# Proof of some Littlewood identities conjectured by Lee, Rains and Warnaar

Seamus P. Albion  Fakultät für Mathematik, Universität Wien, Oskar-
Morgenstern-Platz 1, A-1090, Vienna, Austria<EMAIL_ADDRESS>
###### Abstract.

We prove a novel pair of Littlewood identities for Schur functions, recently
conjecture by Lee, Rains and Warnaar in the Macdonald case, in which the sum
is over partitions with empty 2-core. As a byproduct we obtain a new
Littlewood identity in the spirit of Littlewood’s original formulae.

Keywords: empty $2$-core, Koornwinder polynomials, Littlewood identities,
Schur functions

###### 2020 Mathematics Subject Classification:

05E05, 33D05, 33D52

##  1\. Introduction

The classical Littlewood identities are the following three summation formulae
for Schur functions

(1.1a) $\displaystyle\sum_{\lambda}s_{\lambda}(x)$
$\displaystyle=\prod_{i\geqslant
1}\frac{1}{1-x_{i}}\prod_{i<j}\frac{1}{1-x_{i}x_{j}}$ (1.1b)
$\displaystyle\sum_{\begin{subarray}{c}\lambda\\\ \text{$\lambda$
even}\end{subarray}}s_{\lambda}(x)$ $\displaystyle=\prod_{i\geqslant
1}\frac{1}{1-x_{i}^{2}}\prod_{i<j}\frac{1}{1-x_{i}x_{j}}$ (1.1c)
$\displaystyle\sum_{\begin{subarray}{c}\lambda\\\ \text{$\lambda^{\prime}$
even}\end{subarray}}s_{\lambda}(x)$
$\displaystyle=\prod_{i<j}\frac{1}{1-x_{i}x_{j}}.$

Each sum is over all partitions $\lambda$ satisfying the given condition where
$\lambda$ even (resp. $\lambda^{\prime}$ even) means $\lambda$ has even rows
(resp. even columns). These were first written down together by Littlewood
[15, p. 238], however (1.1a) was already known to Schur [26]. The identities
(1.1) have since afforded many far-reaching generalisations and have found
applications in areas such as combinatorics, representation theory and
elliptic hypergeometric series. In particular there are many generalisations
of (1.1) at the Schur level [3, 7, 10, 11, 12, 20, 21, 27]. Also see [24] for
comprehensive references to the literature.

The purpose of this note is to prove the Schur function case of a pair of
Littlewood identities for Macdonald polynomials recently conjectured by Lee,
Rains and Warnaar [14, Conjecture 9.5]. To state these we need some notation.
Denote the multiset of hook lengths of a partition $\lambda$ by
$\mathcal{H}_{\lambda}$. We refine this by writing
$\mathcal{H}^{\mathrm{e}/\mathrm{o}}_{\lambda}$ for the submultiset of
even/odd hook lengths. The standard infinite $q$-shifted factorial is given by
$(a;q)_{\infty}:=\prod_{i\geqslant 0}(1-aq^{i})$ and we define a statistic

$b(\lambda):=\sum_{(i,j)\in\lambda}(-1)^{\lambda_{i}+\lambda_{j}^{\prime}-i-j+1}(\lambda_{i}-i),$

in terms of the Young diagram of $\lambda$; see Subsection 2.1 below.

######  Theorem 1.1.

There holds

(1.2) $\sum_{\begin{subarray}{c}\lambda\\\
2\textup{-core}(\lambda)=0\end{subarray}}q^{b(\lambda)}\frac{\prod_{h\in\mathcal{H}^{\mathrm{o}}_{\lambda}}(1-q^{h})}{\prod_{h\in\mathcal{H}^{\mathrm{e}}_{\lambda}}(1-q^{h})}s_{\lambda}(x)=\prod_{i\geqslant
1}\frac{(qx_{i}^{2};q^{2})_{\infty}}{(x_{i}^{2};q^{2})_{\infty}}\prod_{i<j}\frac{1}{1-x_{i}x_{j}},$

and

(1.3) $\sum_{\begin{subarray}{c}\lambda\\\
2\textup{-core}(\lambda)=0\end{subarray}}q^{b(\lambda^{\prime})}\frac{\prod_{h\in\mathcal{H}^{\mathrm{o}}_{\lambda}}(1-q^{h})}{\prod_{h\in\mathcal{H}^{\mathrm{e}}_{\lambda}}(1-q^{h})}s_{\lambda}(x)=\prod_{i\geqslant
1}\frac{(q^{2}x_{i}^{2};q^{2})_{\infty}}{(qx_{i}^{2};q^{2})_{\infty}}\prod_{i<j}\frac{1}{1-x_{i}x_{j}}.$

The condition $2\textup{-core}(\lambda)=0$ generalises both the even
row/column conditions of (1.1b) and (1.1c). Indeed, by Lemma 2.2 we have that
$b(\lambda)=0$ if and only if $\lambda$ is even. Thus in the $q\to 0$ limit
(1.2) and (1.3) collapse to (1.1b) and (1.1c) respectively. In this sense
these identities are in the spirit of Kawanaka’s identity [12, Theorem 1.1]

$\sum_{\lambda}\prod_{h\in\mathcal{H}_{\lambda}}\bigg{(}\frac{1+q^{h}}{1-q^{h}}\bigg{)}s_{\lambda}(x)=\prod_{i\geqslant
1}\frac{(-qx_{i};q)_{\infty}}{(x_{i};q)_{\infty}}\prod_{i<j}\frac{1}{1-x_{i}x_{j}},$

since this reduces to (1.1a) when $q=0$. Unlike Kawanaka’s identity one can
make sense of the $q\to 1$ limit of (1.2) and (1.3). In either case we obtain
the following Littlewood identity.

######  Corollary 1.2.

We have

$\sum_{\begin{subarray}{c}\lambda\\\
2\textup{-core}(\lambda)=0\end{subarray}}\frac{\prod_{h\in\mathcal{H}^{\mathrm{o}}_{\lambda}}(h)}{\prod_{h\in\mathcal{H}^{\mathrm{e}}_{\lambda}}(h)}s_{\lambda}(x)=\prod_{i\geqslant
1}\frac{1}{(1-x_{i}^{2})^{1/2}}\prod_{i<j}\frac{1}{1-x_{i}x_{j}}.$

The outline of the paper is as follows. In the next section we give
preliminaries regarding partitions, Schur functions and Koornwinder
polynomials. In Section 3 we prove a pair of vanishing integrals for Schur
functions again conjectured by Lee, Rains and Warnaar in the Macdonald case
[14, Conjecture 9.2]. Then, in Section 4, we follow the techniques of [24] to
prove the bounded analogues of Theorem 1.1 conjectured in [14, Conjecture
9.4]. The theorem then follows by taking an appropriate limit. We conclude
with a derivation of Corollary 1.2.

