The Expansion of Wronskian Hermite Polynomials in the Hermite Basis

The Expansion of Wronskian Hermite Polynomials
in the Hermite Basis

Codruţ GROSU † and Corina GROSU ‡

C. Grosu and C. Grosu

† Google Zürich, Brandschenkestrasse 110, Zürich, Switzerland
<EMAIL_ADDRESS>
‡ Department of Applied Mathematics, Politehnica University of Bucharest,
‡ Splaiul Independentei 313, Bucharest, Romania<EMAIL_ADDRESS>
Received July 08, 2020, in final form January 04, 2021; Published online
January 09, 2021

We express Wronskian Hermite polynomials in the Hermite basis and obtain an
explicit formula for the coefficients. From this we deduce an upper bound for
the modulus of the roots in the case of partitions of length 2. We also derive
a general upper bound for the modulus of the real and purely imaginary roots.
These bounds are very useful in the study of irreducibility of Wronskian
Hermite polynomials. Additionally, we generalize some of our results to a
larger class of polynomials.

Wronskian; Hermite polynomials; Schrödinger operator

26C10; 30C15; 34L40

##  1 Introduction

Let $\\{H_{n}(x)\\}_{n\geq 0}$ be the classical Hermite polynomials, solutions
to the equation $y^{\prime\prime}(x)-2xy^{\prime}(x)+2ny(x)=0$. In this paper
we study the Wronskian of such polynomials: if $n_{1}<n_{2}<\dots<n_{r}$ is a
sequence of non-negative integers, we can define the Wronskian
$\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)]$ as the
determinant

$\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)]=\begin{vmatrix}H_{n_{1}}(x)&H_{n_{2}}(x)&\cdots&H_{n_{r}}(x)\\\
H_{n_{1}}^{\prime}(x)&H_{n_{2}}^{\prime}(x)&\cdots&H_{n_{r}}^{\prime}(x)\\\
\vdots&\vdots&\ddots&\vdots\\\
H_{n_{1}}^{(r-1)}(x)&H_{n_{2}}^{(r-1)}(x)&\cdots&H_{n_{r}}^{(r-1)}(x)\end{vmatrix}.$

Wronskians of Hermite polynomials appear in the study of rational potentials
admitted by the Schrödinger operator $L=-\frac{\partial^{2}}{\partial
x^{2}}+V(x)$. They are also used to define exceptional Hermite polynomials, a
subclass of the widely studied exceptional orthogonal polynomials [14]. When
the sequence is $m,m+1,\dots,m+n-1$, these polynomials are called generalized
Hermite polynomials, and form rational solutions to the fourth Painlevé
equation [6, 20]. Recurrence relations for Wronskian Hermite polynomials were
established in [5, 15], and invariance properties were determined in [12].

Oblomkov [21] characterized rational potentials of monodromy-free Schrödinger
operators that grow as $x^{2}$ at infinity. In this case, the potentials have
the form

$V(x)=-2\frac{\partial^{2}}{\partial
x^{2}}\log\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)]+x^{2}.$
(1.1)

Wronskians of Hermite polynomials also provide rational solutions to the
fourth Painlevé equation and its higher order generalizations. In fact, as
shown in [13], all rational solutions can be expressed using such Wronskians.

From (1.1) the zeros of
$\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)]$ are precisely
the poles of the potential. Because of this relationship it is important to
understand the geometry of the zeros of the Wronskian [3, 6, 7, 9].
Unfortunately not much is known about the set of zeros. Veselov (see [9])
conjectured that all the zeros are simple, except possibly at the origin. This
conjecture is known to be true in a few special cases, but in general it is
still open. In contrast, for the Hermite polynomials $H_{n}(x)$ it is well
known that the zeros are real and simple. The asymptotic behavior of the zeros
was studied in [9] for $2$-term Wronskians, and in [18] in general. However,
the results in [18] for non-real zeros depend on the assumption of simplicity.

It turns out it is useful to define the Wronskian polynomials in terms of
partitions.

Let $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{r})$ be any
partition. We define the degree vector of $\lambda$ as
$n_{\lambda}:=(\lambda_{r},\lambda_{r-1}+1,\dots,\lambda_{1}+r-1)$.
Furthermore, let

$\Delta(x_{1},x_{2},\dots,x_{r}):=\det\big{[}x_{i}^{j-1}\big{]}_{1\leq i,j\leq
r}=\prod_{j>i}(x_{j}-x_{i})$

be the Vandermonde determinant, with $\Delta(x_{1})=1$.

We shall consider a rescaled version of $H_{n}(x)$, defined by
$\textrm{{He}}_{n}(x)=2^{-\frac{n}{2}}H_{n}\big{(}\frac{x}{\sqrt{2}}\big{)}$.
These are the probabilistic Hermite polynomials, solution to the equation
$y^{\prime\prime}(x)-xy^{\prime}(x)+ny(x)=0$.

