mu}}}}\frac{1}{1-q^{\lambda_{i}+j}q^{-\lambda^{\prime}_{j}-i}p}.$

We now simplify the last line. Note that since $\bm{\mu}\prec\bm{\lambda}$ and
$(i,j)\in\bm{\lambda}/\bm{\mu}$, we have $\lambda^{\prime}_{j}=i$ and
$\mu^{\prime}_{j}=i-1$. Hence, we get

$\displaystyle\prod_{i}\left[\prod_{j=\mu_{i}+1}^{\lambda_{i}}\frac{1}{1-q^{\mu_{i}+j}q^{-2i}p}\right]\cdot\left[\prod_{j=\mu_{i}+1}^{\lambda_{i}}\frac{1}{1-q^{\lambda_{i}+j}q^{-2i}p}\right]$
$\displaystyle=\prod_{i}\left[\prod_{j=1}^{2(\lambda_{i}-\mu_{i})}\frac{1}{1-q^{2\mu_{i}+j}q^{-2i}p}\right]$
$\displaystyle=\prod_{i}\frac{(q^{2(\lambda_{i}-i)+1}p;q)_{\infty}}{(q^{2(\mu_{i}-i)+1}p;q)_{\infty}}.$

Let us now rewrite the branching rule for Koornwinder polynomials at $q=t$:

######  Lemma A.1.

Consider Koornwinder polynomials with $t=q$, $t_{0}=r\cdot t^{-n-1}$. Define
the quasi-branching coefficient through:

$K^{(n+1)}_{\bm{\lambda}}(x_{1},\dots,x_{n},r\cdot q^{-n-1};\,q,q;\,r\cdot
q^{-n-1},t_{1},t_{2},t_{3})\\\
=\sum_{\bm{\mu}\prec\bm{\lambda}}\tilde{\psi}_{{\bm{\lambda}}/{\bm{\mu}}}\cdot
K^{(n)}_{\bm{\mu}}(x_{1},\dots,x_{n};\,q,q;\,r\cdot
q^{-n},t_{1},t_{2},t_{3}).$

Then we have:

$\tilde{\psi}_{{\bm{\lambda}}/{\bm{\mu}}}=\frac{g_{n+1}({\bm{\lambda}})}{g_{n}({\bm{\mu}})}\prod_{i=1}^{n+1}\left[q^{-\lambda_{i}}\frac{1-q^{2(\lambda_{i}-i)+n}rt_{1}t_{2}t_{3}}{1-q^{-2i+n}rt_{1}t_{2}t_{3}}\right].$

where $g_{n}({{\bm{\mu}}})$ is

$(r)^{-|{{\bm{\mu}}}|}C^{0}_{{\bm{\mu}}}(rt_{1}t_{2}t_{3}/q^{2})C^{0}_{{\bm{\mu}}}(rt_{1}/q)C^{0}_{{\bm{\mu}}}(rt_{2}/q)C^{0}_{{\bm{\mu}}}(rt_{3}/q)\cdot
q^{(n+1)|{{\bm{\mu}}}|}\prod_{i\geqslant
1}\frac{(q^{2(\mu_{i}-i)+n}rt_{1}t_{2}t_{3};q)_{\infty}}{(q^{-2i+n}rt_{1}t_{2}t_{3};q)_{\infty}}.$
(337)

###### Proof.

We denote $q^{2n}t_{0}t_{1}t_{2}t_{3}=p$. Comparing with Theorem 3.16, we have

$\tilde{\psi}_{{\bm{\lambda}}/{\bm{\mu}}}=\psi^{(i)}_{{\bm{\lambda}}/{\bm{\mu}}}\left(q^{n+1};\,q,q,q^{-1}\sqrt{p}\right)\cdot\frac{k^{0}_{{\bm{\lambda}}}(q,q,q^{n+1};\,r\cdot
q^{-n-1}:t_{1},t_{2},t_{3})}{k^{0}_{{\bm{\mu}}}(q,q,q^{n};\,r\cdot
q^{-n}:t_{1},t_{2},t_{3})}.$

Using the above simplifications for $\psi^{(i)}_{{\bm{\lambda}}/{\bm{\mu}}}$,
the last expression is