##  2\. Partitions and ($\mathrm{BC}_{n}$)-symmetric functions

###  2.1. Partitions

A partition $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\dots)$ is a weakly
decreasing sequence of nonnegative integers such that finitely many
$\lambda_{i}$ are nonzero. The sum of the entries is denoted
$\lvert\lambda\rvert:=\lambda_{1}+\lambda_{2}+\lambda_{3}+\cdots$ and if
$\lvert\lambda\rvert=n$ we say $\lambda$ is a partition of $n$, written
$\lambda\vdash n$. Nonzero entries are called parts, and the number of parts
is called the length, denoted $l(\lambda)$. We denote by $\mathscr{P}$ the set
of all partitions and by $\mathscr{P}_{n}$ the set of all partitions with
length at most $n$. In particular $\mathscr{P}_{0}=\\{0\\}$ where $0$ denotes
the unique partition of zero. If $\lambda\in\mathscr{P}_{n}$ we write
$\lambda+\delta$ for the partition
$(\lambda_{1}+n-1,\lambda_{2}+n-2,\dots,\lambda_{n})$. The number
$m_{i}(\lambda)$ of occurrences of an integer $i$ as a part of $\lambda$ is
called the multiplicity. Sometimes we express a partition in terms of its
multiplicities as
$\lambda=(1^{m_{1}(\lambda)}2^{m_{2}(\lambda)}3^{m_{3}(\lambda)}\cdots)$. We
write $\mu\subset\lambda$ if the partition $\mu$ is contained in $\lambda$,
i.e. if $\mu_{i}\leqslant\lambda_{i}$ for all $i\geqslant 1$. If
$\lambda\subseteq(m^{n})$ for some nonnegative integers $m,n$, then we write
$(m^{n})-\lambda$ for the complement of $\lambda$ inside $(m^{n})$, that is,
$(m^{n})-\lambda:=(m-\lambda_{n},m-\lambda_{n-1},\dots,m-\lambda_{1})$. A
partition is identified with its Young diagram, which is the left-justified
array of squares with $\lambda_{i}$ squares in row $i$ with $i$ increasing
downward. For example

is the Young diagram of $(6,4,3,1)$. The conjugate of a partition, written
$\lambda^{\prime}$, is obtained by reflecting the Young diagram in the main
diagonal, so that $(6,4,3,1)^{\prime}=(4,3,3,2,1,1)$. The arm and leg lengths
of a square $s=(i,j)\in\lambda$ are given by

$a(s):=\lambda_{i}-j\quad\text{and}\quad l(s):=\lambda_{j}^{\prime}-i,$

which is the number of boxes strictly to the right or below $s$ respectively.
The hook length is the sum of these including $s$ itself, so that
$h(s):=a(s)+l(s)+1$. Using the same example as above,

$s$

where we have marked the square $s=(2,2)$ with $a(s)=1$, $l(s)=2$ and
$h(s)=4$. As in the introduction we denote the multiset of hook lengths of
$\lambda$ by $\mathcal{H}_{\lambda}$. This is further refined as
$\mathcal{H}^{\mathrm{e}}_{\lambda}$ and $\mathcal{H}^{\mathrm{o}}_{\lambda}$,
the collection of hook lengths which are even or odd respectively. In terms of
these we define the hook polynomials

$\displaystyle H_{\lambda}(q)$
$\displaystyle:=\prod_{h\in\mathcal{H}_{\lambda}}(1-q^{h})$ $\displaystyle
H_{\lambda}^{\mathrm{e}/\mathrm{o}}(q)$
$\displaystyle:=\prod_{h\in\mathcal{H}^{\mathrm{e}/\mathrm{o}}_{\lambda}}(1-q^{h}),$

which are invariant under conjugation of $\lambda$. For $z\in\mathbb{C}$ we
also need the content polynomials

$\displaystyle C_{\lambda}(z;q)$
$\displaystyle:=\prod_{(i,j)\in\lambda}(1-zq^{j-i})$ $\displaystyle
C_{\lambda}^{\mathrm{e}/\mathrm{o}}(z;q)$
$\displaystyle:=\prod_{\begin{subarray}{c}(i,j)\in\lambda\\\ \text{$i+j$
even/odd}\end{subarray}}(1-zq^{j-i}).$

In this paper we will frequently encounter partitions with empty $2$-core,
written $2\textup{-core}(\lambda)=0$. One definition of such partitions is
that their diagrams may be tiled by dominoes. Our running example of
$(6,4,3,1)$ has empty $2$-core since it admits the following tiling

We will now give some equivalent formulations of partitions with empty
$2$-core, which all easily follow by induction on $\lvert\lambda\rvert$.

######  Lemma 2.1.

For $\lambda\in\mathscr{P}_{2n}$ the following are equivalent:

  1. (1)

$2\textup{-core}(\lambda)=0$.

  2. (2)

$\lvert\mathcal{H}^{\mathrm{o}}_{\lambda}\rvert=\lvert\mathcal{H}^{\mathrm{e}}_{\lambda}\rvert=n$.

  3. (3)

For any integer $m$ such that $2m\geqslant n$, the sequence

$\lambda_{1}+2m-1,\lambda_{2}+2m-2,\dots,\lambda_{2m-1}+1,\lambda_{2m}$

contains $n$ even and $n$ odd integers.

The equivalence of (1) and (3) is the content of [14, Lemma 2.1]. Claim (2)
also follows from Littlewood’s decomposition of a partition into $r$-cores and
$r$-quotients in the case of empty $2$-core [16].

###  2.2. Auxiliary results

Here we prove some properties of the statistic $b(\lambda)$. Firstly, as we
have already used in the introduction, we have the following characterisation
of the vanishing of $b(\lambda)$.

######  Lemma 2.2.

Let $2\textup{-core}(\lambda)=0$. Then $b(\lambda)\geqslant 0$ with
$b(\lambda)=0$ if and only if $\lambda$ is even.

###### Proof.

If $\lambda$ is even then $b(\lambda)=0$ since the number of even/odd hook
lengths in each row is equal. Assume that $\lambda$ is not even. Then the
number of odd $\lambda_{i}$ is even. Let $\lambda_{i_{1}},\lambda_{i_{2}}$ be
the first two odd rows of $\lambda$. Since $2\textup{-core}(\lambda)=0$ these
must be separated by an even number of even rows (possibly zero). Ignoring the
remaining rows, the contribution to $b(\lambda)$ from the rows up to
$\lambda_{i_{2}}$ is thus

$\lambda_{i_{1}}-\lambda_{i_{2}}+i_{2}-i_{1}+2\sum_{j=i_{1}+1}^{i_{2}-1}(-1)^{i_{1}+j-1}(\lambda_{j}-j).$

Since the numbers $\lambda_{i}-i$ are strictly decreasing this sum must be
nonnegative. In fact it is positive since
$\lambda_{i_{1}}-\lambda_{i_{2}}+i_{2}-i_{1}$ is positive. The next nonzero
contribution to $b(\lambda)$ will come from the next pair of odd rows. Thus
repeating the above shows us that if $\lambda$ is not even then
$b(\lambda)>0$. ∎

Note that $b((2,1,1,1))=0$, so that $b(\lambda)$ may vanish for partitions
with nonempty $2$-core.

