#  The best simultaneous approximation in linear 2-normed spaces

Mehmet Acikgoz University of Gaziantep Faculty of Science and Arts, Department
of Mathematics 27310 Gaziantep, TURKEY<EMAIL_ADDRESS>
(Date: May 07, 2012)

###### Abstract.

In this paper, we shall investigate and analyse a new study on the best
simultaneous approximation in the context of linear 2-normed spaces inspired
by Elumalai and his coworkers in [10]. The basis of this investigation is to
extend and refinement the definition of the classical aproximation, best
approximation and some related concepts to linear 2-normed spaces.

###### Key words and phrases:

2-normed space, best approximation, simultaneous best approximation, 2-Banach
space.

###### 2000 Mathematics Subject Classification:

Primary 46A15, Secondary 41A65.

##  1\. Introduction

The problem of best and simultaneous best approximation has been studied by
several mathematicians (for more informations, see [5], [6], [7], [8], [9],
[18], [19], [20], [26]). Most of these works have dealt with the existence,
uniqueness and characterization of best approximations in spaces of continuous
functions with values in Banach spaces. Recently, many works on approximation
has been done on 2-structures such as 2-normed spaces, generalized 2-normed
spaces (for details, see [4], [23]) and 2-Banach spaces (see [2], [3], [10],
[11], [12], [13], [14], [15]). Diaz and McLaughlin [7] and Dunham [9] have
considered simultaneously approximating of two real-valued continuously
functions $f_{1}$, $f_{2}$ defined on $[a,b]$, by elements of set $C[a,b].$
Several results of best simultaneous approximation in the context of linear
space were obtained by Goel et al. (for details, see [19], [20]). The subject
of approximation theory has attracted the attention of several mathematicians
during the last 130 years or so. This theory is now an extremely extensive
branch of mathematical analysis. It has many applications in many areas,
especially in engineering.

The concept of linear 2-normed spaces has been investigated by Gahler in 1965
[17] and given many important properties and examples for these spaces. After,
these spaces have been developed extensively in different subjects by other
researchers from many points of view and then the field has considerably
grown. Z. Lewandowska published some of papers on 2-normed sets and
generalized 2-normed spaces in 1999-2003 (see [23]). In [10], Elumalai and his
coworkers published a series of papers related this subject. They have
developed best approximation theory in the context of linear 2-normed spaces.
These are some works on characterization of 2-normed spaces, extension of
2-functionals and approximation in 2-normed spaces (see [1], [2]). Also, the
author has some works in $\varepsilon$-approximation theory [3] and Rezapour
has also such studies in [25]. The essential aim of this paper is to derive
new different definitions of approximation and obtain some results related to
these definitions. The essential results of the set of best simultaneous
approximation are given in the fourth section of this paper.

Throughout this paper, we first fix some notations. Let $X$ be a linear space
and $L\left\\{y\right\\}$ be the subspace of $Y$ generated by $y$. It is also
let $\left(X,\left\|.\right\|\right)$ and $(X,\|.,.\|)$ denote a normed space
and $2$-normed space with the corresponding norms, respectively. $\mathbb{R}$
denotes the set of real numbers, $\mathbb{N}$ denotes the set of natural
numbers and $\mathbb{C}$ denotes the set of complex numbers. Throughout this
work, $K$ is variously considered as an indeterminate, as a real number
$K\in\mathbb{R}$, or as a complex number $K\in\mathbb{C}$.

We now summarize our work in four section as follows:

In first section, we gave history of normed and $2$-normed spaces and
motivation of our work. In section 2 and 3, we specify definitions and
properties of normed and $2$-normed spaces, respectively. In section 4, we
gave suitable a definition for studying in linear $2$-normed spaces and so we
derived three lemma and two proposition by using our definition.

Thus, we are now ready in order to begin with the second section as follows.

##  2\. Some Definitions of Normed Spaces

######  Definition 1.

Let $\left(X,\left\|.\right\|\right)$ be a normed space and $K\subset X$. For
$u\in X$,

$\underset{v\in K}{\inf}\left\\{\left\|u-v\right\|\right\\}$

is a general best approximation.

Mohebi and Rubinov ([24]) and Rezapour in [25] gave the main preliminaries on
the approximation theory in the usual sense as follows:

######  Definition 2.

Let $\left(X,\left\|.\right\|\right)$ be a linear normed space. For a nonempty
subset $A$ of $X$ and $x\in X$,

$d\left(x,A\right)=\underset{a\in A}{\inf}\left\\{\left\|u-a\right\|\right\\}$

denotes the distance from $x$ to the set $A$. If

$\left\|x-a_{0}\right\|=d\left(x,A\right)\text{.}$

Then, we say that a point $a_{0}\in A$ is called a best approximation for
$x\in X$. If each $x\in X$ has at least one best approximation $a_{0}\in A,$
then $A$ is called a proximinal subset of $X$. If each $x\in X$ has a unique
best approximation $a_{0}\in A$, then $A$ is called a Chebyshev subset of $X$.

######  Definition 3.

Let $A\subset X$. For $x\in X$,

$P_{A}\left(x\right)=\left\\{a\in
A:\left\|x-a\right\|=d\left(x,A\right)\right\\}$

where $P_{A}\left(x\right)$, the set of all best approximations of $x$ in $A$.
We know that $P_{A}\left(x\right)$ is a closed and bounded subset of $X$. For
$x\notin A$, $P_{A}\left(x\right)$ is located in the boundary of $A$.

######  Definition 4.

Let $\left(X,\left\|.\right\|\right)$ be a linear normed space. For a nonempty
subset $A$ of $X$ and a nonempty set $W$ of $X$,

$d\left(A,W\right)=\underset{w\in W}{\inf}\underset{a\in
A}{\sup}\left\\{\left\|a-w\right\|\right\\}$

denotes the distance from the set $A$ to the set $W$. If

$\underset{w\in W}{\inf}\underset{a\in
A}{\sup}\left\\{\left\|a-w\right\|\right\\}=\underset{a\in
A}{\sup}\left\\{\left\|a-w_{0}\right\|\right\\}\text{.}$

Then, we say that a point $w_{0}\in W$ is called a best approximation from $A$
to $W$.

##  3\. Properties of $2$-Normed Spaces

In [16], Cho et al. defined linear 2-normed spaces and gave interesting
properties of them. After, Lewandowska defined generalized $2$-normed spaces
and derived properties of these spaces in [23]. Now, let us give the
definition of 2-normed space.

######  Definition 5.

Let $X$ be a linear space over $F$, where $F$ is the real or complex numbers
field, $\dim X>1$, and let

$\left\|.,.\right\|:X^{2}\rightarrow\mathbb{R}^{+}\cup\left\\{0\right\\}$

be a non-negative real-valued function on $X\times X$ with the following
properties:

N1) $\left\|x,y\right\|=0$ if and only if $x$ and $y$ are linearly dependent
vectors,

N2) $\left\|x,y\right\|=\left\|y,x\right\|$ for all $x,y\in X$,

N3) $\left\|\alpha x,y\right\|=\left|\alpha\right|\left\|x,y\right\|$ for all
$\alpha\in K$ and all $x,y\in X$,

N4) $\left\|x+y,z\right\|\leq\left\|x,z\right\|+\left\|y,z\right\|$ for all
$x,y,z\in X$.

Then, $\left\|.,.\right\|$ is called a 2-norm on $X$ and
$\left(X,\left\|.,.\right\|\right)$ is called a linear 2-normed space.

Every 2-normed space is a locally convex topological linear space. In fact,
for a fixed $b\in X$. For all $x\in X$,

$p_{b}\left(x\right)=\left\|x,b\right\|$

which is a seminorm and the family of $P$, that is

$P=\left\\{p_{b}:b\in X\right\\}$

generates a locally convex topology on $X.$ This space will be denoted by
$\left(X,p_{b}\right)$. In each 2-normed space
$\left(X,\left\|.,.\right\|\right)$. For all $x,y\in X$ and for every real
$\alpha$, we have non-negative norm,

$\left\|x,y\right\|\geq 0\text{ and }\left\|x,y+\alpha
x\right\|=\left\|x,y\right\|\text{.}$

Also, if $x$, $y$ and $z$ are linearly dependent, this occurs for $\dim X=2$.
Then,

$\left\|x,y+z\right\|=\left\|x,y\right\|+\left\|x,z\right\|or\left\|x,y-z\right\|=\left\|x,y\right\|+\left\|x,z\right\|\text{.}$

######  Example 1.

([27])Let $P_{n}$ denotes the set of real polynomials of degree less than or
equal to $n$, on the interval $\left[0,1\right]$. By considering usual
addition and scalar multiplication, $P_{n}$ is a linear vector space over the
reals. Let $\left\\{x_{1},x_{2},\cdots,x_{2n}\right\\}$ be distinct fixed
points in $\left[0,1\right]$ and define the 2-norm on $P_{n}$ as

$\left\|f,g\right\|=\mathop{\displaystyle\sum}\limits_{k=1}^{2n}\left|f\left(x_{k}\right)g^{\prime}\left(x_{k}\right)-f^{\prime}\left(x_{k}\right)g\left(x_{k}\right)\right|\text{.}$

Then, $\left(P_{n},\left\|.,.\right\|\right)$ is a 2-normed space.

Let $\left(X,\left\|.,.\right\|\right)$ be a 2-normed space. Under this
assumption, we can give the following defitinions:

######  Definition 6.

([27])A sequence $\left\\{x_{n}\right\\}_{n\geq 1}$ in a linear 2-normed space
$X$ is called Cauchy sequence if there exist independent elements $y,z\in X$
such that

$\lim_{n,m\rightarrow\infty}\left\|x_{n}-x_{m},y\right\|=0\text{ and
}\lim_{n,m\rightarrow\infty}\left\|x_{n}-x_{m},z\right\|=0\text{.}$

######  Definition 7.

([27])A sequence $\left\\{x_{n}\right\\}_{n\geq 1}$ in a linear 2-normed space
$X$ is called convergent if there exists an element $x\in X$ such that

$\lim_{n\rightarrow\infty}\left\|x_{n}-x,z\right\|=0$

for all $z\in X$.

######  Proposition 1.

([6])Let $\left(X,\left\|.,.\right\|\right)$ be 2-normed space and $W$ be a
subspace of $X$, $b\in X$ and $L\left\\{b\right\\}$ be the subspace of $X$
generated by $b$. If $x_{0}\in X$ is such that

$\delta=\underset{w\in
W}{\inf}\left\\{\left\|x_{0}-w,b\right\|\right\\}>0\text{.}$

Then, there exists a bounded bilinear functional as follows

$f:X\times L\left\\{b\right\\}\rightarrow K$

such that

$F|_{w\times L\left\\{b\right\\}}=0\text{, }F\left(x_{0},b\right)=1\text{ and
}\left\|F\right\|=\frac{1}{\delta}\text{.}$

######  Definition 8.

A 2-normed space $\left(X,\left\|.,.\right\|\right)$ is which every Cauchy
sequence $\left(x_{n}\right)$ converges to some $x\in X$ then $X$ is said to
be complete with respect to the 2-norm.

######  Definition 9.

A complete 2-normed space $\left(X,\left\|.,.\right\|\right)$ is called a
2-Banach space.

The examples 1 and 2 are 2-Banach spaces while the example 3 does not (For
details, see [27]).

######  Lemma 1.

([27]) $\left(i\right)$ Every 2-normed space of dimension 2 is a 2-Banach
space, when the underlying field is complete.