$\frac{q^{n|{{\bm{\lambda}}}|-n({{\bm{\lambda}}})}}{q^{n|{{\bm{\mu}}}|-n({{\bm{\mu}}})}}\frac{C^{0}_{{\bm{\lambda}}}(pq^{-1-n})C^{0}_{{\bm{\mu}}}(q^{n})}{C^{0}_{{\bm{\mu}}}(pq^{-1-n})C^{0}_{{\bm{\lambda}}}(q^{n+1})}\frac{\prod\limits_{(i,j)\in{{\bm{\lambda}}}}(1-q^{\lambda_{i}-j}q^{\lambda^{\prime}_{j}-i+1})(1-q^{\lambda_{i}+j}q^{-\lambda^{\prime}_{j}-i}p)}{\prod\limits_{(i,j)\in{{\bm{\mu}}}}(1-q^{\mu_{i}-j+1}q^{\mu^{\prime}_{j}-i})(1-q^{\mu_{i}+j-1}q^{-\mu^{\prime}_{j}-i}p)}\prod_{i}\frac{(q^{2(\lambda_{i}-i)+1}p;q)_{\infty}}{(q^{2(\mu_{i}-i)+1}p;q)_{\infty}}\\\
\times\frac{(r/q)^{-|{{\bm{\lambda}}}|}q^{n({{\bm{\lambda}}})}}{(r/q)^{-|{{\bm{\mu}}}|}q^{n({{\bm{\mu}}})}}\frac{C^{0}_{{\bm{\lambda}}}(q^{n+1})C^{0}_{{\bm{\lambda}}}(rt_{1}/q)C^{0}_{{\bm{\lambda}}}(rt_{2}/q)C^{0}_{{\bm{\lambda}}}(rt_{3}/q)}{C^{0}_{{\bm{\mu}}}(q^{n})C^{0}_{{\bm{\mu}}}(rt_{1}/q)C^{0}_{{\bm{\mu}}}(rt_{2}/q)C^{0}_{{\bm{\mu}}}(rt_{3}/q)}\frac{\prod\limits_{(i,j)\in{{\bm{\mu}}}}(1-q^{\mu_{i}-j+1}q^{\mu^{\prime}_{j}-i})(1-q^{\mu_{i}+j-1}q^{-\mu^{\prime}_{j}-i}p)}{\prod\limits_{(i,j)\in{{\bm{\lambda}}}}(1-q^{\lambda_{i}-j}q^{\lambda^{\prime}_{j}-i+1})(1-q^{\lambda_{i}+j}q^{-\lambda^{\prime}_{j}-i}p)}\\\
=\frac{q^{(n+1)|{{\bm{\lambda}}}|}}{q^{(n+1)|{{\bm{\mu}}}|}}\frac{C^{0}_{{\bm{\lambda}}}(pq^{-1-n})}{C^{0}_{{\bm{\mu}}}(pq^{-1-n})}\frac{(r)^{-|{{\bm{\lambda}}}|}}{(r)^{-|{{\bm{\mu}}}|}}\frac{C^{0}_{{\bm{\lambda}}}(rt_{1}/q)C^{0}_{{\bm{\lambda}}}(rt_{2}/q)C^{0}_{{\bm{\lambda}}}(rt_{3}/q)}{C^{0}_{{\bm{\mu}}}(rt_{1}/q)C^{0}_{{\bm{\mu}}}(rt_{2}/q)C^{0}_{{\bm{\mu}}}(rt_{3}/q)}\prod_{i}\frac{(q^{2(\lambda_{i}-i)+1}p;q)_{\infty}}{(q^{2(\mu_{i}-i)+1}p;q)_{\infty}},$

with all $C^{0}$ being evaluated at $(\cdot;\,q,q)$. It remains to note that

$p=q^{n-1}rt_{1}t_{2}t_{3}.\qed$

Here is a restatement of Lemma A.1: using the notation (337) we define a
symmetric function $\tilde{K}_{{\bm{\lambda}}}(x_{1},x_{2},\dots,x_{n})$
through

$\tilde{K}_{{\bm{\mu}}}(x_{1},x_{2},\dots,x_{n}):=\frac{K^{(n)}_{{\bm{\mu}}}(x_{1},\dots,x_{n};\,q,q;\,r\cdot
q^{-n},t_{1},t_{2},t_{3})}{g_{n}({{\bm{\mu}}})}.$

(It still depends on $q,r,t_{1},t_{2},t_{3}$, but we are hiding this
dependence.) Then we have

$\tilde{K}_{{\bm{\lambda}}}(x_{1},x_{2},\dots,x_{n},rq^{-n-1})=\sum_{{{\bm{\mu}}}\prec{{\bm{\lambda}}}}\left[\prod_{i=1}^{n+1}\left[q^{-\lambda_{i}}\frac{1-q^{2(\lambda_{i}-i)+n}rt_{1}t_{2}t_{3}}{1-q^{-2i+n}rt_{1}t_{2}t_{3}}\right]\right]\tilde{K}_{{\bm{\mu}}}(x_{1},x_{2},\dots,x_{n}).$

Let us now match to the notations of Theorem 3.19. For that we transform the
weight arising in branching of $n$ (rather than $n+1$) variable polynomial

$q^{-\lambda_{i}}\frac{1-q^{2(\lambda_{i}-i)+n-1}rt_{1}t_{2}t_{3}}{1-q^{-2i+n-1}rt_{1}t_{2}t_{3}}=\frac{\sqrt{rt_{1}t_{2}t_{3}}\,q^{\lambda_{i}+\frac{n-1}{2}-i}-\frac{1}{\sqrt{rt_{1}t_{2}t_{3}}\,q^{\lambda_{i}+(n-1)/2-i}}}{\sqrt{rt_{1}t_{2}t_{3}}\,q^{(n-1)/2-i}-\frac{1}{\sqrt{rt_{1}t_{2}t_{3}}\,q^{(n-1)/2-i}}}.$
(338)

Denoting $x_{i}=\lambda_{i}+\frac{n+1}{2}-i$ and
$\sigma=q^{-1}\sqrt{rt_{1}t_{2}t_{3}}$ and removing the
$\lambda_{i}$–independent normalization, we get the weight