######  Lemma 2.3.

For $\lambda\in\mathscr{P}_{n}$ there holds

$b(\lambda)=\sum_{(i,j)\in\lambda+\delta}(-1)^{\lambda_{i}-i-j+1}(\lambda_{i}-i)-\sum_{1\leqslant
i<j\leqslant 2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{i}-i).$

Moreover, if $2\textup{-core}(\lambda)=0$, then

$b(\lambda^{\prime})=\frac{\lvert\lambda\rvert}{2}-n^{2}-n+\sum_{1\leqslant
i<j\leqslant 2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{j}-j).$

###### Proof.

We interpret the definition of $b(\lambda)$ as a sum over the diagram of
$\lambda$ where each square has weight
$(-1)^{\lambda_{i}+\lambda_{j}^{\prime}-i-j+1}(\lambda_{i}-i)$. In the diagram
of $\lambda+\delta$ place the integer
$(-1)^{\lambda_{i}-i-j+1}(\lambda_{i}-i)$ in box $(i,j)$. Summing over $i,j$
gives the first sum on the right. To identify the second sum, we remove the
columns with index $\lambda_{j}+2n-j+1$ for $2\leqslant j\leqslant 2n$ whose
entries are $(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{i}-i)$. The remaining
diagram is that of $\lambda$ with entries
$(-1)^{\lambda_{i}+\lambda_{j}^{\prime}-i-j+1}(\lambda_{i}-i)$, which shows
the first identity.

The proof of the second identity is similar. Note that $b(\lambda^{\prime})$
may be written as

$b(\lambda^{\prime})=\sum_{(i,j)\in\lambda}(-1)^{\lambda_{i}+\lambda_{j}^{\prime}-i-j+1}(\lambda_{j}^{\prime}-j).$

We thus fill the diagram of $\lambda+\delta$ with integers
$(-1)^{\lambda_{i}-i-j+1}(2n-j)$, so that removing the same columns as before
now gives

$b(\lambda^{\prime})=\sum_{(i,j)\in\lambda+\delta}(-1)^{\lambda_{i}-i-j+1}(2n-j)-\sum_{1\leqslant
i<j\leqslant 2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(j-\lambda_{j}-1).$

A simple calculation shows that for $2\textup{-core}(\lambda)=0$,

$\sum_{(i,j)\in\lambda+\delta}(-1)^{\lambda_{i}-i-j+1}(2n-j)+\sum_{1\leqslant
i<j\leqslant
2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}=\frac{\lvert\lambda\rvert}{2}-n^{2}-n,$

completing the proof. ∎

###  2.3. Schur functions

For completeness we give a definition of the Schur functions in terms of the
classical ratio of alternants. Let $\lambda\in\mathscr{P}_{n}$, then the Schur
function is defined as

$s_{\lambda}(x):=\frac{\det_{1\leqslant i,j\leqslant
n}(x_{i}^{\lambda_{j}+n-j})}{\det_{1\leqslant i,j\leqslant n}(x_{i}^{n-j})},$

and $s_{\lambda}(x):=0$ for $l(\lambda)>n$. The set of the $s_{\lambda}(x)$
indexed over $\mathscr{P}_{n}$ forms a $\mathbb{Z}$-basis for $\Lambda_{n}$,
the ring of symmetric functions in $n$ variables. We also use the Schur
functions in countably many variables $x_{1},x_{2},x_{3},\dots$, such as in
Theorem 1.1. The set of such $s_{\lambda}(x)$ when indexed over all partitions
$\lambda$ form a $\mathbb{Z}$-basis for the ring $\Lambda$ of symmetric
functions in infinitely many variables [18].

Several of the results we need below are best stated in terms of Macdonald
polynomials, which are a $q,t$-analogue of the Schur functions [18]. We simply
note that the Macdonald polynomials $P_{\lambda}(x;q,t)$ are a basis for
$\Lambda_{n}$ with coefficients in $\mathbb{Q}(q,t)$ and reduce to the Schur
functions when $q=t$, i.e., $P_{\lambda}(x;q,q)=s_{\lambda}(x)$.

###  2.4. Koornwinder polynomials and integrals

The Koornwinder polynomials are a family of $\mathrm{BC}_{n}$-symmetric
functions depending on six parameters first introduced by Koornwinder [13] as
a multivariate analogue of the Askey–Wilson polynomials [1]. In what follows
$x=(x_{1},\dots,x_{n}),x^{\pm}=(x_{1},x_{1}^{-1},\dots,x_{n},x_{n}^{-1})$ and
for a single-variable function $g(x_{i})$ we set

$\displaystyle g\big{(}x_{i}^{\pm}\big{)}$
$\displaystyle:=g(x_{i})g\big{(}x_{i}^{-1}\big{)}$ $\displaystyle
g\big{(}x_{i}^{\pm}x_{j}^{\pm}\big{)}$
$\displaystyle:=g(x_{i}x_{j})g\big{(}x_{i}^{-1}x_{j}\big{)}g\big{(}x_{i}x_{j}^{-1}\big{)}g\big{(}x_{i}^{-1}x_{j}^{-1}\big{)}.$

Below the function will be one of $g(x_{i})=(x_{i};q)_{\infty}$ or
$g(x_{i})=(1-x_{i})$. Also for the infinite $q$-shifted factorial we adopt the
usual multiplicative notation

$(a_{1},\dots,a_{n};q)_{\infty}:=(a_{1};q)_{\infty}\cdots(a_{n};q)_{\infty}.$

Let $W:=\mathfrak{S}_{n}\ltimes(\mathbb{Z}/2\mathbb{Z})^{n}$ be the group of
signed permutations on $n$ letters. A Laurent polynomial
$f(x)\in\mathbb{C}[x^{\pm}]$ is called $\mathrm{BC}_{n}$-symmetric if it is
invariant under the natural action of $W$ on the $n$ variables where the
reflections act by $x_{i}\mapsto 1/x_{i}$. For $\lambda\in\mathscr{P}_{n}$
define the orbit-sum indexed by $\lambda$ as

$m_{\lambda}^{\mathrm{BC}}(x):=\sum_{\alpha}x^{\alpha},$

where the sum is over all elements of the $W$-orbit of $\lambda$, the
reflections act on sequences by $\alpha_{i}\mapsto-\alpha_{i}$, and
$x^{\alpha}:=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$. The orbit-sums form
a basis for the ring $\Lambda_{n}^{\mathrm{BC}}$ of
$\mathrm{BC}_{n}$-symmetric functions. For
$q,t,t_{0},t_{1},t_{2},t_{3}\in\mathbb{C}$ with $\lvert q\rvert,\lvert
t\rvert,\lvert t_{0}\rvert,\lvert t_{1}\rvert,\lvert t_{2}\rvert,\lvert
t_{3}\rvert<1$, define the Koornwinder density by