$\left(ii\right)$ If $\left\\{x_{n}\right\\}$ is a sequence in 2-normed space
$\left(X,\left\|.,.\right\|\right)$ and if

$\lim_{n\rightarrow\infty}\left\|x_{n}-x,y\right\|=0\text{{.}}$

then, we have

$\lim_{n\rightarrow\infty}\left\|x_{n},y\right\|=\left\|x,y\right\|\text{.}$

##  4\. Fundamental Results

In this section, let us also consider a definition, and however, we give Lemma
and Proposition for the best simultaneous approximation in linear $2$-normed
spaces.

######  Definition 10.

Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space and $W$
be any bounded subset of $X$. An element $g^{\ast}\in G$ is said to be a best
approximation to the set $W$, if

$\underset{f\in W}{\sup}\left\|f-g^{\ast},b\right\|=\underset{g\in
G}{\inf}\left\\{\underset{f\in W}{\sup}\left\|f-g,b\right\|\right\\}$

where $b\in X\backslash L\left\\{f,g^{\ast}\right\\}$ is the subspace of $X$
generated by $f$ and $g^{\ast}$.

######  Lemma 2.

Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space,
$G\subset X$ and $W$ be bounded subset of $X$. Then,

$\Phi\left(g,b\right)=\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}$

is a continuous functional on $X$, where $b\in X\backslash
L\left\\{f,g^{\ast}\right\\}$.

###### Proof.

Since the norms $\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|$are
continuous functionals of $g$ on $X$, $\phi\left(g,b\right)$ is clearly a
continuous functional. To show this, for any $f_{1},f_{2}\in W$ and
$g,g^{{}^{\prime}}\in X$ , we have

$\displaystyle\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}$
$\displaystyle\leq$
$\displaystyle\left\\{\left\|f_{1}-g^{{}^{\prime}},b\right\|+\left\|g-g^{{}^{\prime}},b\right\|,\left\|f_{2}-g^{{}^{\prime}},b\right\|,\left\|g-g^{{}^{\prime}},b\right\|\right\\}\text{.}$

Then

$\displaystyle\underset{f_{1},f_{2}\in
w}{\sup}\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}$
$\displaystyle\leq$ $\displaystyle\underset{f_{1},f_{2}\in
w}{\sup}\left\\{\left\|f_{1}-g^{{}^{\prime}},b\right\|+\left\|g-g^{{}^{\prime}},b\right\|,\left\|f_{2}-g^{{}^{\prime}},b\right\|,\left\|g-g^{{}^{\prime}},b\right\|\right\\}\text{.}$

Now, if

$\left\|g-g^{{}^{\prime}},b\right\|<\frac{\varepsilon}{2},\text{ then
}\phi\left(g,b\right)\leq\phi\left(g^{{}^{\prime}},b\right)+\varepsilon\text{.}$

By interchanging $g$ and $g^{{\acute{}}}$, proof of Theorem will be completed.

######  Lemma 3.

Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space,
$G\subset X$ and $W$ be bounded subset of $X$. Then there exists a best
simultaneous approximation $g^{\ast}\in G$ to any given compact subset
$W\subset X$.

###### Proof.

By using the proof of Elumalai and his coworkers in same manner, we can make
the proof, using the definition of the continuous functional

$\phi\left(g,b\right)=\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}\text{.}$

######  Lemma 4.

Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space,
$G\subset X$ and $W$ be bounded subset of $X$. If $g_{1},g_{2}\in G$ are best
simultaneous approximations to $W$ by elements of $G$. Then
$g=\lambda_{1}g_{1}+\lambda_{2}g_{2}$ is also a best simultaneous
approximation to $f_{1}$ and $f_{2}$, where $0\leq\lambda\leq 1$ and
$\lambda_{1}+\lambda_{2}=1$.

###### Proof.

By using expression of $\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-\overset{\\_}{g},b\right\|,\left\|f_{2}-\overset{\\_}{g},b\right\|\right\\}$,
we discover the followings

$\displaystyle=$ $\displaystyle\underset{f\in
W}{\sup}\left\\{\left\|f_{1}-\lambda_{1}g_{1}-\lambda_{2}g_{2},b\right\|,\left\|f_{2}-\lambda_{1}g_{1}-\lambda_{2}g_{2},b\right\|\right\\}$
$\displaystyle=$ $\displaystyle\underset{f\in
W}{\sup}\left\\{\left\|\lambda\left(f_{1}-g_{1}\right)+\left(1-\lambda\right)\left(f_{1}-g_{2}\right),b\right\|,\left\|\lambda\left(f_{2}-g_{1}\right)+\left(1-\lambda\right)\left(f_{2}-g_{2}\right),b\right\|\right\\}\text{.}$

From last equality, we easily derive as

$\leq\left[\begin{array}[]{c}\underset{f\in
W}{\sup}\left\\{\left\|\lambda\left(f_{1}-g_{1}\right)+\left(1-\lambda\right)\left(f_{1}-g_{2}\right),b\right\|,\left\|\lambda\left(f_{2}-g_{1}\right)+\left(1-\lambda\right)\left(f_{2}-g_{2}\right),b\right\|\right\\}\\\
+\underset{f\in
W}{\sup}\left\\{\left\|\lambda\left(f_{1}-g_{1}\right)+\left(1-\lambda\right)\left(f_{1}-g_{2}\right),b\right\|,\left\|\lambda\left(f_{2}-g_{1}\right)+\left(1-\lambda\right)\left(f_{2}-g_{2}\right),b\right\|\right\\}\end{array}\right]$

By using definition of 2-norm and definition 10, we deduce as follows

$\underset{g\in G}{\inf}\left\\{\underset{f\in
W}{\sup}\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right\\}\text{.}$

Subsequently, we complete the proof of Lemma.

######  Proposition 2.

Let $\left(X,\left\|.,.\right\|\right)$ be a linear $2$-normed space, $G$ is a
non-empty strictly convex subset of $X$ and $Y$ be a compact subset of $X$.
Then there is only one $y_{0}\in Y$ such that

$\left\|x_{0}-y_{0},z\right\|=\underset{y\in
Y}{\inf}\left\\{\left\|x_{0}-y,z\right\|\right\\}$

for $x_{0}\in X\backslash Y$ and for every $z\in X\backslash L\left\\{x\in
G\right\\}$ and $y_{0}\in Y$.

###### Proof.

If $x_{0}\in Y$. Then, we have $\left\|x_{0}-y_{0},z\right\|=0$. Hence, assume
that

$x_{0}\in Y\ \text{or }x\in X\backslash Y\text{.}$

If we say

$d_{0}=\underset{y\in Y}{\inf}\left\\{\left\|x_{0}-y,z\right\|\right\\}$

and

$d_{0}=\underset{y\in
Y}{\inf}\left\\{\left\|x_{0}-y,y^{{}^{\prime}}\right\|\right\\}\text{.}$

Then, there are linearly independent elements $y^{{}^{\prime}}$ and $z$ in
$X$. So, there is a Cauchy sequence $\left\\{y_{n}\right\\}$ such that

$\underset{n\rightarrow\infty}{\lim}\left\|x_{0}-y_{n},z\right\|=d_{0},\text{
\ \ }\underset{m\rightarrow\infty}{\lim}\left\|x_{0}-y_{m},z\right\|=d_{0}$

and

$\underset{n\rightarrow\infty}{\lim}\left\|x_{0}-y_{n},y^{{}^{\prime}}\right\|=d_{0},\text{
\ \
}\underset{m\rightarrow\infty}{\lim}\left\|x_{0}-y_{m},y^{{}^{\prime}}\right\|=d_{0}\text{.}$

Thus, we procure the following

$\left\|x_{0}-y_{0},y\right\|=d_{0}\text{ \ and \
}\left\|x_{0}-y_{0},z\right\|=d_{0}\text{.}$

By using the following inequalities

$d_{0}\leq\left\|x_{0}-y_{0},z\right\|\leq\left\|x_{0}-y_{n},z\right\|+\left\|y_{n}-y_{0},z\right\|$

and

$d_{0}\leq\left\|x_{0}-y_{0},y\right\|\leq\left\|x_{0}-y_{n},y\right\|+\left\|y_{n}-y_{0},y\right\|\text{.}$

From this, we see that

$\displaystyle\left\|y_{0}-y_{0}^{{}^{\prime}},z\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),z\right\|^{2}$
$\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),z\right\|^{2}+\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),z\right\|^{2}$
$\displaystyle-\left\|y_{0}+y_{0}^{{}^{\prime}}-2x_{0},z\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
2\left(\left\|x_{0}-y_{0},z\right\|^{2}+\left\|y_{0}^{{}^{\prime}}-x_{0},z\right\|^{2}\right)-4\left\|\frac{y_{0}+y_{0}^{{}^{\prime}}}{2}-x_{0},z\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
2\left(2d_{0}^{2}\right)-4d_{0}^{2}=0\text{.}$

We find $y_{0}=y_{0}^{{}^{\prime}}$. In similar way, we again obtain
$y_{0}=y_{0}^{{}^{\prime}}$ with respect to $y$

$\displaystyle\left\|y_{0}-y_{0}^{{}^{\prime}},y\right\|^{2}$ $\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),y\right\|^{2}$
$\displaystyle=$
$\displaystyle\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),y\right\|^{2}+\left\|\left(y_{0}-x_{0}\right)+\left(x_{0}-y_{0}^{{}^{\prime}}\right),y\right\|^{2}$
$\displaystyle-\left\|y_{0}+y_{0}^{{}^{\prime}}-2x_{0},y\right\|^{2}$
$\displaystyle\leq$ $\displaystyle
2\left(\left\|x_{0}-y_{0},y\right\|^{2}+\left\|y_{0}^{{}^{\prime}}-x_{0},y\right\|^{2}\right)-4\left\|\frac{y_{0}+y_{0}^{{}^{\prime}}}{2}-x_{0},y\right\|^{2}$
$\displaystyle\leq$ $\displaystyle 2\left(2d_{0}^{2}\right)-4d_{0}^{2}=0$

Then, we complete the proof of theorem.

Let $X$ be a linear 2-normed space and $W_{1}$ and $W_{2}$ are linear
subspaces in $X$, and $f$ be a 2-functional with domain $W_{1}\times W_{2}$.
If $\left\|.,.\right\|$ denotes 2-norm, then the problem is to find an element
$g^{\ast}\in G$, if it exists for which

$\underset{f_{1},f_{2}\in
W}{\sup}\left\\{\left\|f_{1}-g^{\ast},b\right\|,\left\|f_{2}-g^{\ast},b\right\|\right\\}=\underset{g\in
G}{\inf}\left\\{\underset{f_{1},f_{2}\in
W}{\sup}\left(\left\|f_{1}-g^{\ast},b\right\|,\left\|f_{2}-g^{\ast},b\right\|\right)\right\\}\text{.}$

Thus, we reach the following proposition which is interesting and worthwhile
for studying in linear $2$-normed spaces.

######  Proposition 3.

Let $\left(X,\left\|.,.\right\|\right)$ be a linear 2-normed space over $R$
and $G$ be a linear subspace of $X$. Let $f_{1},f_{2}\in X\backslash G$ such
that $f_{1},f_{2}$ and $b\in X$ are linearly independent. Then there exists a
best simultaneous approximation by elements of $G$ to $f_{1},f_{2}\in W$ such
that

$\underset{g\in G}{\inf}\left\\{\underset{f_{1},f_{2}\in
W}{\sup}\left(\left\|f_{1}-g,b\right\|,\left\|f_{2}-g,b\right\|\right)\right\\}=\underset{f_{1},f_{2}\in
W}{\sup}\left\\{\left\|f_{1}-g^{\ast},b\right\|,\left\|f_{2}-g^{\ast},b\right\|\right\\}$

where $W=\left\\{f_{1},f_{2}\right\\}$.