$\sigma q^{x_{i}}-\frac{1}{\sigma q^{x_{i}}}.$

We are now ready to finish the proof of Theorem 3.19. We compute

$\sum_{\emptyset={{\bm{\lambda}}}^{(0)}\prec{{\bm{\lambda}}}^{(1)}\prec\dots\prec{{\bm{\lambda}}}^{(n)}={{\bm{\lambda}}}}\prod_{k=1}^{n}\prod_{i=1}^{k}\left(\sigma
q^{\lambda^{(k)}_{i}+\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\lambda^{(k)}_{i}+\frac{k+1}{2}-i}}\right),$ (339)

which differs from the left-hand side of (77) by allowing $k=n$. Using Lemma
A.1 recurrently for $n-1$, $n-2$,…, adopting the notation
$\sigma=q^{-1}\sqrt{rt_{1}t_{2}t_{3}}$, and transforming the weights as in
(338), the sum (339) evaluates to

$\left[\prod_{k=1}^{n}\prod_{i=1}^{k}\left(\sigma
q^{\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\frac{k+1}{2}-i}}\right)\right]\frac{K^{(n)}_{{\bm{\mu}}}(rq^{-1},rq^{-2},\dots,rq^{-n};\,q,q;\,r\cdot
q^{-n},t_{1},t_{2},t_{3})}{g_{n}({{\bm{\lambda}}})}.$ (340)

For the principal specialization of Koornwinder polynomials we use another
formula from [98]:

$K^{(n+1)}_{{\bm{\lambda}}}(t_{0},t_{0}t,\dots,t_{0}t^{n};\,q,t;\,t_{0},t_{1},t_{2},t_{3})=(t_{0}t^{n})^{-|{{\bm{\lambda}}}|}t^{n({{\bm{\lambda}}})}\frac{C^{0}_{{\bm{\lambda}}}(t^{n+1},t^{n}t_{0}t_{1},t^{n}t_{0}t_{2},t^{n}t_{0}t_{3};\,q,t)}{C_{{\bm{\lambda}}}^{-}(t;\,q,t)C_{{\bm{\lambda}}}^{+}(t^{2n}t_{0}t_{1}t_{2}t_{3}/q;\,q,t)},$

where $C^{0}_{{\bm{\lambda}}}$ is as in Theorem 3.16 and

$\displaystyle C^{+}_{{\bm{\lambda}}}(x;q,t)$
$\displaystyle:=\prod_{(i,j)\in{{\bm{\lambda}}}}(1-q^{\lambda_{i}+j-1}t^{2-\lambda^{\prime}_{j}-i}x)=\prod_{1\leqslant
i\leqslant
l}\frac{(q^{\lambda_{i}}t^{2-l-i}x;q)}{(q^{2\lambda_{i}}t^{2-2i}x;q)}\prod_{1\leqslant
i<j\leqslant
l}\frac{(q^{\lambda_{i}+\lambda_{j}}t^{3-i-j}x;q)}{(q^{\lambda_{i}+\lambda_{j}}t^{2-i-j}x;q)},$
$\displaystyle C^{-}_{{\bm{\lambda}}}(x;q,t)$
$\displaystyle:=\prod_{(i,j)\in{{\bm{\lambda}}}}(1-q^{\lambda_{i}-j}t^{\lambda^{\prime}_{j}-i}x)=\prod_{1\leqslant
i\leqslant l}\frac{(x;q)}{(q^{\lambda_{i}}t^{l-i}x;q)}\prod_{1\leqslant
i<j\leqslant
l}\frac{(q^{\lambda_{i}-\lambda_{j}}t^{j-i}x;q)}{(q^{\lambda_{i}-\lambda_{j}}t^{j-i-1}x;q)}.$

Hence, (340) becomes

$\left[\prod_{k=1}^{n}\prod_{i=1}^{k}\left(\sigma
q^{\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\frac{k+1}{2}-i}}\right)\right](r/q)^{-|{{\bm{\lambda}}}|}q^{n({{\bm{\lambda}}})}\frac{C^{0}_{{\bm{\lambda}}}(q^{n})C^{0}_{{\bm{\lambda}}}(rt_{1}/q)C^{0}_{{\bm{\lambda}}}(rt_{2}/q)C^{0}_{{\bm{\lambda}}}(rt_{3}/q)}{C_{{\bm{\lambda}}}^{-}(q;\,q,q)C_{{\bm{\lambda}}}^{+}(\sigma
q^{n-1};\,q,q)}\\\
\times(r)^{|{{\bm{\lambda}}}|}\frac{1}{C^{0}_{{\bm{\lambda}}}(\sigma^{2})C^{0}_{{\bm{\lambda}}}(rt_{1}/q)C^{0}_{{\bm{\lambda}}}(rt_{2}/q)C^{0}_{{\bm{\lambda}}}(rt_{3}/q)\cdot
q^{(n+1)|{{\bm{\lambda}}}|}}\prod_{i\geqslant
1}\frac{(q^{-2i+n+2}\sigma^{2};q)_{\infty}}{(q^{2(\lambda_{i}-i)+n+2}\sigma^{2};q)_{\infty}}\\\
=\left[\prod_{k=1}^{n}\prod_{i=1}^{k}\left(\sigma
q^{\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\frac{k+1}{2}-i}}\right)\right]q^{-n|{{\bm{\lambda}}}|}q^{n({{\bm{\lambda}}})}\frac{C^{0}_{{\bm{\lambda}}}(q^{n})}{C_{{\bm{\lambda}}}^{-}(q;\,q,q)C_{{\bm{\lambda}}}^{+}(\sigma^{2}q^{n-1};\,q,q)}\\\
\times\frac{1}{C^{0}_{{\bm{\lambda}}}(\sigma^{2})}\cdot\prod_{i\geqslant
1}\frac{(q^{-2i+n+1}\sigma^{2};q)_{\infty}}{(q^{2(\lambda_{i}-i)+n+1}\sigma^{2};q)_{\infty}}\\\
=\left[\prod_{k=1}^{n}\prod_{i=1}^{k}\left(\sigma
q^{\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\frac{k+1}{2}-i}}\right)\right]q^{-n|{{\bm{\lambda}}}|+n({{\bm{\lambda}}})}\prod_{1\leqslant
i<j\leqslant
n}\bigl{(}1-q^{\lambda_{i}-\lambda_{j}}q^{j-i}\bigr{)}\bigl{(}1-\sigma^{2}q^{\lambda_{i}+\lambda_{j}+n-i-j+1}\bigr{)}\\\
\times\prod_{i=1}^{n}\left[\frac{(q^{\lambda_{i}+n-i+1};q)_{\infty}}{(q;q)_{\infty}}\frac{(\sigma^{2}q^{-2i+n+2};q)_{\infty}}{(\sigma^{2}q^{2(\lambda_{i}-i)+n+2};q)_{\infty}}\frac{(\sigma^{2}q^{2(\lambda_{i}-i)+n+1};q)_{\infty}}{(\sigma^{2}q^{\lambda_{i}-i+1};q)_{\infty}}\cdot\frac{(q^{1-i}q^{n};q)_{\lambda_{i}}}{(\sigma^{2}q^{1-i};q)_{\lambda_{i}}}\right].$