$\Delta(x;q,t;t_{0},t_{1},t_{2},t_{3}):=\prod_{i=1}^{n}\frac{(x_{i}^{\pm
2};q)_{\infty}}{\prod_{r=0}^{3}(t_{r}x_{i}^{\pm};q)_{\infty}}\prod_{1\leqslant
i<j\leqslant
n}\frac{(x_{i}^{\pm}x_{j}^{\pm};q)_{\infty}}{(tx_{i}^{\pm}x_{j}^{\pm};q)_{\infty}}.$

This further allows to define an inner product on $\Lambda_{n}^{\mathrm{BC}}$
by

$\langle
f,g\rangle_{q,t;t_{0},t_{1},t_{2},t_{3}}^{(n)}:=\int\limits_{\mathbb{T}^{n}}f(x)g(x^{-1})\Delta(x;q,t;t_{0},t_{1},t_{2},t_{3})\,\mathrm{d}T(x),$

where $\mathbb{T}^{n}$ is the standard $n$-torus and the measure $T(x)$ is
given by

$\mathrm{d}T(x):=\frac{1}{2^{n}n!(2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt)^{n}}\frac{\mathrm{d}x_{1}}{x_{1}}\cdots\frac{\mathrm{d}x_{n}}{x_{n}}.$

The Koornwinder polynomials are defined to be the unique
$\mathrm{BC}_{n}$-symmetric functions satisfying

$K_{\lambda}=m_{\lambda}^{\mathrm{BC}}+\sum_{\mu<\lambda}c_{\lambda\mu}m_{\mu}^{\mathrm{BC}},$

where $c_{\lambda\mu}\in\mathbb{C}(q,t,t_{0},t_{1},t_{2},t_{3})$, and for
which

$\langle
K_{\lambda},K_{\mu}\rangle_{q,t;t_{0},t_{1},t_{2},t_{3}}^{(n)}=0\qquad\text{if
$\lambda\neq\mu$}.$

Note that $\mu\leqslant\lambda$ denotes the extension of the usual dominance
order to all partitions $\lambda,\mu\in\mathscr{P}$: $\mu\leqslant\lambda$ if
and only if $\mu_{1}+\cdots+\mu_{i}\leqslant\lambda_{1}+\cdots+\lambda_{i}$
for all $i\geqslant 1$. The Koornwinder polynomials satisfy many nice
properties such as the quadratic norm evaluation and evaluation symmetry [4,
25]. The key identity we need is [24, Equation (2.6.9)] (see also [22,
Corollary 7.2.1])

(2.1) $\lim_{m\to\infty}(x_{1}\dots
x_{n})^{m}K_{(m^{n})-\lambda}(x;q,t;t_{0},t_{1},t_{2},t_{3})\\\
=P_{\lambda}(x;q,t)\prod_{i=1}^{n}\frac{(t_{0}x_{i},t_{1}x_{i},t_{2}x_{i},t_{3}x_{i};q)_{\infty}}{(x_{i}^{2};q)_{\infty}}\prod_{1\leqslant
i<j\leqslant n}\frac{(tx_{i}x_{j};q)_{\infty}}{(x_{i}x_{j};q)_{\infty}}.$

We will only use this for $\lambda=0$, in which case $P_{0}(x;q,t)=1$.

For a basis $\\{f_{\lambda}\\}$ of $\Lambda_{n}^{\mathrm{BC}}$ we write
$[f_{\lambda}]g$ for the coefficient of $f_{\lambda}$ in the expansion
$g=\sum_{\lambda}c_{\lambda}f_{\lambda}$ where the $c_{\lambda}$ lie in some
coefficient ring. The virtual Koornwinder integral of a
$\mathrm{BC}_{n}$-symmetric function $f$ is defined as

$I^{(n)}_{K}(f;q,t;t_{0},t_{1},t_{2},t_{3}):=[K_{0}(x;q,t;t_{0},t_{1},t_{2},t_{3})]f.$

This is extended to allow for symmetric function arguments via the
homomorphism $\Lambda_{2n}\longrightarrow\Lambda_{n}^{\mathrm{BC}}$ for which
$f(x_{1},\dots,x_{n})\mapsto f(x_{1},x_{1}^{-1},\dots,x_{n},x_{n}^{-1})$. Of
course since $K_{0}=1$ the orthogonality of the Koornwinder polynomials allows
us to express this as

$I^{(n)}_{K}(f;q,t;t_{0},t_{1},t_{2},t_{3})=\frac{\langle
f,1\rangle_{q,t;t_{0},t_{1},t_{2},t_{3}}^{(n)}}{\langle
1,1\rangle_{q,t;t_{0},t_{1},t_{2},t_{3}}^{(n)}}.$

Note that the denominator has the explicit evaluation

$\langle
1,1\rangle_{q,t;t_{0},t_{1},t_{2},t_{3}}^{(n)}=\prod_{i=1}^{n}\frac{(t,t_{0}t_{1}t_{2}t_{3}t^{n+i-2};q)_{\infty}}{(q,t^{i};q)_{\infty}\prod_{0\leqslant
r<s\leqslant 3}(t_{r}t_{s}t^{i-1};q)_{\infty}},$

which is Gustafson’s generalised Askey–Wilson integral [9]. The virtual
Koornwinder integral can be evaluated for many choices of the argument $f$,
see [14, 22, 23, 24]. In particular, the vanishing integrals of the next
section may be expressed in terms of virtual Koornwinder integrals. We need
one final identity involving virtual Koornwinder integrals [24, Propostion
4.2]. To state this conveniently, let

$f^{(m)}_{\lambda}(q,t;t_{0},t_{1},t_{2},t_{3}):=[P_{\lambda}(x;q,t)](x_{1}\cdots
x_{n})^{m}K_{(m^{n})}(x;q,t;t_{0},t_{1},t_{2},t_{3}).$

######  Proposition 2.4.

For nonnegative integers $n,m$ and $\lambda\subseteq(2m)^{n}$,

$f_{\lambda}^{(m)}(q,t;t_{0},t_{1},t_{2},t_{3})=(-1)^{\lvert\lambda\rvert}I_{K}^{(m)}\big{(}P_{\lambda^{\prime}}(t,q);t,q;t_{0},t_{1},t_{2},t_{3}\big{)}.$

##  3\. Vanishing integrals

In this section we evaluate a pair of vanishing integrals for Schur functions
conjectured by Lee, Rains and Warnaar in the Macdonald case [14, Conjecture
9.2].