######  Acknowledgement 1.

Author would like to thank to Serkan Araci for his help in this paper.

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d^{R}_{0}\cdot\mathsf{U}^{1}_{0j}}\cdot{d^{A}_{i}\over d^{\prime A}_{j}}={}$
$\displaystyle{(1/\alpha,1/\beta,w/z,\alpha\beta/zw;q)_{n}\over(\alpha/z,\beta/z,1/w,w/\alpha\beta;q)_{n}}{(\beta/z,w/\alpha\beta,q^{-n+1}z/\alpha,q^{-n+1}w;q)_{i}\over(1/\alpha,w/z,q^{-n+1}\beta,q^{-n+1}zw/\alpha\beta;q)_{i}}$
$\displaystyle\cdot{(\beta/z,1/w,q^{-n+1}z/\alpha,q^{-n+1}\alpha\beta/w;q)_{j}\over(1/\alpha,\alpha\beta/zw,q^{-n+1}\beta,q^{-n+1}z/w;q)_{j}}.$
(D.46)

###### Proof.

For (D.44), we have from (D.31), (D.40), (D.41), and (D.47), (D.51) below that

$\displaystyle{1\over\mathsf{L}^{1}_{i0}\cdot
d^{R}_{0}\cdot\mathsf{U}^{1}_{0j}}(\mathsf{L}^{1}\cdot
D_{R}\cdot\mathsf{U}^{1})_{ij}=\sum_{k=0}^{\min(i,j)}{\mathsf{L}^{1}_{ik}\cdot
d^{R}_{k}\cdot\mathsf{U}^{1}_{kj}\over\mathsf{L}^{1}_{i0}\cdot
d^{R}_{0}\cdot\mathsf{U}^{1}_{0j}}$ $\displaystyle={}$
$\displaystyle\sum_{k=0}^{\min(i,j)}{{\mathcal{B}}_{ik}(q^{-n}w/\alpha,q^{-n}z/\alpha)\over{\mathcal{B}}_{i0}(q^{-n}w/\alpha,q^{-n}z/\alpha)}{h^{R}_{k}\over
q^{k^{2}}g^{L}_{k}}{d^{R}_{k}\over
d^{R}_{0}}{(q^{-n}z/\alpha)^{k}h^{R}_{k}\over
g^{U}_{k}}{{\mathcal{B}}_{jk}(q^{-n}\beta/w,q^{-n}z/\alpha)\over{\mathcal{B}}_{j0}(q^{-n}\beta/w,q^{-n}z/\alpha)}$
$\displaystyle={}$
$\displaystyle{}_{10}W_{9}\left(q^{-n}z/\alpha;q^{-i},q^{-j},q^{-n+i}w/\alpha,q^{-n+j}\beta/w,z/\alpha\beta,qz/\alpha,q^{-n+1}z;q,q\right).$

For (D.45), in a similar manner, we have from (D.33), (D.42), (D.43), and
(D.48), (D.52) below that

$\displaystyle{1\over\mathsf{U}^{2}_{in}\cdot d^{\prime
R}_{n}\cdot\mathsf{L}^{2}_{nj}}(\mathsf{U}^{2}\cdot
D^{\prime}_{R}\cdot\mathsf{L}^{2})_{ij}=\sum_{k=\max(i,j)}^{n}{\mathsf{U}^{2}_{ik}\cdot
d^{\prime R}_{k}\cdot\mathsf{L}^{2}_{kj}\over\mathsf{U}^{2}_{in}\cdot
d^{\prime
R}_{n}\cdot\mathsf{L}^{2}_{nj}}=\sum_{l=0}^{\min(i,j)}{\mathsf{U}^{2}_{i,n-l}\cdot
d^{\prime R}_{n-l}\cdot\mathsf{L}^{2}_{n-l,j}\over\mathsf{U}^{2}_{in}\cdot
d^{\prime R}_{n}\cdot\mathsf{L}^{2}_{nj}}$ $\displaystyle={}$
$\displaystyle\sum_{l=0}^{\min(i,j)}{\mathcal{B}_{n-l,i}(q^{-n}\beta/z,q^{-n}w/\alpha)\over\mathcal{B}_{ni}(q^{-n}\beta/z,q^{-n}w/\alpha)}{g^{\prime
U}_{n-l}\over g^{\prime U}_{n}}{(q^{-n}\beta/z)^{n}h^{\prime
R}_{n}\over(q^{-n}\beta/z)^{n-l}h^{\prime R}_{n-l}}{d^{\prime R}_{n-l}\over
d^{\prime R}_{n}}$ $\displaystyle\qquad\cdot{q^{(n-l)^{2}}g^{\prime
L}_{n-l}\over q^{n^{2}}g^{\prime L}_{n}}{h^{\prime R}_{n}\over h^{\prime
R}_{n-l}}{\mathcal{B}_{n-l,j}(q^{-n}\beta/z,q^{-n}\beta/w)\over\mathcal{B}_{nj}(q^{-n}\beta/z,q^{-n}\beta/w)}$
$\displaystyle={}$
$\displaystyle{}_{10}W_{9}\left(q^{-n}z/\beta;q^{-n+i},q^{-n+j},q^{-i}\alpha/w,q^{-j}w/\beta,z/\alpha\beta,qz/\beta,q^{-n+1}z;q,q\right).$

Concerning (D.46), one can proceed as

$\displaystyle{\mathsf{U}^{2}_{in}\cdot d^{\prime
R}_{n}\cdot\mathsf{L}^{2}_{nj}\over\mathsf{L}^{1}_{i0}\cdot
d^{R}_{0}\cdot\mathsf{U}^{1}_{0j}}\cdot{d^{A}_{i}\over d^{\prime A}_{0}}$
$\displaystyle={}$
$\displaystyle\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}w/\alpha)\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}\beta/w){d^{A}_{0}\over
d^{\prime A}_{0}}{d^{\prime R}_{n}\over d^{R}_{0}}{g^{\prime
U}_{n}\over(q^{-n}\beta/z)^{n}h^{\prime R}_{n}}{q^{n^{2}}g^{\prime L}_{n}\over
h^{\prime R}_{n}}$
$\displaystyle\cdot{\mathcal{B}_{ni}(q^{-n}\beta/z,q^{-n}w/\alpha)\over\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}w/\alpha)}{1\over\mathcal{B}_{i0}(q^{-n}w/\alpha,q^{-n}z/\alpha)}{(q^{-n}w/\alpha)^{i}h^{A}_{i}h^{A}_{i}\over
q^{i^{2}}g^{L}_{i}w^{i}g^{U}_{i}}{d^{A}_{i}\over d^{A}_{0}}$
$\displaystyle\cdot{\mathcal{B}_{nj}(q^{-n}\beta/z,q^{-n}\beta/w)\over\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}\beta/w)}{1\over\mathcal{B}_{j0}(q^{-n}\beta/w,q^{-n}z/\alpha)}{(q^{-n}\beta/w)^{j}h^{\prime
A}_{j}w^{j}h^{\prime A}_{j}\over g^{\prime U}_{j}q^{j^{2}}g^{\prime
L}_{j}}{d^{\prime A}_{0}\over d^{\prime A}_{j}}.$

Then from (D.53), (D.54), and (D.55) below, we obtain (D.46). ∎

######  Lemma D.20.

We have

$\displaystyle{\mathcal{B}_{ik}(a,b)\over\mathcal{B}_{i0}(a,b)}=(a^{-1}q)^{k}{(aq^{i};q)_{k}(q^{-i};q)_{k}\over(bq^{i+1};q)_{k}(a^{-1}bq^{1-i};q)_{k}},$
(D.47)
$\displaystyle{\mathcal{B}_{n-l,i}(a,b)\over\mathcal{B}_{n,i}(a,b)}=(a^{-2}b)^{l}{(a^{-1}q^{1-2n};q)_{2l}\over(a^{-1}q^{-2n};q)_{2l}}{(b^{-1}q^{-n-i};q)_{l}(q^{-n+i};q)_{l}\over(a^{-1}q^{1-n-i};q)_{l}(a^{-1}bq^{1-n+i};q)_{l}},$
(D.48)
$\displaystyle\mathcal{B}_{i0}(a,b)=b^{i}{(aq;q)_{2i}\over(a;q)_{2i}}{(a;q)_{i}(a/b;q)_{i}\over(bq;q)_{i}(q;q)_{i}},$
(D.49)
$\displaystyle{\mathcal{B}_{n,i}(a,b)\over\mathcal{B}_{n,0}(a,b)}=(a^{-1}q)^{i}{(aq^{n};q)_{i}(q^{-n};q)_{i}\over(bq^{1+n};q)_{i}(a^{-1}bq^{1-n};q)_{i}}.$
(D.50)

######  Lemma D.21.

We have

$\displaystyle{h^{R}_{k}\over
q^{k^{2}}g^{L}_{k}}{(q^{-n}z/\alpha)^{k}h^{R}_{k}\over
g^{U}_{k}}=q^{-k^{2}}(q^{-n-1}\beta)^{k}{(q^{-n+1}z/\alpha;q)_{2k}(q^{-n+1}z/\alpha;q)_{2k}\over(1/\alpha;q)_{k}(q^{-n};q)_{k}(q^{-n+1}\beta;q)_{k}(q;q)_{k}},$
(D.51) $\displaystyle{g^{\prime U}_{n-l}\over g^{\prime
U}_{n}}{(q^{-n}\beta/z)^{n}h^{\prime R}_{n}\over(q^{-n}\beta/z)^{n-l}h^{\prime
R}_{n-l}}{q^{(n-l)^{2}}g^{\prime L}_{n-l}\over q^{n^{2}}g^{\prime
L}_{n}}{h^{\prime R}_{n}\over h^{\prime R}_{n-l}}$
$\displaystyle\qquad=q^{-l^{2}}(q^{-n+1}\beta^{3}/z^{4})^{l}{(q^{-n}z/\beta;q)_{2l}(q^{-n}z/\beta;q)_{2l}\over(1/\beta;q)_{l}(q^{-n};q)_{l}(q^{-n+1}\alpha;q)_{l}(q;q)_{l}}.$
(D.52)

######  Lemma D.22.

We have

$\displaystyle\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}w/\alpha)\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}\beta/w){d^{A}_{0}\over
d^{\prime A}_{0}}{d^{\prime R}_{n}\over d^{R}_{0}}{g^{\prime
U}_{n}\over(q^{-n}\beta/z)^{n}h^{\prime R}_{n}}{q^{n^{2}}g^{\prime L}_{n}\over
h^{\prime R}_{n}}$ $\displaystyle={}$
$\displaystyle{(1/\alpha,1/\beta,w/z,\alpha\beta/zw;q)_{n}\over(\alpha/z,\beta/z,1/w,w/\alpha\beta;q)_{n}},$
(D.53)
$\displaystyle{\mathcal{B}_{ni}(q^{-n}\beta/z,q^{-n}w/\alpha)\over\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}w/\alpha)}{1\over\mathcal{B}_{i0}(q^{-n}w/\alpha,q^{-n}z/\alpha)}{(q^{-n}w/\alpha)^{i}h^{A}_{i}h^{A}_{i}\over
q^{i^{2}}g^{L}_{i}w^{i}g^{U}_{i}}{d^{A}_{i}\over d^{A}_{0}}$
$\displaystyle={}$
$\displaystyle{(\beta/z,w/\alpha\beta,q^{-n+1}z/\alpha,q^{-n+1}w;q)_{i}\over(1/\alpha,w/z,q^{-n+1}\beta,q^{-n+1}zw/\alpha\beta;q)_{i}},$
(D.54)
$\displaystyle{\mathcal{B}_{nj}(q^{-n}\beta/z,q^{-n}\beta/w)\over\mathcal{B}_{n0}(q^{-n}\beta/z,q^{-n}\beta/w)}{1\over\mathcal{B}_{j0}(q^{-n}\beta/w,q^{-n}z/\alpha)}{(q^{-n}\beta/w)^{j}h^{\prime
A}_{j}w^{j}h^{\prime A}_{j}\over g^{\prime U}_{j}q^{j^{2}}g^{\prime
L}_{j}}{d^{\prime A}_{0}\over d^{\prime A}_{j}}$ $\displaystyle={}$
$\displaystyle{(\beta/z,1/w,q^{-n+1}z/\alpha,q^{-n+1}\alpha\beta/w;q)_{j}\over(1/\alpha,\alpha\beta/zw,q^{-n+1}\beta,q^{-n+1}z/w;q)_{j}}.$
(D.55)