######  Definition 1.1.

For any partition $\lambda\vdash n$ we define the Wronskian Hermite polynomial
associated to $\lambda$ as

$\textrm{{He}}_{\lambda}(x):=\frac{\textrm{{Wr}}[\textrm{{He}}_{n_{1}}(x),\textrm{{He}}_{n_{2}}(x),\dots,\textrm{{He}}_{n_{r}}(x)]}{\Delta(n_{\lambda})},$

where $n_{\lambda}=(n_{1},n_{2},\dots,n_{r})$ is the degree vector of
$\lambda$.

Then $\textrm{{He}}_{\lambda}(x)$ is a monic polynomial of degree $n$.
Furthermore, $\textrm{{He}}_{\lambda}(x)$ has up to scaling the same set of
zeros as the Wronskian of
$\\{H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)\\}$.

Recently, Bonneux, Dunning and Stevens [3], following earlier work [4], found
an explicit formula for the coefficients of $\textrm{{He}}_{\lambda}(x)$ in
terms of the characters of the symmetric group.

######  Theorem 1.2 ([3, Theorem 2]).

Let $\lambda\vdash n$. Then

$\textrm{{He}}_{\lambda}(x)=\sum_{k=0}^{\lfloor
n/2\rfloor}(-1)^{k}H(\lambda)\frac{\chi^{\lambda}\big{(}2^{k}1^{n-2k}\big{)}}{2^{k}k!(n-2k)!}x^{n-2k},$

where $\chi^{\lambda}$ is the irreducible character associated to the
partition $\lambda$, $\big{(}2^{k}1^{n-2k}\big{)}$ is the conjugacy class of
$k$ disjoint transpositions, and $H(\lambda):=\frac{n!}{\chi^{\lambda}(1)}$.

Our main contribution in this paper is to establish a dual version of Theorem
1.2, where we determine the coefficients of $\textrm{{He}}_{\lambda}(x)$ in
the Hermite basis.

######  Theorem 1.3.

Let $\lambda\vdash n$. Then

$\textrm{{He}}_{\lambda}(x)=\sum_{k=0}^{\lfloor
n/2\rfloor}\frac{H(\lambda)}{k!(n-2k)!}K_{\lambda^{\prime}\mu^{(k)}}\textrm{{He}}_{n-2k}(x),$

where $\mu^{(k)}:=\big{(}2^{k}1^{n-2k}\big{)}$, $\lambda^{\prime}$ is the
conjugate partition of $\lambda$, and $K_{\lambda\mu}$ are the Kostka numbers.

In [17], exceptional Hermite polynomials are written as a linear combination
of Hermite polynomials (see formula $(101)$). However, the coefficients
$v_{j}^{(\lambda)}(n)$ of the expansion are determined using an algorithm.
Theorem 1.3 shows that
$v_{j}^{(\lambda)}(n)=(-1)^{j}2^{j}\frac{H(\lambda)}{j!(n-2j)!}K_{\lambda^{\prime}\mu^{(j)}}$
(compare with Examples 7.1 and 7.2 in [17]).

Our interest in Theorem 1.3 stems from the need to find good bounds for the
modulus of the roots, proportional to $\sqrt{n}$. For Hermite polynomials,
Szegő proved the following.

######  Theorem 1.4 (Szegő, [24]).

If $z$ is a root of $H_{n}(x)$ then $|z|\leq\frac{\sqrt{2}(n-1)}{\sqrt{n+2}}$.

Theorem 1.3 gives asymptotically good bounds for the modulus of zeros, in the
case when $\lambda=(\lambda_{1},\lambda_{2})$ and $\lambda_{2}$ is fixed.

######  Corollary 1.5.

Let $\lambda=(\lambda_{1},\lambda_{2})\vdash n$. If $z$ is a root of
$\textrm{{He}}_{\lambda}(x)$ then $|z|<2\big{(}\sqrt{{\rm
e}}\lambda_{2}+\sqrt{2}\big{)}\sqrt{\lambda_{1}+1}$.

We made no effort to optimize the constant in Corollary 1.5, as the correct
bound is likely close to $2\sqrt{\lambda_{1}}$.

One can also obtain bounds for the real and purely imaginary roots from
Theorem 1.3. The next corollary gives such bounds for an arbitrary partition
$\lambda$.

######  Corollary 1.6.

Let $\lambda\vdash n$. If $z$ is a real or purely imaginary root of
$\textrm{{He}}_{\lambda}(x)$ then $|z|\leq x_{n}$, where $x_{n}$ is the
largest root of $\textrm{{He}}_{n}(x)$.