For $a,b,q\in\mathbb{C}$ with $\lvert a\rvert,\lvert b\rvert,\lvert q\rvert<1$
we define

$I^{(n)}_{\lambda}(a,b;q):=\frac{1}{Z_{n}(a,b;q)}\int\limits_{\mathbb{T}^{n}}s_{\lambda}\big{(}x_{1}^{\pm},\dots,x_{n}^{\pm}\big{)}\prod_{i=1}^{n}\frac{(x_{i}^{\pm
2};q)_{\infty}}{(ax_{i}^{\pm 2},bx_{i}^{\pm 2};q^{2})_{\infty}}\\\
\times\prod_{1\leqslant i<j\leqslant
n}\big{(}1-x_{i}^{\pm}x_{j}^{\pm}\big{)}\,\mathrm{d}T(x),$

where $\lambda$ is a partition with length at most $2n$ and the normalising
factor is given by

$\displaystyle Z_{n}(a,b;q)$
$\displaystyle:=\int\limits_{\mathbb{T}^{n}}\prod_{i=1}^{n}\frac{(x_{i}^{\pm
2};q)_{\infty}}{(ax_{i}^{\pm 2},bx_{i}^{\pm
2};q^{2})_{\infty}}\prod_{1\leqslant i<j\leqslant
n}\big{(}1-x_{i}^{\pm}x_{j}^{\pm}\big{)}\,\mathrm{d}T(x)$
$\displaystyle\hphantom{:}=\prod_{i=1}^{n}\frac{(abq^{n+i-2};q)_{\infty}}{(q^{i},-aq^{i-1},-bq^{i-1};q)_{\infty}(abq^{2i-2};q^{2})_{\infty}^{2}}.$

Note that in terms of virtual Koornwinder integrals this is

$I_{\lambda}^{(n)}(a,b;q)=I^{(n)}_{K}(s_{\lambda};q,q,a^{1/2},-a^{1/2},b^{1/2},-b^{1/2}).$

Lee, Rains and Warnaar prove the following properties of the above integral
[14, Propostion 9.3].

######  Proposition 3.1.

For $a,b,q\in\mathbb{C}$ with $\lvert a\rvert,\lvert b\rvert,\lvert q\rvert<1$
and $\lambda$ a partition of length at most $2n$ the integral
$I^{(n)}_{\lambda}(a,b;q)$ vanishes unless $2\textup{-core}(\lambda)=0$.
Furthermore

(3.1a) $\displaystyle I^{(n)}_{\lambda}(q,q;q)$
$\displaystyle=\prod_{i=1}^{n}\frac{(1-q^{2i-1})^{2n-2i+1}}{(1-q^{2i})^{2n-2i}}$
$\displaystyle\quad\times\operatorname*{Pf}_{1\leqslant i,j\leqslant
2n}\bigg{(}\frac{q^{(\lambda_{i}-\lambda_{j}+j-i-1)/2}}{1-q^{\lambda_{i}-\lambda_{j}+j-i}}\chi(\text{$\lambda_{i}-\lambda_{j}+j-i$
odd})\bigg{)},$ and (3.1b) $\displaystyle I^{(n)}_{\lambda}(1,q^{2};q)$
$\displaystyle=\frac{1}{2^{n-1}(1+q^{n})}\prod_{i=1}^{n}\frac{(1-q^{2i-1})^{2n-2i+1}}{(1-q^{2i})^{2n-2i}}$
$\displaystyle\quad\times\operatorname*{Pf}_{1\leqslant i,j\leqslant
2n}\bigg{(}\frac{1+q^{\lambda_{i}-\lambda_{j}+j-i}}{1-q^{\lambda_{i}-\lambda_{j}+j-i}}\chi(\text{$\lambda_{i}-\lambda_{j}+j-i$
odd})\bigg{)}.$

Lee, Rains and Warnaar also give a conjectural Macdonald polynomial analogue
of this proposition [14, Conjecture 9.2]. There the generalisations of (3.1)
are explicit products. Our next proposition gives the evaluation of the
Pfaffians in the previous proposition, verifying the conjecture of Lee, Rains
and Warnaar for $q=t$.

######  Proposition 3.2.

For $\lambda$ with length at most $2n$ and $2\text{-core}(\lambda)=0$,

(3.2)
$I^{(n)}_{\lambda}(q,q;q)=q^{b(\lambda^{\prime})}\frac{C_{\lambda}^{\mathrm{e}}(q^{2n};q)H_{\lambda}^{\mathrm{o}}(q)}{C_{\lambda}^{\mathrm{o}}(q^{2n};q)H_{\lambda}^{\mathrm{e}}(q)}$

and

(3.3)
$I^{(n)}_{\lambda}(1,q^{2};q)=q^{b(\lambda)}\frac{1+q^{n+2(b(\lambda^{\prime})-b(\lambda))}}{1+q^{n}}\,\frac{C_{\lambda}^{\mathrm{e}}(q^{2n};q)H_{\lambda}^{\mathrm{o}}(q)}{C_{\lambda}^{\mathrm{o}}(q^{2n};q)H_{\lambda}^{\mathrm{e}}(q)}.$

###### Proof.

Since the structure of the Pfaffians is similar, we focus on the second
identity, and evaluate (3.1b).

Fix a partition $\lambda\in\mathscr{P}_{2n}$ with empty $2$-core. Define the
set $J\subseteq\\{1,\dots,2n\\}$ as the collection of integers $j$ for which
column $j$ has a nonzero entry in the first row, and set
$I:=\\{1,\dots,2n\\}\setminus J$. Since $2\textup{-core}(\lambda)=0$ it
follows that $\lvert I\rvert=\lvert J\rvert=n$. The elements of $I$ and $J$
are labeled by $i_{k}$ and $j_{k}$ respectively, where $1\leqslant k\leqslant
n$ and ordered naturally. With this established we define the $n\times n$
matrix

$M_{k,\ell}:=\frac{1+q^{\lambda_{i_{k}}-\lambda_{j_{\ell}}+j_{\ell}-i_{k}}}{1-q^{\lambda_{i_{k}}-\lambda_{j_{\ell}}+j_{\ell}-i_{k}}}.$

The Pfaffian in (3.1b) may be expressed in terms of the determinant of $M$.
Indeed, by rearranging rows and columns we see that

$\displaystyle\operatorname*{Pf}_{1\leqslant i,j\leqslant
2n}\bigg{(}\frac{1+q^{\lambda_{i}-\lambda_{j}+j-i}}{1-q^{\lambda_{i}-\lambda_{j}+j-i}}\chi(\text{$\lambda_{i}-\lambda_{j}+j-i$
odd})\bigg{)}$
$\displaystyle\qquad\qquad\qquad\qquad=(-1)^{\binom{n+1}{2}+\sum_{j\in
J}j}\operatorname*{Pf}\begin{pmatrix}0&-M^{t}\\\ M&0\end{pmatrix}$
$\displaystyle\qquad\qquad\qquad\qquad=(-1)^{n+\sum_{j\in J}j}\det M.$