###  D.8. Final step of proof with Bailey’s transformation formula for
terminating very-well-poised balanced ${}_{10}\phi_{9}$ series

Recall that Bailey’s transformation formula for terminating very-well-poised
balanced ${}_{10}\phi_{9}$ series [28, (2.9.4), (2.9.5)] reads:

$\displaystyle{}_{10}W_{9}(a;b,c,d,e,f,g,h;q,q)$
$\displaystyle={(aq,aq/ef,aq/eg,aq/eh,aq/fg,aq/fh,aq/gh,aq/efgh;q)_{\infty}\over(aq/e,aq/f,aq/g,aq/h,aq/efg,aq/efh,aq/egh,aq/fgh;q)_{\infty}}$
(D.56)
$\displaystyle\cdot{}_{10}W_{9}(qa^{2}/bcd;aq/bc,aq/bd,aq/cd,e,f,g,h;q,q),$

where at least one of the parameters $e,f,g,h$ is of the form
$q^{-n},n=0,1,2,\ldots$, and the balancing condition is satisfied, namely

$\displaystyle q^{2}a^{3}=bcdefgh.$ (D.57)

In particular, by setting $h=q^{-n}$, we have

$\displaystyle{}_{10}W_{9}(a;b,c,d,e,f,g,q^{-n};q,q)$ (D.58)
$\displaystyle={(aq,aq/ef,aq/eg,aq/fg;q)_{n}\over(aq/e,aq/f,aq/g,aq/efg;q)_{n}}{}_{10}W_{9}(qa^{2}/bcd;aq/bc,aq/bd,aq/cd,e,f,g,q^{-n};q,q).$

######  Proposition D.23.

Let $n,i,j\in\mathbb{Z}_{\geq 0}$ and $i,j\leq n$. Applying Bailey’s
transformation twice, we have

$\displaystyle{}_{10}W_{9}\left(q^{-n}z/\alpha;q^{-i},q^{-j},q^{-n+i}w/\alpha,q^{-n+j}\beta/w,z/\alpha\beta,qz/\alpha,q^{-n+1}z;q,q\right)$
$\displaystyle={}$
$\displaystyle{(1/\alpha,1/\beta,w/z,\alpha\beta/zw;q)_{n}\over(\alpha/z,\beta/z,1/w,w/\alpha\beta;q)_{n}}{(\beta/z,w/\alpha\beta,q^{-n+1}z/\alpha,q^{-n+1}w;q)_{i}\over(1/\alpha,w/z,q^{-n+1}\beta,q^{-n+1}zw/\alpha\beta;q)_{i}}$
$\displaystyle\cdot{(\beta/z,1/w,q^{-n+1}z/\alpha,q^{-n+1}\alpha\beta/w;q)_{j}\over(1/\alpha,\alpha\beta/zw,q^{-n+1}\beta,q^{-n+1}z/w;q)_{j}}$
(D.59)
$\displaystyle\cdot{}_{10}W_{9}\left(q^{-n}z/\beta;q^{-n+i},q^{-n+j},q^{-i}\alpha/w,q^{-j}w/\beta,z/\alpha\beta,qz/\beta,q^{-n+1}z;q,q\right).$

###### Proof.

Observe that the balancing condition as in (D.57) is satisfied:

$\displaystyle
q^{2}\left({q^{-n}z\over\alpha}\right)^{3}=q^{-i}q^{-j}{q^{-n+i}w\over\alpha}{q^{-n+j}\beta\over
w}{z\over\alpha\beta}{qz\over\alpha}{q^{-n+1}z}.$

Hence Bailey’s transformation formula (D.58) applies and we have

$\displaystyle{}_{10}W_{9}\left(q^{-n}z/\alpha;q^{-i},q^{-j},q^{-n+i}w/\alpha,q^{-n+j}\beta/w,z/\alpha\beta,qz/\alpha,q^{-n+1}z;q,q\right)$
$\displaystyle={}$
$\displaystyle{}_{10}W_{9}\left(q^{-n}z/\alpha;q^{-j},q^{-n+j}\beta/w,qz/\alpha,q^{-n+i}w/\alpha,z/\alpha\beta,q^{-n+1}z,q^{-i};q,q\right)$
$\displaystyle={}$
$\displaystyle{(q^{-n+1}z/\alpha,q^{-i+1}\alpha\beta/w,q^{-i+n}/w,\beta/z;q)_{i}\over(q^{-i+1}z/w,q^{-n+1}\beta,1/\alpha,q^{-i+n}\alpha\beta/zw;q)_{i}}$
$\displaystyle\cdot{}_{10}W_{9}\left(q^{-n}zw/\alpha\beta;qzw/\alpha\beta,q^{-n+j},q^{-j}w/\beta,q^{-n+i}w/\alpha,z/\alpha\beta,q^{-n+1}z,q^{-i};q,q\right).$

Applying Bailey’s transformation once again, we have

$\displaystyle{}_{10}W_{9}\left(q^{-n}zw/\alpha\beta;qzw/\alpha\beta,q^{-n+j},q^{-j}w/\beta,q^{-n+i}w/\alpha,z/\alpha\beta,q^{-n+1}z,q^{-i};q,q\right)$
$\displaystyle={}$
$\displaystyle{}_{10}W_{9}\left(q^{-n}zw/\alpha\beta;q^{-i},q^{-n+i}w/\alpha,qzw/\alpha\beta,q^{-j}w/\beta,z/\alpha\beta,q^{-n+1}z,q^{-n+j};q,q\right)$
$\displaystyle={}$
$\displaystyle{(q^{-n+1}zw/\alpha\beta,q^{-n+j+1}\beta,q^{j}/\alpha,w/z;q)_{n-j}\over(q^{-n+j+1}z/\alpha,q^{-n+1}w,w/\alpha\beta,q^{j}\beta/z;q)_{n-j}}$
$\displaystyle\cdot{}_{10}W_{9}\left(q^{-n}z/\beta;qz/\beta,q^{-n+i},q^{-i}\alpha/w,q^{-j}w/\beta,z/\alpha\beta,q^{-n+1}z,q^{-n+j};q,q\right).$

Simplifying the prefactor we have (D.59). ∎

Now we are ready to state our proof of Theorem D.3, therewith its
reformulation Theorem D.13.

Proof of Theorem D.3. In view of (D.44), (D.45), (D.46) in Proposition D.19,
and (D.59) in Proposition D.23, we have the equality of matrices

$\displaystyle{1\over\mathsf{L}^{1}_{i0}\cdot
d^{R}_{0}\cdot\mathsf{U}^{1}_{0j}}\mathsf{L}^{1}\cdot
D_{R}\cdot\mathsf{U}^{1}={1\over\mathsf{L}^{1}_{i0}\cdot
d^{R}_{0}\cdot\mathsf{U}^{1}_{0j}}D_{A}\cdot\mathsf{U}^{2}\cdot
D^{\prime}_{R}\cdot\mathsf{L}^{2}\cdot{D^{\prime}_{A}}^{\\!\\!-1},$

which is equivalent to the commutativity $RD_{2}A=ARD_{2}$. ∎

##  Appendix E Truncation by tuning mass parameters

In [10] we employed the following orbifolded Nekrasov factor for the affine
Laumon space;

$\displaystyle{\mathsf{N}}^{(k|N)}_{\lambda,\mu}(u|q,\kappa)={\mathsf{N}}^{(k)}_{\lambda,\mu}(u|q,\kappa)$
$\displaystyle=$ $\displaystyle\prod_{j\geq i\geq 1\atop j-i\equiv k\,\,({\rm
mod}\,N)}[uq^{-\mu_{i}+\lambda_{j+1}}\kappa^{-i+j};q]_{\lambda_{j}-\lambda_{j+1}}\cdot\prod_{\beta\geq\alpha\geq
1\atop\beta-\alpha\equiv-k-1\,\,({\rm
mod}\,N)}[uq^{\lambda_{\alpha}-\mu_{\beta}}\kappa^{\alpha-\beta-1};q]_{\mu_{\beta}-\mu_{\beta+1}},$
(E.1)

with $[u;q]_{n}:=u^{-n/2}q^{-n(n-1)/4}(u;q)_{n}$. To work out the tuning
condition of mass parameters required for the truncation of the Young
diagrams, it is convenient to recast (E) to the “Nakajima” form, which is
expressed as a product over the boxes of the Young diagrams $(\lambda,\mu)$.
In the following we temporally replace $[u;q]_{n}$ by $(u;q)_{n}$, since we
can neglect the monomial factors for our purpose.

By taking the product over ${\mathsf{N}}^{(0)}$ to ${\mathsf{N}}^{(N-1)}$, we
can remove the selection rule;

$\displaystyle{\mathsf{N}}_{\lambda,\mu}(u|q,\kappa):=$
$\displaystyle\prod_{k=0}^{N-1}{\mathsf{N}}^{(k)}_{\lambda,\mu}(u|q,\kappa)$
$\displaystyle=$ $\displaystyle\prod_{j\geq i\geq
1}(uq^{-\mu_{i}+\lambda_{j+1}}\kappa^{-i+j};q)_{\lambda_{j}-\lambda_{j+1}}\cdot\prod_{\beta\geq\alpha\geq
1}(uq^{\lambda_{\alpha}-\mu_{\beta}}\kappa^{\alpha-\beta-1};q)_{\mu_{\beta}-\mu_{\beta+1}}.$
(E.2)

Namely we can obtain ${\mathsf{N}}^{(k|N)}_{\lambda,\mu}(u|q,\kappa)$ by
factorizing the total Nekrasov factor ${\mathsf{N}}_{\lambda,\mu}(u|q,\kappa)$
according to the power of $\kappa$. Now we can transform the total Nekrasov
factor as follows;

$\displaystyle{\mathsf{N}}_{\lambda,\mu}(u|q,\kappa)=$
$\displaystyle\prod_{j\geq i\geq
1}\frac{(uq^{-\mu_{i}+\lambda_{j+1}}\kappa^{-i+j};q)_{\infty}}{(uq^{-\mu_{i}+\lambda_{j}}\kappa^{-i+j};q)_{\infty}}\cdot\prod_{i\geq
j\geq
1}\frac{(uq^{\lambda_{j}-\mu_{i}}\kappa^{j-i-1};q)_{\infty}}{(uq^{\lambda_{j}-\mu_{i+1}}\kappa^{j-i-1};q)_{\infty}}$
$\displaystyle=$
$\displaystyle\prod_{i,j=1}^{\infty}\frac{(uq^{\lambda_{j}-\mu_{i}}\kappa^{j-i-1};q)_{\infty}}{(uq^{\lambda_{j}-\mu_{i}}\kappa^{j-i};q)_{\infty}}\cdot\frac{(u\kappa^{j-i};q)_{\infty}}{(u\kappa^{j-i-1};q)_{\infty}}$
$\displaystyle=$
$\displaystyle\prod_{(i,j)\in\lambda}(1-uq^{\lambda_{i}-j}\kappa^{-\mu^{\vee}_{j}+i-1})\cdot\prod_{(i,j)\in\mu}(1-uq^{-\mu_{i}+j-1}\kappa^{\lambda^{\vee}_{j}-i}).$
(E.3)

where we have used the combinatorial identity (cf. Proposition in [11],
(2.12)=(2.9) with $\kappa=t^{-1}$) for the last equality. When one of the
partitions is empty as in the case of fundamental matter contributions, the
Nekrasov factor is simplified considerably.