However, Corollary 1.6 does not give the full picture. By exploiting the
Schrödinger equation it is possible to obtain a better bound for the real
roots.

######  Proposition 1.7.

Let $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$. If $z$ is a real
root of $\textrm{{He}}_{\lambda}(x)$ then $|z|\leq x_{\lambda_{1}+r-1}$, where
$x_{\lambda_{1}+r-1}$ is the largest root of
$\textrm{{He}}_{\lambda_{1}+r-1}(x)$.

This result is already apparent in the work of García-Ferrero and Gómez-Ullate
[11, Section 2]. However, the proof in [11] is done under the additional
assumption of semi-degeneracy, which is so far unproven for Wronskian Hermite
polynomials. Therefore, we include a proof of Proposition 1.7.

These bounds are very useful in the study of irreducibility of
$\textrm{{He}}_{\lambda}(x)$. In [16], we use Corollary 1.6 to show that
$\textrm{{He}}_{n,2}(x)$ is irreducible for all $n\geq 2$. Our initial proof
relied on the stronger Corollary 1.5, but we were eventually able to replace
it with an application of Corollary 1.6.

The rest of this note is organized as follows. In Section 2, we fix the
notation and state the auxiliary results that we will need. In Section 3, we
prove Theorem 1.3. In Section 4, we derive the two corollaries. In Section 5,
we prove Proposition 1.7. Finally, in Section 6, we generalize the results to
a larger class of polynomials.

##  2 Notation and auxiliary results

In this section we define the notation that we use throughout the paper. We
also state several results that will be needed for the proofs.

###  2.1 Partitions and the symmetric group

If $n\geq 0$ is an integer, a partition $\lambda$ of $n$, denoted
$\lambda\vdash n$, is a sequence of non-negative integers
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{r}$ such that
$\sum_{i=1}^{r}\lambda_{i}=n$. We denote $|\lambda|=n$ and call
$\ell(\lambda):=r$ the length of the partition $\lambda$. We say that
$\lambda_{i}$ are the parts of the partition.

We shall frequently use the notation
$(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ for $\lambda$. Sometimes we will
also use the notation $\mu=\big{(}m^{r_{m}}\cdots
3^{r_{3}}2^{r_{2}}1^{r_{1}}\big{)}$, meaning that the partition $\mu$ has
$r_{i}$ parts of size $i$. We will drop the parentheses if they are clear from
the context.

The degree vector of the partition $\lambda$ is defined as
$n_{\lambda}:=(\lambda_{r},\lambda_{r-1}+1,\dots,\lambda_{1}+r-1)$. We denote
by $n_{\lambda,i}$ the $i$-th entry of this vector.

The Ferrers diagram of $\lambda$ is $D_{\lambda}=\\{(i,j)\colon 1\leq i\leq
r,\,1\leq j\leq\lambda_{i}\\}$. This can be represented as a collection of
unit squares arranged in rows, with the $i$-th row having $\lambda_{i}$
squares. For example,

$\Yvcentermath 1D_{(3,2,2,1)}=\young(~{}~{}~{},~{}~{},~{}~{},~{})$

The conjugate partition $\lambda^{\prime}$ is obtained from $\lambda$ by
transposing the Ferrers diagram: $\lambda_{j}^{\prime}$ is the largest index
$i$ such that $\lambda_{i}\geq j$.

We can define a partial order on the set of partitions by saying that
$\lambda\leq\mu$ if $\ell(\lambda)=\ell(\mu)$, and $D_{\lambda}\subseteq
D_{\mu}$ or equivalently if $\lambda_{i}\leq\mu_{i}$ for all
$i\leq\ell(\lambda)$. If $\lambda\leq\mu$ then $\mu/\lambda$ denotes the skew
shape $D_{\mu/\lambda}:=D_{\mu}\setminus D_{\lambda}$. We let $|\mu/\lambda|$
denote the number of squares in the skew shape.

A $q$-hook $R$ is any connected set of squares in $D_{\lambda}$ of size $q$,
whose removal produces a valid partition. The height $ht(R)$ is defined as one
less than the number of rows spanned by $R$. We let $\mathcal{R}(\lambda,q)$
be the set of partitions $\mu\geq\lambda$ such that $\lambda$ can be obtained
from $\mu$ by removing a $q$-hook. In this case $\mu/\lambda$ is the removed
$q$-hook. The condition $\mu\geq\lambda$ means $\mu$ and $\lambda$ have the
same length, so $\mathcal{R}(\lambda,q)$ does not include two partitions which
differ only in the number of zero parts.