The determinant may be simply evaluated by applying a generalisation of
Cauchy’s double alternant which may be found in [5, Example 3.1; $a=0$]:

$\displaystyle\det_{1\leqslant i,j\leqslant
n}\bigg{(}\frac{bx_{i}+cy_{j}}{x_{i}+y_{j}}\bigg{)}$
$\displaystyle=(b-c)^{n-1}\bigg{(}b\prod_{i=1}^{n}x_{i}+(-1)^{n-1}c\prod_{i=1}^{n}y_{i}\bigg{)}$
$\displaystyle\qquad\qquad\times\frac{\prod_{1\leqslant i<j\leqslant
n}(x_{i}-x_{j})(y_{i}-y_{j})}{\prod_{i,j=1}^{n}(x_{i}+y_{j})}.$

We wish apply this with
$(b,c,x_{i},y_{i})\mapsto(1,-1,q^{\lambda_{i}-i},-q^{\lambda_{i}-i})$ for
$1\leqslant i\leqslant n$. After doing so and cancelling minus signs, the
evaluation may be expressed as

$\displaystyle I_{\lambda}^{(n)}(1,q^{2};q)$
$\displaystyle\quad=\frac{\prod_{i\in I}q^{\lambda_{i}-i}+\prod_{j\in
J}q^{\lambda_{j}-j}}{1+q^{n}}\prod_{i=1}^{n}\frac{(1-q^{2i-1})^{2n-2i+1}}{(1-q^{2i})^{2n-2i}}$
$\displaystyle\quad\quad\times\prod_{\begin{subarray}{c}1\leqslant
i<j\leqslant 2n\\\ \text{$\lambda_{i}-\lambda_{j}+j-i$
even}\end{subarray}}\frac{1-q^{\lambda_{i}-\lambda_{j}+j-i}}{q^{\lambda_{j}-j}}\prod_{\begin{subarray}{c}1\leqslant
i<j\leqslant 2n\\\ \text{$\lambda_{i}-\lambda_{j}+j-i$
odd}\end{subarray}}\frac{q^{\lambda_{j}-j}}{1-q^{\lambda_{i}-\lambda_{j}+j-i}}.$

The terms of the form $1-q^{x}$ can be simplified thanks to the identity [18,
p. 10–11]

$\frac{C_{\lambda}(q^{2n};q)}{H_{\lambda}(q)}=\prod_{s\in\lambda}\frac{1-q^{n+c(s)}}{1-q^{h(s)}}=\frac{\prod_{1\leqslant
i<j\leqslant n}1-q^{\lambda_{i}-\lambda_{j}+j-i}}{\prod_{i=1}^{n}(q;q)_{i}},$

where $l(\lambda)\leqslant n$. Restricting all products to even/odd exponents
implies that

$\displaystyle\frac{C_{\lambda}^{\mathrm{e}}(q^{2n};q)H_{\lambda}^{\mathrm{o}}(q)}{C_{\lambda}^{\mathrm{o}}(q^{2n};q)H_{\lambda}^{\mathrm{e}}(q)}$
$\displaystyle=\prod_{\begin{subarray}{c}1\leqslant i<j\leqslant 2n\\\
\text{$\lambda_{i}-\lambda_{j}+j-i$
even}\end{subarray}}(1-q^{\lambda_{i}-\lambda_{j}+j-i})\prod_{\begin{subarray}{c}1\leqslant
i<j\leqslant 2n\\\ \text{$\lambda_{i}-\lambda_{j}+j-i$
odd}\end{subarray}}\frac{1}{1-q^{\lambda_{i}-\lambda_{j}+j-i}}$
$\displaystyle\qquad\quad\times\prod_{i=1}^{n}\frac{(1-q^{2i-1})^{2n-2i+1}}{(1-q^{2i})^{2n-2i}}.$

It remains to show that the powers of $q$ agree in the prefactor. Since

$\prod_{i\in I}q^{\lambda_{i}-i}+\prod_{j\in
J}q^{\lambda_{j}-j}=\prod_{\begin{subarray}{c}i=1\\\ \text{$\lambda_{i}-i$
even}\end{subarray}}^{2n}q^{\lambda_{i}-i}+\prod_{\begin{subarray}{c}i=1\\\
\text{$\lambda_{i}-i$ odd}\end{subarray}}^{2n}q^{\lambda_{i}-i},$

this may be reduced to the pair of identities

$\displaystyle b(\lambda)=\sum_{\begin{subarray}{c}i=1\\\
\text{$\lambda_{i}-i$
even}\end{subarray}}^{2n}(\lambda_{i}-i)+\sum_{1\leqslant i<j\leqslant
2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{j}-j),$ $\displaystyle
n+2(b(\lambda^{\prime})-b(\lambda))=\sum_{\begin{subarray}{c}i=1\\\
\text{$\lambda_{i}-i$
odd}\end{subarray}}^{2n}(\lambda_{i}-i)-\sum_{\begin{subarray}{c}i=1\\\
\text{$\lambda_{i}-i$ even}\end{subarray}}^{2n}(\lambda_{i}-i).$

In the first of these write

$\displaystyle\sum_{\begin{subarray}{c}i=1\\\ \text{$\lambda_{i}-i$
even}\end{subarray}}^{2n}(\lambda_{i}-i)$
$\displaystyle=\sum_{(i,j)\in\lambda+\delta}(-1)^{\lambda_{i}-i-j+1}(\lambda_{i}-i)+\sum_{i=1}^{2n}(\lambda_{i}-i)$
$\displaystyle=b(\lambda)+\sum_{1\leqslant i<j\leqslant
2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{i}-i)+\sum_{i=1}^{2n}(\lambda_{i}-i),$

where in the second equality we have applied Lemma 2.3. Since

$\displaystyle\sum_{1\leqslant i<j\leqslant
2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{i}-i)+\sum_{1\leqslant
i<j\leqslant
2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{i}-i)+\sum_{i=1}^{2n}(\lambda_{i}-i)$
$\displaystyle\qquad\qquad=\sum_{i,j=1}^{2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{i}-i)$
$\displaystyle\qquad\qquad=0,$

the first identity follows. For the second identity, a similar rewriting using
Lemma 2.3 shows us that

$\displaystyle\sum_{\begin{subarray}{c}i=1\\\ \text{$\lambda_{i}-i$
odd}\end{subarray}}^{2n}(\lambda_{i}-i)-\sum_{\begin{subarray}{c}i=1\\\
\text{$\lambda_{i}-i$ even}\end{subarray}}^{2n}(\lambda_{i}-i)$
$\displaystyle\qquad=-2\sum_{(i,j)\in\lambda+\delta}(-1)^{\lambda_{i}-i-j+1}(\lambda_{i}-i)-\sum_{i=1}^{2n}(\lambda_{i}-i)$
$\displaystyle\qquad=-2b(\lambda)-\lvert\lambda\rvert+2n^{2}+n-2\sum_{1\leqslant
i<j\leqslant 2n}(-1)^{\lambda_{i}-\lambda_{j}+j-i}(\lambda_{i}-i)$
$\displaystyle\qquad=n+2(b(\lambda^{\prime})-b(\lambda)).$

This finishes the evaluation of (3.1b). The evaluation of (3.1a) is almost
identical except one directly applies the second identity of Lemma 2.3 to
compute the exponent of $q$ in the prefactor. ∎

##  4\. Bounded Littlewood identities

Here we use the integral evaluations of the previous section to prove a
bounded analogue of Theorem 1.1. This is followed by proofs of the theorem and
Corollary 1.2.