${\mathsf{N}}_{\lambda,\emptyset}(u|q,\kappa)=\prod_{(i,j)\in\lambda}(1-uq^{\lambda_{i}-j}\kappa^{i-1}),\qquad{\mathsf{N}}_{\emptyset,\mu}(u|q,\kappa)=\prod_{(i,j)\in\mu}(1-uq^{-\mu_{i}+j-1}\kappa^{-i}).$
(E.4)

When $N=2$, the factors from the first rows of $\lambda$ and $\mu$ are
contained in ${\mathsf{N}}_{\lambda,\emptyset}^{(0)}(u|q,\kappa)$ and
${\mathsf{N}}_{\emptyset,\mu}^{(1)}(u|q,\kappa)$, respectively, which are
explicitly

$\prod_{k=0}^{\lambda_{1}-1}(1-uq^{k}),\qquad\prod_{k=1}^{\mu_{1}}(1-uq^{-k}\kappa^{-1}).$
(E.5)

Hence, we can see that if $u=q^{-m}$,
${\mathsf{N}}_{\lambda,\emptyset}^{(0)}(u|q,\kappa)$ vanishes for
$m\leq\lambda_{1}-1$. Similarly, if $u\kappa^{-1}=q^{n+1}$,
${\mathsf{N}}_{\emptyset,\mu}^{(1)}(u|q,\kappa)$ vanishes for
$n+1\leq\mu_{1}$.

In the localization formula of the partition function, the numerator of the
contribution from a fixed point $(\lambda^{(1)},\lambda^{(2)})$, which is
identified with the matter contribution is explicitly given by

$\displaystyle\prod_{i,j=1}^{2}\mathsf{N}^{(j-i|2)}_{\emptyset,\lambda^{(j)}}(u_{i}/v_{j}|q,\kappa)\mathsf{N}^{(j-i|2)}_{\lambda^{(i)},\emptyset}(v_{i}/w_{j}|q,\kappa)$
$\displaystyle=\mathsf{N}^{(0)}_{\emptyset,\lambda^{(1)}}(u_{1}/v_{1}|q,\kappa)\mathsf{N}^{(1)}_{\emptyset,\lambda^{(2)}}(u_{1}/v_{2}|q,\kappa)\mathsf{N}^{(1)}_{\emptyset,\lambda^{(1)}}(u_{2}/v_{1}|q,\kappa)\mathsf{N}^{(0)}_{\emptyset,\lambda^{(2)}}(u_{2}/v_{2}|q,\kappa)$
$\displaystyle~{}~{}\mathsf{N}^{(0)}_{\lambda^{(1)},\emptyset}(v_{1}/w_{1}|q,\kappa)\mathsf{N}^{(1)}_{\lambda^{(1)},\emptyset}(v_{1}/w_{2}|q,\kappa)\mathsf{N}^{(1)}_{\lambda^{(2)},\emptyset}(v_{2}/w_{1}|q,\kappa)\mathsf{N}^{(0)}_{\lambda^{(2)},\emptyset}(v_{2}/w_{2}|q,\kappa).$
(E.6)

Let us look at the mass parameter tuning conditions such that the columns of
the Young diagrams of $\lambda^{(1)}$ and $\lambda^{(2)}$ are at most $m$ and
$n$, respectively. We need a zero coming from
$\mathsf{N}^{(0)}_{\lambda^{(1)},\emptyset}(v_{1}/w_{1}|q,\kappa)$ or
$\mathsf{N}^{(1)}_{\emptyset,\lambda^{(1)}}(u_{2}/v_{1}|q,\kappa)$, namely

$(v_{1}/w_{1})=q^{-m},\qquad(u_{2}/v_{1})\kappa^{-1}=q^{m+1}.$ (E.7)

Similarly we need a zero coming from
$\mathsf{N}^{(0)}_{\lambda^{(2)},\emptyset}(v_{2}/w_{2}|q,\kappa)$ or
$\mathsf{N}^{(1)}_{\emptyset,\lambda^{(2)}}(u_{1}/v_{2}|q,\kappa)$, namely

$(v_{2}/w_{2})=q^{-n},\qquad(u_{1}/v_{2})\kappa^{-1}=q^{n+1}.$ (E.8)

Recall that we use the following specialization (see (4.11)) in this paper;

$u_{1}=\frac{qQ}{d_{3}},\quad u_{2}=\frac{\kappa q}{d_{1}};\qquad
v_{1}=1,\quad v_{2}=\frac{Q}{\kappa};\qquad w_{1}=\frac{1}{d_{2}},\quad
w_{2}=\frac{Q}{d_{4}\kappa}.$ (E.9)

Hence, the above conditions (E.7) and (E.8) are translated to

$d_{2}=q^{-m},\qquad d_{1}=q^{-m},$ (E.10)

and

$d_{4}=q^{-n},\qquad d_{3}=q^{-n}.$ (E.11)

The conditions for $d_{1}$ and $d_{2}$ are the same and those for $d_{3}$ and
$d_{4}$ are also the same. Thus, there are four equivalent choices for the
truncation of the Young diagrams. In the main text of the paper, we choose
$d_{2}=q^{-m}$ and $d_{3}=q^{-n}$.

##  Appendix F Affine Laumon space and orbifolded Nekrasov factor

The affine Laumon space is the moduli space of parabolic torsion free sheaves
on $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Originally in [35],[36], a partial
compactification of the space of quasi-maps from $\mathbb{P}^{1}$ to the flag
variety of $GL_{N}$ was defined. The geometric representation theory of the
compactification is controlled by $\mathfrak{gl}_{N}$. There is an action of
the $\mathfrak{gl}_{N}$-Yangian on the equivariant cohomology of the Laumon
space [25]. The affine Laumon space is an affine analogue, where the base
manifold is replaced by $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and we have an
action of the affine Yangian of $\mathfrak{gl}_{N}$. In [44] it was shown that
$U_{\mathsf{v}}(\widehat{\mathfrak{gl}}_{n})$ acts on the equivariant
$K$-theory of the affine Laumon space, which cannot be irrelevant to the
present work. In algebraic geometry such “affine” quasi-maps are described by
torsion free sheaves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with a canonical
framing at $\\{\infty\\}\times\\{\infty\\}$ and a parabolic structure on
$\mathbb{P}^{1}\times\\{0\\}$.

In the context of four dimensional $\mathcal{N}=2$ supersymmetric gauge
theories the affine Laumon space was first featured in [15],[16], where the
conjecture on the relation of the Nekrasov partition function and the Seiberg-
Witten prepotential was proved. Then as a generalization of the AGT
correspondence for Virasoro and $\mathcal{W}_{N}$ conformal blocks, Alday-
Tachikawa [6] pointed out that we could employ the affine Laumon space to
describe the theory with a surface defect whose classification by the
embedding of $\mathfrak{sl}_{2}$ subalgebra into $\mathfrak{sl}_{N}$ is
exactly the same as that of the parabolic structure on
$\mathbb{P}^{1}\times\\{0\\}$. In [6] it is conjectured when the defect is of
the full type the instanton partition function gives a conformal block for the
$\mathfrak{sl}_{N}$ current algebra212121For the codimension two surface
defect of general type the conjecture is that the corresponding chiral algebra
is the $\mathcal{W}$ algebra obtained from Drinfeld-Sokolov reduction of
$\widehat{\mathfrak{sl}}_{N}$ current algebra, see for example [33]..

It is known that the parabolic sheaves are equally described by the sheaves
with the $\mathbb{Z}_{M}$ orbifold action
$\mathbb{P}^{1}\times\mathbb{P}^{1}\ni(z,w)\to(z,\omega w)$ with
$\omega^{M}=1$. Moreover a natural toric action on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ induces that on the affine Laumon space.
The fixed points of the toric action are labelled by $N$-tuples of Young
diagrams, which allows us to use the technique of the equivariant
localization. In practice the relevant quiver is called the chain-saw quiver
[26]. In particular, the equivariant character of the Yangian action on the
equivariant cohomology of the affine Laumon space was computed in [25]. It is
straightforward to up-lift the formula in [25] to the $K$ theory version,
which we use in the present paper. On the method of orbifolding for the
computation of the instanton partition function with a surface defect, see for
example [33],[19],[45],[46] and [47].

In [17] the relation of the asymptotically free Macdonald polynomials and the
geometric representation theory of the Laumon space was clarified. In an
attempt at generalizing this result to an affine version, a formula of the
non-stationary Ruijsenaars function is obtained in [52] based on the affine
screening operators. The non-stationary Ruijsenaars function proposed in [52]
agrees with the instanton partition function obtained from the equivariant
Euler character of of the affine Laumon space222222Physically this corresponds
to the gauge theory with adjoint hypermultiplet, sometimes called
$\mathcal{N}=2^{*}$ theory.. For the relation of the formula in [52], which we
employ in the present paper, and the formula of the equivariant character in
[25], see for example, Appendix B of [12]. See also [43] on the relations of
the geometry of affine Laumon space, intertwines of the quantum toroidal
algebra and associated integrable systems.

As we have seen in Appendix E, the total Nekrasov factor

$\displaystyle{\mathsf{N}}_{\lambda,\mu}(u|q,\kappa)=$
$\displaystyle\prod_{(i,j)\in\lambda}(1-uq^{\lambda_{i}-j}\kappa^{-\mu^{\vee}_{j}+i-1})\cdot\prod_{(i,j)\in\mu}(1-uq^{-\mu_{i}+j-1}\kappa^{\lambda^{\vee}_{j}-i}),$
$\displaystyle=$
$\displaystyle\prod_{i,j=1}^{\infty}\frac{(uq^{\lambda_{j}-\mu_{i}}\kappa^{j-i-1};q)_{\infty}}{(uq^{\lambda_{j}-\mu_{i}}\kappa^{j-i};q)_{\infty}}\cdot\frac{(u\kappa^{j-i};q)_{\infty}}{(u\kappa^{j-i-1};q)_{\infty}}$
(F.1)

can be factorized into
${\mathsf{N}}^{(k|n)}_{\lambda,\mu}(u|q,\kappa)~{}(0\leq k\leq n-1)$ according
to the power of $\kappa$. The combinatorial identities in [11] (see (2.8) and
(2.13)) also show the following “transposed” formula for the total Nekrasov
factor;

$\displaystyle{\mathsf{N}}_{\lambda,\mu}(u|q,t=\kappa^{-1})=$
$\displaystyle\prod_{(i,j)\in\mu}(1-uq^{\lambda_{i}-j}t^{\mu^{\vee}_{j}-i+1})\cdot\prod_{(i,j)\in\lambda}(1-uq^{-\mu_{i}+j-1}t^{-\lambda^{\vee}_{j}+i})$
$\displaystyle=$
$\displaystyle\prod_{i,j=1}^{\infty}\frac{(ut^{\mu_{i}^{\vee}-\lambda_{j}^{\vee}+1}q^{j-i};t)_{\infty}}{(ut^{\mu_{i}^{\vee}-\lambda_{j}^{\vee}+1}q^{j-i-1};t)_{\infty}}\cdot\frac{(utq^{j-i-1};t)_{\infty}}{(utq^{j-i};t)_{\infty}}.$
(F.2)

Compared with (F), the partitions $\lambda$ and $\mu$ in the formula (F) are
exchanged in the product over the boxes of Young diagram, while the form of
each factor is kept intact. Factorization of (F) according to the power of
$\kappa$ gives the following formula of the orbifolded Nekrasov factor;

$\mathsf{N}_{\lambda\mu}^{(k|n)}=\prod_{i\leq
j}\prod_{\ell=0\atop\lambda_{j+1}^{\vee}-\mu_{i}^{\vee}+\ell\equiv
k}^{\lambda_{j}^{\vee}-\lambda_{j+1}^{\vee}-1}(1-q^{j-i}\kappa^{\lambda_{j+1}^{\vee}-\mu_{i}^{\vee}+\ell})\cdot\prod_{i\leq
j}\prod_{\ell=0\atop\lambda_{i}^{\vee}-\mu_{j}^{\vee}+\ell\equiv
k}^{\mu_{j}^{\vee}-\mu_{j+1}^{\vee}-1}(1-q^{i-j-1}\kappa^{\lambda_{i}^{\vee}-\mu_{j}^{\vee}+\ell}),$
(F.3)

where we have substituted $u=1$ for simplicity.