It remains to transform the last formula. First, let us simplify the factor in
$\prod_{i=1}^{n}$:

$\frac{(q^{\lambda_{i}+n-i+1};q)_{\infty}}{(q;q)_{\infty}}\frac{(\sigma^{2}q^{-2i+n+2};q)_{\infty}}{(\sigma^{2}q^{2(\lambda_{i}-i)+n+2};q)_{\infty}}\frac{(\sigma^{2}q^{2(\lambda_{i}-i)+n+1};q)_{\infty}}{(\sigma^{2}q^{\lambda_{i}-i+1};q)_{\infty}}\cdot\frac{(q^{1-i}q^{n};q)_{\lambda_{i}}}{(\sigma^{2}q^{1-i};q)_{\lambda_{i}}}\\\
=\frac{(q^{n-i+1};q)_{\infty}}{(q;q)_{\infty}}\frac{(\sigma^{2}q^{-2i+n+2};q)_{\infty}}{(\sigma^{2}q^{1-i};q)_{\infty}}\bigl{(}1-\sigma^{2}q^{2(\lambda_{i}-i)+n+1}\bigr{)}=\frac{1-\sigma^{2}q^{2(\lambda_{i}-i)+n+1}}{(q;q)_{n-i}\,(\sigma^{2}q^{1-i};q)_{n-i+1}}$
(341)

Next, let us modify the $\prod_{k=1}^{n}\prod_{i=1}^{k}$ factor:

$\left[\prod_{k=1}^{n}\prod_{i=1}^{k}\left(\sigma
q^{\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\frac{k+1}{2}-i}}\right)\right]=(-\sigma)^{-n(n+1)/2}\prod_{1\leqslant
i\leqslant k\leqslant n}\left(1-\sigma^{2}q^{k+1-2i}\right)\\\
=(-\sigma)^{-n(n+1)/2}\prod_{i=1}^{n}(\sigma^{2}q^{1-i};q)_{n-i+1}.$ (342)

We observe a nice cancellation between $q$-Pochammer symbols involving
$\sigma$ in the last two formulas. Finally, we rewrite the remaining factors:

$q^{-n|{{\bm{\lambda}}}|+n({{\bm{\lambda}}})}\prod_{1\leqslant i<j\leqslant
n}\bigl{(}1-q^{\lambda_{i}-\lambda_{j}}q^{j-i}\bigr{)}\bigl{(}1-\sigma^{2}q^{\lambda_{i}+\lambda_{j}+n-i-j+1}\bigr{)}\\\
=q^{\sum_{i=1}^{n}(i-1-n)\lambda_{i}}\prod_{1\leqslant i<j\leqslant
n}\bigl{(}1-q^{\lambda_{i}-\lambda_{j}+j-i}-\sigma^{2}q^{\lambda_{i}+\lambda_{j}+n-i-j+1}+\sigma^{2}q^{2\lambda_{i}+n-2i+1}\bigr{)}\\\
=q^{\sum_{i=1}^{n}(i-1-n)\lambda_{i}}\prod_{1\leqslant i<j\leqslant
n}q^{\lambda_{i}-i+\frac{n+1}{2}}\left(q^{i-\lambda_{i}-\frac{n+1}{2}}+\sigma^{2}q^{\lambda_{i}-i+\frac{n+1}{2}}-q^{j-\lambda_{j}-\frac{n+1}{2}}-\sigma^{2}q^{\lambda_{j}-j+\frac{n+1}{2}}\right)\\\
=\left[\prod_{i=1}^{n}q^{(\frac{n+1}{2}-i)(n-i)}\right]q^{-|{{\bm{\lambda}}}|}\prod_{1\leqslant
i<j\leqslant
n}\left(q^{i-\lambda_{i}-\frac{n+1}{2}}+\sigma^{2}q^{\lambda_{i}-i+\frac{n+1}{2}}-q^{j-\lambda_{j}-\frac{n+1}{2}}-\sigma^{2}q^{\lambda_{j}-j+\frac{n+1}{2}}\right)$
(343)