###  4.1. A bounded analogue of Theorem 1.1

Bounded Littlewood identities are generalisations of ordinary Littlewood
identities in which the largest part of the indexing partition has an upper
bound, say $m$, such that sending $m$ to infinity recovers an ordinary
(unbounded) Littlewood identity. The first example of such an identity was
discovered by Macdonald [17] where he used a bounded analogue of (1.1a) to
prove the MacMahon and Bender–Knuth conjectures on plane partitions [2, 19].
Bounded analogues of the remaining two classical identities (1.1b) and (1.1c)
were obtained by Désarménien, Proctor and Stembridge [7, 21, 27] and Okada
[20] respectively. A host of other bounded identities for Hall–Littlewood and
Macdonald polynomials may be found in [24] and references therein. We now
state the bounded analogue of Theorem 1.1.

######  Theorem 4.1.

For $m,n$ nonnegative integers,

(4.1) $\sum_{\begin{subarray}{c}\lambda\\\
2\textup{-core}(\lambda)=0\end{subarray}}q^{b(\lambda^{\prime})}\frac{C_{\lambda}^{\mathrm{e}}(q^{-2m};q)H_{\lambda}^{\mathrm{o}}(q)}{C_{\lambda}^{\mathrm{o}}(q^{-2m};q)H_{\lambda}^{\mathrm{e}}(q)}s_{\lambda}(x)\\\
=(x_{1}\cdots
x_{n})^{m}K_{(m^{n})}\big{(}x;q,q;q^{1/2},-q^{1/2},q^{1/2},-q^{1/2}\big{)}$

and

(4.2) $\sum_{\begin{subarray}{c}\lambda\\\
2\textup{-core}(\lambda)=0\end{subarray}}\frac{q^{2b(\lambda^{\prime})-b(\lambda)}+q^{m+b(\lambda)}}{1+q^{m}}\,\frac{C_{\lambda}^{\mathrm{e}}(q^{-2m};q)H_{\lambda}^{\mathrm{o}}(q)}{C_{\lambda}^{\mathrm{o}}(q^{-2m};q)H_{\lambda}^{\mathrm{e}}(q)}s_{\lambda}(x)\\\
=(x_{1}\cdots x_{n})^{m}K_{(m^{n})}(x;q,q;1,-1,q,-q).$

These identities are indeed bounded since
$C_{\lambda}^{\mathrm{e}}(q^{-2m};q)$ vanishes if $\lambda_{1}>2m$. Since, by
[14, Lemma 4.1], the Koornwinder polynomials on the right reduce to classical
group characters for $q=0$, one recovers the previously mentioned
Désarménien–Proctor–Stembridge and Okada identities respectively in this case.
The Koornwinder polynomials for $q=t$ on the right-hand side may alternatively
be expressed as a ratio of determinants of Askey–Wilson polynomials [1]; see
e.g. [6, Definition 4.1]. This, however, does not seem to shed light on a more
explicit expression for the evaluation of these sums. In particular, the
specialisations of $K_{(m^{n})}$ above are not contained in [14, Lemma 4.1].

The following argument is sketched in [14, §9], but we give the details in the
Schur case. Assuming the Macdonald polynomial version of the vanishing
integrals [14, Conjecture 9.2], the same argument gives the conjectural
Littlewood identities.

###### Proof of Theorem 4.1.

The goal is to find an expression for the coefficient of $s_{\lambda}(x)$ in
the Schur expansion of the right-hand side. By Proposition 2.4 this
coefficient is

$f_{\lambda}^{(m)}(x;q,q,t_{0},t_{1},t_{2},t_{3})=I_{K}^{(m)}(s_{\lambda^{\prime}}(x);q,q;t_{0},t_{1},t_{2},t_{3}).$

If we specialise
$(t_{0},t_{1},t_{2},t_{3})=(q^{1/2},-q^{1/2},q^{1/2},-q^{1/2})$ then this
reduces to

$f_{\lambda}^{(m)}\big{(}x;q,q;q^{1/2},-q^{1/2},q^{1/2},-q^{1/2}\big{)}=(-1)^{\lvert\lambda\rvert}I_{\lambda^{\prime}}^{(m)}(q,q;q).$

The integral on the right is (3.2), as desired, and vanishes unless
$2\textup{-core}(\lambda)=0$. In this case the sign cancels since
$\lvert\lambda\rvert$ is even and we obtain

$(-1)^{\lvert\lambda\rvert}I_{\lambda^{\prime}}^{(m)}(q,q;q)=q^{b(\lambda)}\frac{C_{\lambda^{\prime}}^{\mathrm{e}}(q^{2m};q)H_{\lambda^{\prime}}^{\mathrm{o}}(q)}{C_{\lambda^{\prime}}^{\mathrm{o}}(q^{2m};q)H_{\lambda^{\prime}}^{\mathrm{e}}(q)}.$

By [14, Lemma 2.3] we may alternatively express this as

(4.3)
$q^{b(\lambda)}\frac{C_{\lambda^{\prime}}^{\mathrm{e}}(q^{2m};q)H_{\lambda^{\prime}}^{\mathrm{o}}(q)}{C_{\lambda^{\prime}}^{\mathrm{o}}(q^{2m};q)H_{\lambda^{\prime}}^{\mathrm{e}}(q)}=q^{b(\lambda^{\prime})}\frac{C_{\lambda}^{\mathrm{e}}(q^{-2m};q)H_{\lambda}^{\mathrm{o}}(q)}{C_{\lambda}^{\mathrm{o}}(q^{-2m};q)H_{\lambda}^{\mathrm{e}}(q)}$

This establishes (4.1). For (4.2) the same procedure applies with the
substitution $(t_{0},t_{1},t_{2},t_{3})=(1,-1,q,-q)$ and by using the integral
(3.3). ∎

###  4.2. Proof of Theorem 1.1

With the bounded identities established we may take the $m\to\infty$ limit of
both identities to obtain their unbounded counterparts. For the Koornwinder
side we use (2.1) with $(\lambda,q,t)=(0,q,q)$ and
$(t_{0},t_{1},t_{2},t_{3})=(q^{1/2},-q^{1/2},q^{1/2},-q^{1/2})$ or
$(t_{0},t_{1},t_{2},t_{3})=(1,-1,q,-q)$. In the case of (4.1) this yields