Usually the Nekrasov factor ${\mathsf{N}}_{\lambda,\mu}(u|q,\kappa)$ is
expressed in terms of $(u;q)_{n}$. But we find in the case of the affine
Laumon space it is more appropriate to use

$\displaystyle[u;q]_{n}=u^{-n/2}q^{-n(n-1)/4}(u;q)_{n}$
$\displaystyle=(u^{-1/2}-u^{1/2})(q^{-1/2}u^{-1/2}-q^{1/2}u^{1/2})\cdots(q^{-(n-1)/2}u^{-1/2}-q^{(n-1)/2}u^{1/2}).$
(F.4)

When $n\to\infty$ we have to regularize the monomial factor which connects
$[u;q]_{n}$ and $(u;q)_{n}$. One of the ways of the regularization is to
define $[u;q]_{\infty}$ by

$[u;q]_{\infty}:=\frac{(u;q)_{\infty}}{\vartheta_{q^{1/2}}(-u^{1/2})}=\frac{(u^{1/2};q^{1/2})_{\infty}}{(-q^{1/2}u^{-1/2};q^{1/2})_{\infty}},$
(F.5)

where $\vartheta_{p}(z):=(z;p)_{\infty}(pz^{-1};p)_{\infty}$. One can check

$\displaystyle\frac{[u;q]_{\infty}}{[q^{n}u;q]_{\infty}}$
$\displaystyle=\frac{(u^{1/2};q^{1/2})_{\infty}}{(-q^{1/2}u^{-1/2};q^{1/2})_{\infty}}\frac{(-q^{(1-n)/2}u^{-1/2};q^{1/2})_{\infty}}{(q^{n/2}u^{1/2};q^{1/2})_{\infty}}$
$\displaystyle=(u^{1/2};q^{1/2})_{n}(-q^{(1-n)/2}u^{-1/2};q^{1/2})_{n}=[u;q]_{n}.$
(F.6)

In particular we have $[u;q]_{\infty}=(u^{-1/2}-u^{1/2})[qu;q]_{\infty}$. We
also find

$\frac{[u;q]_{\infty}}{[q/u;q]_{\infty}}=\frac{(u^{1/2};q^{1/2})_{\infty}}{(-q^{1/2}u^{-1/2};q^{1/2})_{\infty}}\frac{(-u^{1/2};q^{1/2})_{\infty}}{(q^{1/2}u^{-1/2};q^{1/2})_{\infty}}=\frac{(u;q)_{\infty}}{(q/u;q)_{\infty}}.$
(F.7)

It is possible to express the result of the selection rule in the orbifolded
Nekrasov factor (F.3) in terms of the floor function $\lfloor\bullet\rfloor$.
In the following manipulations we use the formulas for the floor function;

$\lfloor\frac{\ell}{n}\rfloor+1=-\lfloor\frac{-\ell-1}{n}\rfloor,\qquad\lfloor\frac{\ell+m}{n}\rfloor=\lfloor\frac{\ell}{n}\rfloor+\lfloor\frac{(\ell)+m}{n}\rfloor,\qquad\ell,m\in\mathbb{Z},$
(F.8)

where $(\ell)$ denotes the residue of an integer $\ell$ modulo $n$.

######  Proposition F.1.

The orbifolded Nekrasov factor (F.3) is given by the following formula;

$\displaystyle\mathsf{N}_{\lambda\mu}^{(k|n)}$ $\displaystyle=\prod_{i\leq
j}[q^{j-i}\kappa^{k+n\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor-n\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor};\kappa^{n}]_{\lfloor\frac{\lambda_{j}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor}$
$\displaystyle\times\prod_{i\leq
j}[q^{i-j-1}\kappa^{k+n\lfloor\frac{\lambda_{i}^{\vee}+n-1-k-(\mu_{j}^{\vee})}{n}\rfloor-n\lfloor\frac{\mu_{j}^{\vee}}{n}\rfloor};\kappa^{n}]_{\lfloor\frac{\mu_{j}^{\vee}+k+(-\lambda_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\mu_{j+1}^{\vee}+k+(-\lambda_{i}^{\vee})}{n}\rfloor}$
$\displaystyle=\prod_{i\leq
j}[q^{j-i}\kappa^{k+n\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor-n\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor};\kappa^{n}]_{\lfloor\frac{\lambda_{j}^{\vee}-1-k-(\mu_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\lambda_{j+1}^{\vee}-1-k-(\mu_{i}^{\vee})}{n}\rfloor}$
$\displaystyle\times\prod_{i\leq
j}[q^{i-j-1}\kappa^{k+n\lfloor\frac{\lambda_{i}^{\vee}+n-1-k-(\mu_{j}^{\vee})}{n}\rfloor-n\lfloor\frac{\mu_{j}^{\vee}}{n}\rfloor};\kappa^{n}]_{\lfloor\frac{\mu_{j}^{\vee}+k+(-\lambda_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\mu_{j+1}^{\vee}+k+(-\lambda_{i}^{\vee})}{n}\rfloor}.$
(F.9)

###### Proof.

Since the selection rule on $\ell$ is imposed by $\equiv$ (the equality mod
$n$), $\mathsf{N}_{\lambda\mu}^{(k|n)}$ can be rewritten in terms of
$\kappa^{n}$-shifted factorial $(z_{0};\kappa^{n})_{m}$. Let us check that the
initial value $z_{0}$ and the number $m$ of factors for each shifted factorial
agree with those in the formula (F.1). To compute them for the first shifted
factorial, let us assume

$\mu_{i}^{\vee}=q_{1}n+r_{1},\qquad\lambda_{j+1}^{\vee}=q_{2}n+r_{2},\qquad\lambda_{i}^{\vee}=q_{3}n+r_{3},\qquad(0\leq
r_{i}\leq n-1).$ (F.10)

Namely we may write, for example,
$q_{1}=\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor$ and $r_{1}=(\mu_{i}^{\vee})$.
But in the following we will use $q_{i}$ and $r_{i}$ for simplicity of
notations. The selection rule in (F.3) tells

$\ell\equiv k+r_{1}-r_{2}.$ (F.11)

Since $0\leq k,r_{1},r_{2}\leq n-1$, we have $1-n\leq k+r_{1}-r_{2}\leq 2n-2$.
Accordingly we can write the initial value of $\ell$ uniformly as

$\ell_{0}=k+r_{1}-r_{2}-n\lfloor\frac{k+r_{1}-r_{2}}{n}\rfloor.$ (F.12)

Hence, we see

$z_{0}=q^{j-i}\kappa^{\lambda_{j+1}^{\vee}-\mu_{i}^{\vee}+\ell_{0}}=q^{j-i}\kappa^{k+n\lfloor\frac{\lambda_{j+1}^{\vee}-\mu_{i}^{\vee}}{n}\rfloor-n\lfloor\frac{k+r_{1}-r_{2}}{n}\rfloor}.$
(F.13)

Now let us count the number of factors in the first shifted factorial. Since
the initial value of $\ell$ is given by (F.12) and the upper bound is
$\lambda_{j}^{\vee}-\lambda_{j+1}^{\vee}-1$, we see

$\displaystyle m$ $\displaystyle=$
$\displaystyle\lfloor\frac{\lambda_{j}^{\vee}-\lambda_{j+1}^{\vee}-1-\ell_{0}}{n}\rfloor+1$
(F.14) $\displaystyle=$
$\displaystyle(q_{3}-q_{2})+1+\lfloor\frac{r_{3}-r_{1}-k-1}{n}\rfloor+\lfloor\frac{k+r_{1}-r_{2}}{n}\rfloor$
$\displaystyle=$
$\displaystyle\lfloor\frac{\lambda_{j}^{\vee}-r_{1}-k-1}{n}\rfloor+\lfloor\frac{k+r_{1}-\lambda_{j+1}^{\vee}}{n}\rfloor+1$

Since

$\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor=q_{1},$ (F.15)

and

$\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor=\begin{cases}q_{2}+1,\qquad
0\leq r_{2}-r_{1}-1-k\\\ q_{2},\qquad-n\leq r_{2}-r_{1}-1-k<0\\\
q_{2}-1,\qquad r_{2}-r_{1}-1-k<-n\end{cases}$ (F.16)

the power of $\kappa$ is

$\begin{cases}\lambda_{j+1}^{\vee}-\mu_{i}^{\vee}+k+r_{1}-r_{2}+n,\qquad 1\leq
r_{2}-r_{1}-k\\\ \lambda_{j+1}^{\vee}-\mu_{i}^{\vee}+k+r_{1}-r_{2},\qquad
1-n\leq r_{2}-r_{1}-k<1\\\
\lambda_{j+1}^{\vee}-\mu_{i}^{\vee}+k+r_{1}-r_{2}-n,\qquad
r_{2}-r_{1}-k<1-n.\end{cases}$ (F.17)

Comparing this with (F.12), we can confirm an agreement. For the number of
factors, what we have to check is

$\lfloor\frac{k+r_{1}-\lambda_{j+1}^{\vee}}{n}\rfloor+1=-\lfloor\frac{\lambda_{j+1}^{\vee}-1-k-(\mu_{i}^{\vee})}{n}\rfloor.$
(F.18)

But this is a special case of the first formulas of (F.8). We have checked a
complete agreement of the first factorial.