It remains to multiply (341), (342), (343) and divide by
$\prod\limits_{i=1}^{k}\left(\sigma
q^{\lambda_{i}+\frac{k+1}{2}-i}-\frac{1}{\sigma
q^{\lambda_{i}+\frac{k+1}{2}-i}}\right)$ to get the result. ∎

[Acknowledgments] The authors thank A. Aggarwal, E. Dimitrov, and N. Nekrasov
for helpful discussions. We are grateful to the Galileo Galilei Institute for
Theoretical Physics, where some of these discussions took place.We thank
anonymous referees for proofreading the manuscript and useful suggestions.

The work of V.G. was partially supported by NSF Grants DMS-1664619,
DMS-1949820, DMS-2152588, and by the Office of the Vice Chancellor for
Research and Graduate Education at the University of Wisconsin–Madison with
funding from the Wisconsin Alumni Research Foundation. The research of J.H. is
supported by the Simons Foundation as a Junior Fellow at the Simons Society of
Fellows, and NSF Grants DMS-2054835 and DMS-2331096.

## References

  * [1] [author] Adhikari, ArkaA. and Huang, JiaoyangJ. (2020). Dyson Brownian motion for general $\beta$ and potential at the edge. Probability Theory and Related Fields 1–58.
  * [2] [author] Ahn, AndrewA. (2020). Global universality of Macdonald plane partitions. In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 56 1641–1705. Institut Henri Poincaré.
  * [3] [author] Ambjørn, JanJ. and Makeenko, Yu MY. M. (1990). Properties of loop equations for the Hermitian matrix model and for two-dimensional quantum gravity. Modern Physics Letters A 5 1753–1763.
  * [4] [author] Anderson, Greg W.G. W., Guionnet, AliceA. and Zeitouni, OferO. (2010). An introduction to random matrices. Cambridge Studies in Advanced Mathematics 118\. Cambridge University Press, Cambridge. 2760897
  * [5] [author] Atkinson, KendallK., Han, WeiminW. and Stewart, David ED. E. (2011). Numerical solution of ordinary differential equations. John Wiley & Sons.
  * [6] [author] Behrndt, JussiJ., Hassi, SeppoS. and De Snoo, HenkH. (2020). Boundary value problems, Weyl functions, and differential operators. Springer Nature.
  * [7] [author] Bekerman, F.F., Figalli, A.A. and Guionnet, A.A. (2015). Transport maps for $\beta$-matrix models and universality. Comm. Math. Phys. 338 589–619. 10.1007/s00220-015-2384-y 3351052
  * [8] [author] Bender, MartinM. (2008). Global fluctuations in general $\beta$ Dyson’s Brownian motion. Stochastic Process. Appl. 118 1022–1042. 10.1016/j.spa.2007.07.010 2418256
  * [9] [author] Berestycki, NathanaëlN. and Powell, EllenE. (2021). Lecture notes on the Gaussian Free Field. Preprint.
  * [10] [author] Betea, DanD. (2012). Elliptic Combinatorics and Markov Processes. Dissertation (Ph.D.), California Institute of Technology.
  * [11] [author] Biane, PhilippeP. (1997). On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46 705–718. 10.1512/iumj.1997.46.1467 1488333
  * [12] [author] Borodin, AlexeiA. (2014). CLT for spectra of submatrices of Wigner random matrices. Moscow Mathematical Journal 14 29-38.
  * [13] [author] Borodin, AlexeiA. (2014). CLT for spectra of submatrices of Wigner random matrices II. Stochastic evolution. Random Matrices, MSRI Publications 65 57-69.
  * [14] [author] Borodin, AlexeiA. and Corwin, IvanI. (2014). Macdonald processes. Probability Theory and Related Fields 158 225–400.
  * [15] [author] Borodin, AlexeiA. and Ferrari, Patrik LP. L. (2014). Anisotropic growth of random surfaces in 2+1 dimensions. Communications in Mathematical Physics 325 603–684.
  * [16] [author] Borodin, AlexeiA. and Gorin, VadimV. (2015). General $\beta$-Jacobi Corners Process and the Gaussian Free Field. Communications on Pure and Applied Mathematics 68 1774–1844.
  * [17] [author] Borodin, AlexeiA. and Gorin, VadimV. (2016). Lectures on integrable probability. Probability and statistical physics in St. Petersburg 91\.
  * [18] [author] Borodin, AlexeiA., Gorin, VadimV. and Guionnet, AliceA. (2017). Gaussian asymptotics of discrete $\beta$-ensembles. Publ. Math. Inst. Hautes Études Sci. 125 1–78. 10.1007/s10240-016-0085-5 3668648
  * [19] [author] Borodin, AlexeiA., Gorin, VadimV. and Rains, Eric ME. M. (2010). q-Distributions on boxed plane partitions. Selecta Mathematica 16 731–789.
  * [20] [author] Borodin, AlexeiA. and Kuan, JeffreyJ. (2010). Random surface growth with a wall and Plancherel measures for O(infinity). Communications on Pure and Applied Mathematics 63 831–894.
  * [21] [author] Borodin, AlexeiA. and Petrov, LeonidL. (2014). Integrable probability: from representation theory to Macdonald processes. Probab. Surv. 11 1–58. 10.1214/13-PS225 3178541