$\displaystyle\lim_{m\to\infty}(x_{1}\dots
x_{n})^{m}K_{(m^{n})}\big{(}x;q,q;q^{1/2},-q^{1/2},q^{1/2},-q^{1/2}\big{)}$
$\displaystyle\quad=\prod_{i=1}^{n}\frac{(q^{1/2}x_{i},-q^{1/2}x_{i},q^{1/2}x_{i},-q^{1/2}x_{i};q)_{\infty}}{(x_{i}^{2};q)_{\infty}}\prod_{1\leqslant
i<j\leqslant n}\frac{1}{1-x_{i}x_{j}}$
$\displaystyle\quad=\prod_{i=1}^{n}\frac{(qx_{i}^{2};q^{2})_{\infty}}{(x_{i}^{2};q^{2})_{\infty}}\prod_{1\leqslant
i<j\leqslant n}\frac{1}{1-x_{i}x_{j}},$

where we have used

$(a,-a;q)_{\infty}=(a^{2};q^{2})_{\infty}.$

For the limit of the summand we use it in conjugate form (4.3) so that

$\lim_{m\to\infty}q^{b(\lambda)}\frac{C_{\lambda^{\prime}}^{\mathrm{e}}(q^{2m};q)H_{\lambda}^{\mathrm{o}}(q)}{C_{\lambda^{\prime}}^{\mathrm{o}}(q^{2m};q)H_{\lambda}^{\mathrm{e}}(q)}=q^{b(\lambda)}\frac{H_{\lambda}^{\mathrm{o}}(q)}{H_{\lambda}^{\mathrm{e}}(q)}.$

Thus we have proved (1.2). As before the same procedure yields (1.3).

###  4.3. Proof of Corollary 1.2

In order to obtain Corollary 1.2 we take $q\to 1$ in either (1.2) or (1.3).
Let $(a;q)_{n}=\prod_{k=0}^{n}(1-aq^{k})$. Then we may take the limit of the
product-side of (1.2) by using

$\displaystyle\lim_{q\to
1}\frac{(qx_{i}^{2};q^{2})_{\infty}}{(x_{i}^{2};q^{2})_{\infty}}$
$\displaystyle=\lim_{q\to
1}\sum_{n=0}^{\infty}\frac{(q;q^{2})_{n}}{(q^{2};q^{2})_{n}}x_{i}^{2n}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1\cdot 3\cdots(2n-1)}{2\cdot 4\cdots
2n}x_{i}^{2n}$ $\displaystyle=\frac{1}{(1-x_{i}^{2})^{1/2}},$

where in the first line we have applied the $q$-binomial theorem [8, Equation
(1.3.2)]

$\sum_{n=0}^{\infty}\frac{(a;q)_{n}}{(q;q)_{n}}z^{n}=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}.$

The $q\to 1$ limit of the product-side of (1.3) gives the same result. The
limit of either sum follows from the characterisation of partitions with empty
$2$-core in Lemma 2.1, namely that
$\lvert\mathcal{H}^{\mathrm{e}}_{\lambda}\rvert=\lvert\mathcal{H}^{\mathrm{o}}_{\lambda}\rvert$.

### Acknowledgements

I thank Christian Krattenthaler and Ole Warnaar for many useful discussions
and suggestions, and for their encouragement.

## References

  * [1] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalise Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 219, iv+55.
  * [2] E. A. Bender and D. E. Knuth, Enumeration of plane partitions, J. Combin. Theory Ser. A 13 (1972), 40–54.
  * [3] D. M. Bressoud, Elementary proofs of identities for Schur functions and plane partitions, Ramanujan J. 4 (2000), no. 1, 69–80.
  * [4] J. F. van Diejen, Self-dual Koornwinder–Macdonald polynomials, Invent. math. 126 (1996), 319–339.
  * [5] W. Chu, Generalizations of the Cauchy determinant, Publ. Math. Debrecen 58 (2001), no. 3, 353–365.
  * [6] S. Corteel, O. Mandelshtam, L. Williams, Combinatorics of the two-species ASEP and Koornwinder moments, Adv. Math. 321 (2017), 160–204.
  * [7] J. Désarménien, La démonstration des identités de Gordon et MacMahon et de deux identités nouvelles, Sém Lothar. Combin. 15 (1986), Art. B15a, 11pp.
  * [8] G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed., Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  * [9] R. A. Gustafson, A generalization of Selberg’s beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97–105.
  * [10] M. Ishikawa and M. Wakayama, Applications of minor-summation formula II. Pfaffians and Schur polynomials, J. Combin. Theory Ser. A 88 (1999), 136–157.
  * [11] F. Jouhet and J. Zeng, Some new identities for Schur functions, Adv. Appl. Math. 27 (2001), 493–509.
  * [12] N. Kawanaka, A $q$-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999), 157–176.
  * [13] T. H. Koornwinder, Askey–Wilson polynomials for root systems of type $BC$, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math. 138, Amer Math. Soc., Providence, RI, 1992, 189–204.
  * [14] C.-h. Lee, E. M. Rains and S. O. Warnaar, An elliptic hypergeometric function approach to branching rules, SIGMA 16 (2020), paper 142, 52 pp.
  * [15] D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford University Press, New York, 1940.
  * [16] D. E. Littlewood, Modular representations of symmetric groups, Proc. Roy. Soc. London. Ser. A 209 (1951), 333–353.
  * [17] I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979.
  * [18] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1995.
  * [19] P. A. MacMahon, Partitions of numbers whose graphs possess symmetry, Trans. Cambridge Phil. Soc. 17 (1899), 149–170.
  * [20] S. Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra 205 (1998), no. 2, 337–367,
  * [21] R. A. Proctor, New symmetric plane partition identities from invariant theory work of De Concini and Procesi, European J. Combin. 11 (1990), no. 3, 289–300.
  * [22] E. M. Rains, $\mathrm{BC}_{n}$-symmetric polynomials, Transform. Groups 10 (2005), no. 1, 63–132.
  * [23] E. M. Rains and M. J. Vazirani, Quadratic transformations of Macdonald and Koornwinder polynomials Transform. Groups 12 (2007), no. 4, 725–759.
  * [24] E. M. Rains and S. O. Warnaar, Bounded Littlewood identities, Mem. Amer. Math. Soc. 270 (2021), no. 1317, vii+115 pp.
  * [25] S. Sahi Nonsymmetric Koornwinder polynomials and duality, Ann. Math. 150 (1999), 267–282.
  * [26] I. Schur, Aufgabe 569, Arch. Math. Phys. 27 (3) (1918), 163; Ges. Abhandlungen, Vol. 3, p. 456.
  * [27] J. R. Stembridge, Hall–Littlewood functions, plane partitions, and the Rogers–Ramanujan identities, Trans. Amer. Math. Soc. 319 (1990), no. 2, 469–498.