Let us proceed to the second factorial. Concerning the initial condition, we
can see the formula for the second is obtained from the first simply by the
replacement;

$(\mu_{i}^{\vee},\lambda_{j+1}^{\vee})\longrightarrow(\mu_{j}^{\vee},\lambda_{i}^{\vee})$
(F.19)

thus an agreement is proved by appropriate changes of variables. However,
counting the number of factors is more involved. Let us assume

$\lambda_{i}^{\vee}=q_{4}n+r_{4},\qquad\mu_{j}^{\vee}=q_{5}n+r_{5},\qquad\mu_{j+1}^{\vee}=q_{6}n+r_{6},\qquad(0\leq
r_{i}\leq n-1).$ (F.20)

Then a similar computation as before gives the number of factors in the
formula (F.3)

$m^{\prime}=\lfloor\frac{\mu_{j}^{\vee}-\mu_{j+1}^{\vee}-1-\ell_{0}^{\prime}}{n}\rfloor+1,$
(F.21)

where

$\ell_{0}^{\prime}=k+r_{5}-r_{4}-n\lfloor\frac{k+r_{5}-r_{4}}{n}\rfloor.$
(F.22)

Hence

$\displaystyle m^{\prime}$ $\displaystyle=$
$\displaystyle(q_{5}-q_{6})+1+\lfloor\frac{r_{4}-r_{6}-k-1}{n}\rfloor+\lfloor\frac{k+r_{5}-r_{4}}{n}\rfloor$
(F.23) $\displaystyle=$
$\displaystyle\lfloor\frac{r_{4}-\mu_{j+1}^{\vee}-k-1}{n}\rfloor+\lfloor\frac{\mu_{j}^{\vee}+k-r_{4}}{n}\rfloor+1.$

We note $(-\lambda_{i}^{\vee})=n-r_{4}$ for $r_{4}\neq 0$, but
$(-\lambda_{i}^{\vee})=-r_{4}=0$ for $r_{4}\neq 0$. But such a difference does
not matter, since we take a difference of the floor function. Hence we can
safely use $(-\lambda_{i}^{\vee})=-r_{4}$ and what we have to check is

$\lfloor\frac{r_{4}-\mu_{j+1}^{\vee}-k-1}{n}\rfloor+1=-\lfloor\frac{\mu_{j+1}^{\vee}+k+(-\lambda_{i}^{\vee})}{n}\rfloor,$
(F.24)

which follows from the first formula of (F.8). ∎

We can also write the formula of Proposition F.1 in terms of the ratio of the
regularized infinite products $[u;q]_{\infty}$ defined by (F.5). Using the
formulas (F) and (F.8), we find

$\displaystyle\mathsf{N}_{\lambda,\mu}^{(k|n)}(u|q,\kappa)$ $\displaystyle=$
$\displaystyle\frac{\displaystyle{\prod_{j>i\geq
1}}[uq^{j-i-1}\kappa^{k-n\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor+n\lfloor\frac{\lambda_{j}^{\vee}}{n}\rfloor-n\lfloor\frac{-(\lambda_{j}^{\vee})+k+(\mu_{i}^{\vee})}{n}\rfloor};\kappa^{n}]_{\infty}}{\displaystyle{\prod_{j\geq
i\geq
1}}[uq^{j-i}\kappa^{k-n\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor+n\lfloor\frac{\lambda_{j}^{\vee}}{n}\rfloor-n\lfloor\frac{-(\lambda_{j}^{\vee})+k+(\mu_{i}^{\vee})}{n}\rfloor};\kappa^{n}]_{\infty}}$
(F.25) $\displaystyle~{}~{}\times\frac{\displaystyle{\prod_{i\geq j\geq
1}}[uq^{j-i-1}\kappa^{k+n\lfloor\frac{\lambda_{j}^{\vee}}{n}\rfloor-n\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor-n\lfloor\frac{(\mu_{i}^{\vee})+k-(\lambda_{j}^{\vee})}{n}\rfloor};\kappa^{n}]_{\infty}}{\displaystyle{\prod_{i>j\geq
1}}[uq^{j-i}\kappa^{k+n\lfloor\frac{\lambda_{j}^{\vee}}{n}\rfloor-n\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor-n\lfloor\frac{(\mu_{i}^{\vee})+k-(\lambda_{j}^{\vee})}{n}\rfloor};\kappa^{n}]_{\infty}}$
$\displaystyle=$
$\displaystyle\prod_{i,j=1}^{\infty}\frac{[uq^{j-i-1}\kappa^{k-n\lfloor\frac{\mu_{i}^{\vee}+k-\lambda_{j}^{\vee}}{n}\rfloor};\kappa^{n}]_{\infty}}{[uq^{j-i}\kappa^{k-n\lfloor\frac{\mu_{i}^{\vee}+k-\lambda_{j}^{\vee}}{n}\rfloor};\kappa^{n}]_{\infty}}\frac{[uq^{j-i}\kappa^{k};\kappa^{n}]_{\infty}}{[uq^{j-i-1}\kappa^{k};\kappa^{n}]_{\infty}}.$

For later convenience, let us express the orbifolded Nekrasov factor in terms
of $t=\kappa^{-n}$-shifted factorials. We have

$\displaystyle\mathsf{N}_{\lambda,\mu}^{(k|n)}(u|q,t)$ $\displaystyle=$
$\displaystyle\prod_{j\geq i\geq
1}[uq^{j-i}t^{-\frac{k}{n}+\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor-\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor};t^{-1}]_{\lfloor\frac{\lambda_{j}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor}$
(F.26) $\displaystyle~{}~{}\times\prod_{j\geq i\geq
1}[uq^{i-j-1}t^{-\frac{k}{n}-\lfloor\frac{\lambda_{i}^{\vee}}{n}\rfloor+\lfloor\frac{\mu_{j}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor};t^{-1}]_{\lfloor\frac{\mu_{j}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\mu_{j+1}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor}$
$\displaystyle=$ $\displaystyle\prod_{j\geq i\geq
1}[uq^{j-i}t^{1-\frac{k}{n}+\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor-\lfloor\frac{\lambda_{j}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor};t]_{\lfloor\frac{\lambda_{j}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor}$
$\displaystyle~{}~{}\times\prod_{j\geq i\geq
1}[uq^{i-j-1}t^{1-\frac{k}{n}-\lfloor\frac{\lambda_{i}^{\vee}}{n}\rfloor+\lfloor\frac{\mu_{j+1}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor};t]_{\lfloor\frac{\mu_{j}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor-\lfloor\frac{\mu_{j+1}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor}$
$\displaystyle=$ $\displaystyle\prod_{j\geq i\geq
1}\frac{[uq^{j-i}t^{1-\frac{k}{n}+\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor-\lfloor\frac{\lambda_{j}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor};t]_{\infty}}{[uq^{j-i}t^{1-\frac{k}{n}+\lfloor\frac{\mu_{i}^{\vee}}{n}\rfloor-\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k-(\mu_{i}^{\vee})}{n}\rfloor};t]_{\infty}}$
$\displaystyle~{}~{}\times\prod_{j\geq i\geq
1}\frac{[uq^{i-j-1}t^{1-\frac{k}{n}-\lfloor\frac{\lambda_{i}^{\vee}}{n}\rfloor+\lfloor\frac{\mu_{j+1}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor};t]_{\infty}}{[uq^{i-j-1}t^{1-\frac{k}{n}-\lfloor\frac{\lambda_{i}^{\vee}}{n}\rfloor+\lfloor\frac{\mu_{j}^{\vee}+k-(\lambda_{i}^{\vee})}{n}\rfloor};t]_{\infty}},$

where we have used

$[u;t^{-1}]_{n}=[ut^{-n+1};t]_{n},\qquad n>0.$ (F.27)

Finally the same manipulation as the last equality of (F.25) gives

$\displaystyle\mathsf{N}_{\lambda,\mu}^{(k|n)}(u|q,t)$ $\displaystyle=$
$\displaystyle\prod_{i,j=1}^{\infty}\frac{[uq^{j-i}t^{1-\frac{k}{n}+\lfloor\frac{\mu_{i}^{\vee}+k-\lambda_{j}^{\vee}}{n}\rfloor};t]_{\infty}}{[uq^{j-i-1}t^{1-\frac{k}{n}+\lfloor\frac{\mu_{i}^{\vee}+k-\lambda_{j}^{\vee}}{n}\rfloor};t]_{\infty}}\frac{[uq^{j-i-1}t^{1-\frac{k}{n}};t]_{\infty}}{[uq^{j-i}t^{1-\frac{k}{n}};t]_{\infty}}.$
(F.28)

When one of the partitions is empty, $\mathsf{N}_{\lambda,\mu}^{(k|n)}$
simplifies as follows;

$\displaystyle\mathsf{N}_{\lambda,\emptyset}^{(k|n)}(u|q,\kappa)$
$\displaystyle=$ $\displaystyle\frac{\displaystyle{\prod_{j\geq i\geq
1}}[uq^{j-i}\kappa^{k};\kappa^{n}]_{\lfloor\frac{\lambda_{j}^{\vee}+n-1-k}{n}\rfloor}}{\displaystyle{\prod_{j\geq
i\geq
1}}[uq^{(j+1)-(i+1)}\kappa^{k};\kappa^{n}]_{\lfloor\frac{\lambda_{j+1}^{\vee}+n-1-k}{n}\rfloor}}=\prod_{i\geq
1}[uq^{i-1}\kappa^{k};\kappa^{n}]_{\lfloor\frac{\lambda_{i}^{\vee}+n-1-k}{n}\rfloor},$
(F.29) $\displaystyle\mathsf{N}_{\emptyset,\mu}^{(k|n)}(u|q,\kappa)$
$\displaystyle=$ $\displaystyle\frac{\displaystyle{\prod_{j\geq i\geq
1}}[uq^{i-j-1}\kappa^{k-n\lfloor\frac{\mu_{j}^{\vee}+k}{n}\rfloor};\kappa^{n}]_{\lfloor\frac{\mu_{j}^{\vee}+k}{n}\rfloor}}{\displaystyle{\prod_{j\geq
i\geq
1}}[uq^{i-(j+1)}\kappa^{k-n\lfloor\frac{\mu_{j+1}^{\vee}+k}{n}\rfloor};\kappa^{n}]_{\lfloor\frac{\mu_{j+1}^{\vee}+k}{n}\rfloor}}=\prod_{i\geq
1}[uq^{-i}\kappa^{k-n\lfloor\frac{\mu_{i}^{\vee}+k}{n}\rfloor};\kappa^{n}]_{\lfloor\frac{\mu_{i}^{\vee}+k}{n}\rfloor}.$

##  Appendix G (Anti-)symmetrization in a factorized form

A basis of the space of cohomology classes for the $N$-tuple Jackson integral
is defined by using the anti-symmetrization of the variables
$z=(z_{1},\ldots,z_{N})$ with the weight function $\Delta(q,z)$ (see (3.2)).
In general the result of (anti-)symmetrization is expanded in terms of
symmetric polynomials (divided by the Vandermonde determinant for anti-
symmetrization). However, from the viewpoint of the correspondence of the
Jackson integral and the Bethe vector for quantum integrable system [48], what
we need is the (anti-)symmetrization that keeps the factorization. One can
rephrase it as a summation over partitions of the index set
$[N]:=\\{1,2,\ldots,N\\}$. We give a proof for the following well-known
formulae for completeness.

######  Proposition G.1.

Let $I_{1}\sqcup\cdots\sqcup I_{s}$ be a partition of $\\{1,2,\ldots,N\\}$
with $|I_{a}|=k_{a}\geq 1$ and ${\mathcal{A}}$ be the anti-symmetrization
operator of $z=(z_{1},\ldots,z_{N})$. For any functions $f_{a}(x)~{}(1\leq
a\leq s)$, we have

$\begin{array}[]{l}\displaystyle{\displaystyle\frac{1}{\Delta(1,z)}}{\mathcal{A}}\Big{(}\prod_{i_{1}=1}^{k_{1}}f_{1}(z_{i_{1}})\prod_{i_{2}=1}^{k_{2}}f_{2}(z_{k_{1}+i_{2}})\cdots\prod_{i_{s}=1}^{k_{s}}f_{s}(z_{k_{1}+\cdots+k_{s-1}+i_{s}})\Delta(q,z)\Big{)}\\\
\displaystyle=\prod_{a=1}^{s}[k_{a}]_{q^{-1}}!\sum_{I_{1}\sqcup\cdots\sqcup
I_{s}}\Big{\\{}\prod_{a=1}^{s}\prod_{i_{a}\in
I_{a}}f_{a}(z_{i_{a}})\prod_{1\leq a<b\leq s}\prod_{i\in I_{a}}\prod_{j\in
I_{b}}{\displaystyle\frac{z_{i}-q^{-1}z_{j}}{z_{i}-z_{j}}}\Big{\\}},\end{array}$
(G.1)

where the sum $\displaystyle\sum_{I_{1}\sqcup\cdots\sqcup I_{s}}$ on the right
hand side is taken over the partitions $I_{1}\sqcup\cdots\sqcup I_{s}$.