  * [22] [author] Borot, GaëtanG., Eynard, BertrandB. and Orantin, NicolasN. (2015). Abstract loop equations, topological recursion and new applications. Commun. Num. Theor. Phys. 09 51–187. 10.4310/CNTP.2015.v9.n1.a2
  * [23] [author] Borot, GaëtanG. and Guionnet, AliceA. (2013). Asymptotic expansion of $\beta$ matrix models in the one-cut regime. Comm. Math. Phys. 317 447–483. 10.1007/s00220-012-1619-4 3010191
  * [24] [author] Borot, GaëtanG. and Guionnet, Alice.A. (2013). Asymptotic Expansion of $\beta$ Matrix Models in the multi-cut Regime. preprint, arXiv:1303.1045.
  * [25] [author] Bourgade, PaulP., Mody, KrishnanK. and Pain, MichelM. (2021). Optimal local law and central limit theorem for $\beta$-ensembles. Preprint, arXiv:2103.06841.
  * [26] [author] Breuer, JonathanJ. and Duits, MauriceM. (2017). Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients. J. Amer. Math. Soc. 30 27–66. 10.1090/jams/854 3556288
  * [27] [author] Brézin, EdouardE., Itzykson, ClaudeC., Parisi, GiorgioG. and Zuber, Jean-BernardJ.-B. (1978). Planar diagrams. Communications in Mathematical Physics 59 35–51.
  * [28] [author] Bufetov, AlexeyA. and Gorin, VadimV. (2015). Representations of classical Lie groups and quantized free convolution. Geom. Funct. Anal. 25 763–814. 10.1007/s00039-015-0323-x 3361772
  * [29] [author] Bufetov, AlexeyA. and Gorin, VadimV. (2018). Fluctuations of particle systems determined by Schur generating functions. Advances in Mathematics 338 702–781.
  * [30] [author] Bufetov, AlexeyA. and Gorin, VadimV. (2019). Fourier transform on high-dimensional unitary groups with applications to random tilings. Duke Mathematical Journal 168 2559–2649.
  * [31] [author] Bufetov, AlexeyA. and Knizel, AlisaA. (2018). Asymptotics of random domino tilings of rectangular Aztec diamonds. In Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 54 1250–1290. Institut Henri Poincaré.
  * [32] [author] Burkholder, Donald LD. L. (1973). Distribution function inequalities for martingales. the Annals of Probability 1 19–42.
  * [33] [author] Cabanal-Duvillard, ThierryT. (2001). Fluctuations de la loi empirique de grandes matrices aléatoires. In Annales de l’IHP Probabilités et statistiques 37 373–402.
  * [34] [author] Cerf, RaphaëlR. and Kenyon, RichardR. (2001). The low-temperature expansion of the Wulff crystal in the 3D Ising model. Communications in Mathematical Physics 222 147–179.
  * [35] [author] Chekhov, LeonidL. and Eynard, BertrandB. (2006). Matrix eigenvalue model: Feynman graph technique for all genera. Journal of High Energy Physics 2006 026\.
  * [36] [author] Chekhov, Leonid OlegovichL. O., Eynard, BertrandB. and Marchal, OlivierO. (2011). Topological expansion of the $\beta$-ensemble model and quantum algebraic geometry in the sectorwise approach. Theoretical and Mathematical Physics 166 141–185.
  * [37] [author] Cohn, HenryH., Kenyon, RichardR. and Propp, JamesJ. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297–346. 10.1090/S0894-0347-00-00355-6 1815214
  * [38] [author] Cohn, HenryH., Larsen, MichaelM. and Propp, JamesJ. (1998). The shape of a typical boxed plane partition. New York J. Math. 4 137–165. 1641839
  * [39] [author] Dimitrov, EvgeniE. and Knizel, AlisaA. (2019). Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions. Journal of Functional Analysis 276 3067–3169.
  * [40] [author] Dimitrov, EvgeniE. and Knizel, AlisaA. (2019). Asymptotics of discrete $\beta$-corners processes via two-level discrete loop equations. Preprint, arXiv:1905.02338.
  * [41] [author] Dimitrov, EvgeniE. and Knizel, AlisaA. (2021). Multi-level loop equations for $\beta$-corners processes. Preprint, arXiv:2108.07710.
  * [42] [author] Dolega, MaciejM. and Féray, ValentinV. (2016). Gaussian fluctuations of Young diagrams and structure constants of Jack characters. Duke Mathematical Journal 165 1193–1282.
  * [43] [author] Duits, MauriceM. (2018). On global fluctuations for non-colliding processes. The Annals of Probability 46 1279–1350.
  * [44] [author] Duits, MauriceM. and Liu, W.W. Limit shape and global fluctuations for $q$-orthogonal polynomial ensembles. In preparation.
  * [45] [author] Duse, EricE., Johansson, KurtK. and Metcalfe, AnthonyA. (2016). The Cusp-Airy Process. Electronic Journal of Probability 21 paper 57.
  * [46] [author] Duse, EricE. and Metcalfe, AnthonyA. (2015). Asymptotic geometry of discrete interlaced patterns: Part I. International Journal of Mathematics 26 1550093\.
  * [47] [author] Duse, EricE. and Metcalfe, AnthonyA. (2020). Asymptotic geometry of discrete interlaced patterns: Part II. Annales de l’Institut Fourier 70 375-436.