###### Proof.

For a permutation $\sigma\in S_{N}$, we define
$I_{a}=\\{\sigma(k_{1}+\cdots+k_{a-1}+i)|\ 1\leq i\leq k_{a}\\}$. Since

${\displaystyle\frac{1}{\Delta(1,z)}}{\mathcal{A}}\Big{(}F(z)\Delta(q,z)\Big{)}=\sum_{\sigma\in
S_{N}}\sigma\Big{(}F(z){\displaystyle\frac{\Delta(q,z)}{\Delta(1,z)}}\Big{)},$

for any $F(z)=F(z_{1},\ldots,z_{N})$, we can recast the left hand side of
(G.1) to

$\sum_{\sigma\in S_{N}}\Big{(}\prod_{a=1}^{s}\prod_{i\in
I_{a}}f_{a}(z_{i}){\displaystyle\frac{\Delta(q,z_{I_{a}})}{\Delta(z_{I_{a}})}}\prod_{1\leq
a<b\leq s}\prod_{i\in I_{a}}\prod_{j\in
I_{b}}{\displaystyle\frac{z_{i}-q^{-1}z_{j}}{z_{i}-z_{j}}}\Big{)}.$

Let $S_{I}$ be the symmetric group of a set $I$. Then according to the coset
decomposition $\displaystyle S_{N}=\mathop{\sqcup}_{I_{1}\sqcup\cdots\sqcup
I_{s}}S_{I_{1}}\times\cdots\times S_{I_{s}}$, we take the sum in the following
manner;

$\sum_{\sigma\in S_{N}}=\sum_{I_{1}\sqcup\cdots\sqcup I_{s}}\sum_{w_{1}\in
S_{I_{1}}}\cdots\sum_{w_{s}\in S_{I_{s}}}.$

By applying the formula ([37]: Chap.III eq.(1.4))

$\sum_{w_{a}\in
S_{I_{a}}}w_{a}\Big{(}{\displaystyle\frac{\Delta(q,z_{I_{a}})}{\Delta(1,z_{I_{a}})}}\Big{)}=[k_{a}]_{q^{-1}}!,$

for each sum over $S_{I_{a}}$, we obtain the desired result (G.1). ∎

##  Appendix H Shakirov’s equation as a coupled system

Consider Shakirov’s equation in the form

$\displaystyle\psi=$ $\displaystyle\SS T_{t,\Lambda}^{-1}T_{qtQ,x}^{-1}\psi,$
(H.1) $\displaystyle\SS=$
$\displaystyle{\displaystyle\frac{1}{\varphi(qx)\varphi(\frac{\Lambda}{x})}}\mathcal{B}{\displaystyle\frac{\varphi(\Lambda)\varphi(q^{-1}d_{1}d_{2}d_{3}d_{4}\Lambda)}{\varphi(-d_{1}x)\varphi(-d_{2}x)\varphi(-d_{3}\frac{\Lambda}{x})\varphi(-d_{4}\frac{\Lambda}{x})}}\mathcal{B}{\displaystyle\frac{1}{\varphi(\frac{d_{1}d_{2}}{q}x)\varphi(d_{3}d_{4}\frac{\Lambda}{x})}}.$

It has a unique formal series solution normalized as
$\psi=1+{\mathcal{O}}(x,\frac{\Lambda}{x})$. In terms of the affine Laumon
partition function $\mathcal{Z}_{\mathrm{AL}}$, the solution is given by

$\displaystyle\psi(\Lambda,x)=$
$\displaystyle~{}\mathcal{Z}_{\mathrm{AL}}\left(\left.\left.\begin{array}[]{c}u_{1},u_{2}\\\
v_{1},v_{2}\\\
w_{1},w_{2}\end{array}\right|x_{1},\frac{\Lambda_{1}}{x_{1}}\right|q,t\right)$
(H.5) $\displaystyle=$
$\displaystyle~{}\mathcal{Z}_{\mathrm{AL}}\left(\left.\left.\begin{array}[]{c}\frac{qQ}{d_{3}},\frac{q\kappa}{d_{1}}\\\
1,\frac{Q}{\kappa}\\\ \frac{1}{d_{2}},\frac{Q}{\kappa
d_{4}}\end{array}\right|-\frac{\sqrt{Qd_{1}d_{2}}}{\kappa}x,-\sqrt{\frac{d_{3}d_{4}}{q^{2}Q}}\frac{\Lambda}{x}\right|q,\kappa^{-2}\right).$
(H.9)

We will give a coupled form of Shakirov’s equation and its solution. To do
this, we define operators $T$ and $K$ by232323One may use $d_{1}$ (or $d_{3}$)
instead of $d_{2}$ (or $d_{4}$).

$\displaystyle T=$
$\displaystyle\left\\{d_{2}\mapsto{\displaystyle\frac{q}{tQd_{2}}},\quad
d_{4}\mapsto{\displaystyle\frac{qQ}{d_{4}}},\quad\Lambda\mapsto{\displaystyle\frac{d_{2}d_{4}\Lambda}{q}},\quad
x\mapsto-{\displaystyle\frac{d_{2}x}{q}}\right\\},$ $\displaystyle K=$
$\displaystyle{\displaystyle\frac{1}{\varphi(qx)\varphi(\frac{\Lambda}{x})}}\mathcal{B}{\displaystyle\frac{1}{\varphi(-d_{1}x)\varphi(-d_{3}\frac{\Lambda}{x})}}.$
(H.10)

On the parameters $u_{i},v_{i},w_{i},x_{1},\Lambda_{1}$, the operator $T$ acts
as

$T=\left\\{w_{i}\mapsto\frac{tQ}{qw_{i}},\quad
x_{1}\mapsto-\frac{x_{1}}{\sqrt{qtQ}},\quad\Lambda_{1}\mapsto\frac{\Lambda_{1}}{\sqrt{t}}\right\\}.$
(H.11)

In our previous paper [10], we used the renormalized $q$-Borel transformation
$\widetilde{\mathcal{B}}:=T^{-1}_{(qt^{1/2}Q)^{1/2},x}\mathcal{B}$ in writing
down the coupled system in order to make the correspondence with the
$qq$-Painlevé VI equation clear. If this is taken into account, the action of
$T$ on the expansion parameters becomes $x_{1}=x\mapsto-t^{-1/4}x_{1}$ and
hence $x_{2}=\Lambda_{1}/x_{1}\mapsto-t^{-1/4}x_{2}$. The $T$ action is the
same for $x_{1}$ and $x_{2}$ and coincides with the action of the shift
operator $\widetilde{T}_{\mathsf{p},b}$ with $\mathsf{p}=t^{1/4}$, which is
regarded as a square root of the time evolution of the parameters $b_{i}$ in
the $qq$-Painlevé VI equation (See eq.(6.5) in [10]).

######  Proposition H.1.

The normalized solution $\psi$ of (H.1) and its parameter transform
$\chi:=T\psi$ satisfy the following coupled system of equations;

$\begin{cases}\psi=gK\chi,\\\ \chi=TgKT\psi,\end{cases}$ (H.12)

where

$g={\displaystyle\frac{(\frac{td_{2}d_{4}}{q}\Lambda,d_{1}d_{3}\Lambda;q,t)_{\infty}}{(t\Lambda,\frac{td_{1}d_{2}d_{3}d_{4}}{q}\Lambda;q,t)_{\infty}}}.$
(H.13)

Conversely the equation (H.1) follows from the coupled system (H.12).

###### Proof.

From (H.1) and (H), one can verify the following relations:

$\displaystyle T^{2}=T_{t,\Lambda}^{-1}T_{qQt,x}^{-1},$ (H.14) $\displaystyle
TK={\displaystyle\frac{1}{\varphi(-d_{2}x)\varphi(-d_{4}\frac{\Lambda}{x})}}\mathcal{B}{\displaystyle\frac{1}{\varphi(\frac{d_{1}d_{2}}{q}x)\varphi(d_{3}d_{4}\frac{\Lambda}{x})}}T,$
(H.15) $\displaystyle
KTKT=K{\displaystyle\frac{1}{\varphi(-d_{2}x)\varphi(-d_{4}\frac{\Lambda}{x})}}\mathcal{B}{\displaystyle\frac{1}{\varphi(\frac{d_{1}d_{2}}{q}x)\varphi(d_{3}d_{4}\frac{\Lambda}{x})}}T^{2}={\displaystyle\frac{1}{\varphi(\Lambda)\varphi(\frac{d_{1}d_{2}d_{3}d_{4}}{q}\Lambda)}}\SS
T^{2},$ (H.16) $\displaystyle KT\SS
T^{2}=\varphi(\frac{d_{2}d_{4}}{q}\Lambda)\varphi(\frac{d_{1}d_{3}}{t}\Lambda)(KT)^{3}={\displaystyle\frac{\varphi(\frac{d_{2}d_{4}}{q}\Lambda)\varphi(\frac{d_{1}d_{3}}{t}\Lambda)}{\varphi(\Lambda)\varphi(\frac{d_{1}d_{2}d_{3}d_{4}}{q}\Lambda)}}\SS
T^{2}KT.$ (H.17)

From (H.14), (H.17) and (H.1), we have

$\displaystyle KT\psi=KT\SS
T^{2}\psi={\displaystyle\frac{\varphi(\frac{d_{2}d_{4}}{q}\Lambda)\varphi(\frac{d_{1}d_{3}}{t}\Lambda)}{\varphi(\Lambda)\varphi(\frac{d_{1}d_{2}d_{3}d_{4}}{q}\Lambda)}}\SS
T^{2}KT\psi.$ (H.18)

Noting that
$T_{t,\Lambda}^{-1}(g)={\displaystyle\frac{\varphi(\frac{d_{2}d_{4}}{q}\Lambda)\varphi(\frac{d_{1}d_{3}}{t}\Lambda)}{\varphi(\Lambda)\varphi(\frac{d_{1}d_{2}d_{3}d_{4}}{q}\Lambda)}}g$,
we see the function $gKT\psi$ satisfies the same equation as (H.1) with the
same initial condition as $\psi$. By the uniqueness of the solution, we obtain
the first relation $\psi=gKT\psi=gK\chi$. The second relation is the
$T$-transform of the first one. To check the converse we note that

$g\cdot T\cdot
g=\frac{(\Lambda,\frac{d_{1}d_{2}d_{3}d_{4}}{q}\Lambda;q,t)_{\infty}}{(t\Lambda,\frac{td_{1}d_{2}d_{3}d_{4}}{q}\Lambda;q,t)_{\infty}}\cdot
T=\varphi(\Lambda)\varphi(\frac{d_{1}d_{2}d_{3}d_{4}}{q}\Lambda)\cdot T.$
(H.19)

Hence

$\psi=g\cdot KT\cdot g\cdot
KT\psi=\varphi(\Lambda)\varphi(\frac{d_{1}d_{2}d_{3}d_{4}}{q}\Lambda)\cdot
KTKT\psi=\SS T^{2}\psi.$ (H.20)

∎

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