  * [48] [author] Dyson, Freeman J.F. J. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Mathematical Phys. 3 1191–1198. 0148397
  * [49] [author] Erdős, LászlóL., Schlein, BenjaminB. and Yau, Horng-TzerH.-T. (2011). Universality of random matrices and local relaxation flow. Invent. Math. 185 75–119. 10.1007/s00222-010-0302-7 2810797
  * [50] [author] Erdős, LászlóL., Schlein, BenjaminB., Yau, Horng-TzerH.-T. and Yin, JunJ. (2012). The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 1–46. 10.1214/10-AIHP388 2919197
  * [51] [author] Erdős, LászlóL. and Yau, Horng-TzerH.-T. (2015). Gap universality of generalized Wigner and $\beta$-ensembles. J. Eur. Math. Soc. (JEMS) 17 1927–2036. 10.4171/JEMS/548 3372074
  * [52] [author] Eynard, BertrandB. (2005). Topological expansion for the 1-hermitian matrix model correlation functions. Journal of High Energy Physics 2004 031\.
  * [53] [author] Eynard, BertrandB. (2016). Counting surfaces. Progress in Mathematical Physics 70 414\.
  * [54] [author] Eynard, BertrandB. and Orantin, NicolasN. (2007). Invariants of algebraic curves and topological expansion. Commun. Num. Theor. Phys. 1 347–452. 10.4310/CNTP.2007.v1.n2.a4
  * [55] [author] Forrester, Peter J.P. J. and Rains, Eric M.E. M. (2005). Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probability Theory and Related Fields 131 1–61.
  * [56] [author] Gelfand, I. M.I. M. and Naimark, M. A.M. A. (1950 (German transl.:Academie-Verlag, Berlin, 1957.)). Unitary representations of the classical groups. Trudy Mat. Inst. Steklov, Leningrad, Moscow (In Russian).
  * [57] [author] Gorin, VadimV. (2021). Lectures on random lozenge tilings. Cambridge University Press.
  * [58] [author] Gorin, VadimV. and Panova, GretaG. (2015). Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Annals of Probability 43 3052–3132.
  * [59] [author] Gorin, VadimV. and Petrov, LeonidL. (2019). Universality of local statistics for noncolliding random walks. The Annals of Probability 47 2686–2753.
  * [60] [author] Gorin, VadimV. and Shkolnikov, MykhayloM. (2015). Multilevel Dyson Brownian motions via Jack polynomials. Probab. Theory Related Fields 163 413–463. 10.1007/s00440-014-0596-2 3418747
  * [61] [author] Gorin, VadimV. and Shkolnikov, MykhayloM. (2015). Limits of Multilevel TASEP and similar processes. Limits of Multilevel TASEP and similar processes 51 18-27.
  * [62] [author] Guionnet, AliceA. (2002). Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. In Annales de l’Institut Henri Poincare (B) Probability and Statistics 38 341–384. Elsevier.
  * [63] [author] Guionnet, AliceA. (2019). Asymptotics of random matrices and related models: the uses of Dyson-Schwinger equations 130\. American Mathematical Soc.
  * [64] [author] Hall, PeterP. and Heyde, Christopher CC. C. (2014). Martingale limit theory and its application. Academic press.
  * [65] [author] Hitczenko, PawelP. (1990). Best constants in martingale version of Rosenthal’s inequality. The Annals of Probability 18 1656–1668.
  * [66] [author] Holte, John MJ. M. (2009). Discrete Gronwall lemma and applications. In MAA-NCS meeting at the University of North Dakota 24 1–7.
  * [67] [author] Huang, JiaoyangJ. (2017). $\beta$-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems. International Mathematics Research Notices.
  * [68] [author] Huang, JiaoyangJ. (2020). Height Fluctuations of Random Lozenge Tilings Through Nonintersecting Random Walks. Preprint, arXiv:2011.01751.
  * [69] [author] Huang, JiaoyangJ. (2020). Edge Universality for Nonintersecting Brownian Bridges. Preprint, arXiv:2011.01752.
  * [70] [author] Huang, JiaoyangJ. (2021). Law of large numbers and central limit theorems through Jack generating functions. Advances in Mathematics 380 107545\.
  * [71] [author] Huang, JiaoyangJ. and Landon, BenjaminB. (2019). Rigidity and a mesoscopic central limit theorem for Dyson Brownian motion for general $\beta$ and potentials. Probab. Theory Related Fields 175 209–253. 10.1007/s00440-018-0889-y 4009708
  * [72] [author] Israelsson, StefanS. (2001). Asymptotic fluctuations of a particle system with singular interaction. Stochastic Process. Appl. 93 25–56. 10.1016/S0304-4149(00)00100-9 1819483
  * [73] [author] Johansson, KurtK. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151–204. 10.1215/S0012-7094-98-09108-6 1487983
  * [74] [author] Johansson, KurtK. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476. 10.1007/s002200050027 1737991
  * [75] [author] Johansson, KurtK. (2002). Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 225–280. 10.1007/s004400100187 1900323
  * [76] [author] Kenyon, RichardR. and Okounkov, AndreiA. (2007). Limit shapes and the complex Burgers equation. Acta Math. 199 263–302. 10.1007/s11511-007-0021-0 2358053
  * [77] [author] Kerov, Sergei Vasil?evichS. V. and Tsilevich, NSN. (2003). Asymptotic representation theory of the symmetric group and its applications in analysis 219\. American Mathematical Society Providence, RI.
  * [78] [author] König, WolfgangW., O’Connell, NeilN. and Roch, SébastienS. (2002). Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles. Electronic Journal of Probability 7\.
  * [79] [author] Kriecherbauer, T.T. and Shcherbina, M.M. (2010). Fluctuations of eigenvalues of matrix models and their applications. preprint, arXiv:1003.6121.
  * [80] [author] Landon, BenjaminB., Sosoe, PhilippeP. and Yau, Horng-TzerH.-T. (2019). Fixed energy universality of Dyson Brownian motion. Adv. Math. 346 1137–1332. 10.1016/j.aim.2019.02.010 3914908
  * [81] [author] Landon, BenjaminB. and Yau, Horng-TzerH.-T. (2017). Convergence of local statistics of Dyson Brownian motion. Comm. Math. Phys. 355 949–1000. 10.1007/s00220-017-2955-1 3687212
  * [82] [author] Macdonald, Ian GrantI. G. (1998). Symmetric functions and Hall polynomials. Oxford university press.
  * [83] [author] Matveev, KonstantinK. (2019). Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture. Annals of Mathematics 189 277–316.
  * [84] [author] Migdal, AlexanderA. (1983). Loop Equations and $1/N$ Expansion. Physical Reports 102 199-290.
  * [85] [author] Moll, AlexanderA. (2015). Random partitions and the quantum Benjamin-Ono hierarchy. Preprint, arXiv:1508.03063.
  * [86] [author] Nekrasov, NikitaN. (2016). BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters. JHEP 03 181\. 10.1007/JHEP03(2016)181
  * [87] [author] Nekrasov, NikitaN. (2017). BPS/CFT correspondence IV: sigma models and defects in gauge theory. Preprint, arXiv:1711.11011.
  * [88] [author] Nekrasov, NikitaN. (2017). BPS/CFT correspondence V: BPZ and KZ equations from qq-characters. Preprint, arXiv:1711.11582.
  * [89] [author] Nekrasov, N.N. and Pestun, VV. (2011). Seiberg-Witten geometry of four dimensional $N=2$ quiver gauge theories. arXiv:1211.2240.
  * [90] [author] Nekrasov, N.N., Pestun, VV. and Shatashvili, S.S. (2013). Quantum geometry and quiver gauge theories. High Energy Physics - Theory 1-83.
  * [91] [author] Neretin, Yu. A.Y. A. (2003). Rayleigh triangles and non-matrix interpolation of matrix beta integrals. Sbornik: Mathematics 194 515–540.
  * [92] [author] O’Connel, NeilN. (2003). A path-transformation for random walks and the Robinson-Schensted correspondence. Transactions of the American Mathematical Society 355 3669-3697.
  * [93] [author] Okounkov, AndreiA. and Olshanski, GrigoriG. (1998). Asymptotics of Jack polynomials as the number of variables tends to infinity. International Mathematics Research Notices 13 641-682.
  * [94] [author] Okounkov, AndreiA. and Reshetikhin, NikolaiN. (2003). Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 581–603. 10.1090/S0894-0347-03-00425-9 1969205
  * [95] [author] Osekowski, ADAMA. (2012). A note on Burkholder-Rosenthal inequality. Bull. Pol. Acad. Sci. Math 60 177–185.
  * [96] [author] Petrov, LeonidL. (2014). Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes. Probab. Theory Related Fields 160 429–487. 10.1007/s00440-013-0532-x 3278913
  * [97] [author] Petrov, LeonidL. (2015). Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field. The Annals of Probability 43 1–43.
  * [98] [author] Rains, Eric ME. M. (2005). $BC_{n}$-symmetric polynomials. Transformation groups 10 63–132.
  * [99] [author] Sheffield, ScottS. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541. 10.1007/s00440-006-0050-1 2322706
  * [100] [author] Stanley, RichardR. (1989). Some combinatorial properties of Jack symmetric functions. Advances in Mathematics 77 76-115.
  * [101] [author] t Hooft, GerardusG. (1974). Magnetic monopoles in unified theories. Nucl. Phys. B 79 276–284.
  * [102] [author] Tutte, William ThomasW. T. (1963). A census of planar maps. Canadian Journal of Mathematics 15 249–271.
  * [103] [author] Unterberger, JérémieJ. (2018). Global fluctuations for 1D log-gas dynamics. Stochastic Processes and their Applications 128 4104–4153.
  * [104] [author] Vershik, Anatolii MoiseevichA. M. (1996). Statistical mechanics of combinatorial partitions, and their limit shapes. Functional Analysis and Its Applications 30 90–105.
  * [105] [author] Werner, WendelinW. and Powell, EllenE. (2020). Lecture notes on the Gaussian Free Field. Preprint, arXiv:2004.04720.