If $\lambda$ and $\mu$ are two partitions, a semistandard Young tableau of
shape $\lambda$ and type $\mu$ is a filling of the Ferrers diagram of
$\lambda$ with the numbers $1,2,\dots,\ell(\mu)$ such that the number $i$
appears $\mu_{i}$ times, the numbers weakly increase along rows, and strictly
increase along columns. The Kostka number $K_{\lambda\mu}$ is the number of
semistandard Young tableaux of shape $\lambda$ and type $\mu$.

Clearly $K_{\lambda\mu}\geq 0$. Furthermore, $K_{\lambda\mu}>0$ if and only if
$\lambda$ dominates $\mu$, written $\lambda\triangleright\mu$, that is when
$|\lambda|=|\mu|$ and

$\lambda_{1}+\dots+\lambda_{i}\geq\mu_{1}+\dots+\mu_{i}\qquad\textrm{for}\quad
i=1,\dots,\ell(\mu).$

We denote by $\chi^{\lambda}$ the irreducible character of $S_{n}$ associated
to $\lambda$. Let $F_{\lambda}:=\chi^{\lambda}(1)$ be the degree of the
irreducible representation. Then this is given by the formula (see [10,
equation (4.11)])

$F_{\lambda}=\frac{|\lambda|!}{H(\lambda)},\qquad\textrm{where}\quad
H(\lambda):=\frac{n_{\lambda,1}!n_{\lambda,2}!\cdots
n_{\lambda,r}!}{\Delta(n_{\lambda})}.$ (2.1)

###  2.2 Schur polynomials

Let $x_{1},x_{2},\dots,x_{k}$ be $k$ variables. We let
$e_{i}(x_{1},\dots,x_{k})$ be the elementary symmetric polynomials:

$e_{i}(x_{1},\dots,x_{k})=\sum_{1\leq j_{1}<\dots<j_{i}\leq
k}x_{j_{1}}x_{j_{2}}\cdots x_{j_{i}}.$

By convention $e_{i}(x_{1},\dots,x_{k})=0$ when $i>k$.

Similarly, we let $h_{i}(x_{1},\dots,x_{k})$ be the complete symmetric
polynomials:

$h_{i}(x_{1},\dots,x_{k})=\sum_{1\leq j_{1}\leq\dots\leq j_{i}\leq
k}x_{j_{1}}x_{j_{2}}\cdots x_{j_{i}}.$

Finally, we let $p_{i}(x_{1},\dots,x_{k})$ be the power-sum polynomials:

$p_{i}(x_{1},\dots,x_{k})=x_{1}^{i}+\dots+x_{k}^{i}.$

We are not going to write the variables if they are clear from the context.
For our purposes we will also use the convention $e_{0}=h_{0}=p_{0}=1$.

If $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{r})$ is a partition, we
define

$\displaystyle e_{\lambda}=e_{\lambda_{1}}e_{\lambda_{2}}\cdots
e_{\lambda_{r}},\qquad h_{\lambda}=h_{\lambda_{1}}h_{\lambda_{2}}\cdots
h_{\lambda_{r}},\qquad p_{\lambda}=p_{\lambda_{1}}p_{\lambda_{2}}\cdots
p_{\lambda_{r}}.$

Now assume that $k\geq r$. By adding parts of size $0$, we can further assume
that $r=k$. In this case we define
$W_{\lambda}(x_{1},\dots,x_{k}):=\det\big{[}x_{i}^{\lambda_{j}+k-j}\big{]}_{i,j=1}^{k}$.
Then $W_{\lambda}$ is an alternating polynomial, and hence it is divisible by
the Vandermonde determinant
$W_{0}(x_{1},\dots,x_{k}):=\det\big{[}x_{i}^{k-j}\big{]}_{i,j=1}^{k}$. Set
$s_{\lambda}(x_{1},\dots,x_{k}):=\frac{W_{\lambda}}{W_{0}}$. We call
$s_{\lambda}$ the Schur polynomial for $\lambda$. Then $s_{\lambda}$ is a
symmetric polynomial, and is defined for any partition $\lambda$ with
$\ell(\lambda)\leq k$. In the case $\ell(\lambda)>k$ we define
$s_{\lambda}(x_{1},\dots,x_{k}):=0$.

We will need the following consequence of Pieri’s rule.

######  Theorem 2.1 ([10, Appendix A.1]).

For any partition $\mu$ we have

$e_{\mu}=\sum_{\lambda}K_{\lambda^{\prime}\mu}s_{\lambda},$

where the sum is taken over all partitions $\lambda$ with non-zero parts.

We shall also need the following (see [10, Lecture 4, equation (4.10)] with
$C_{i}$ the conjugacy class of $\mu$).

######  Theorem 2.2 (Frobenius character formula).

Let $\lambda,\mu\vdash n$ be any partitions such that $\ell(\lambda)=n$. Then
$\chi^{\lambda}(\mu)$ is the coefficient of the monomial
$x_{1}^{\lambda_{1}+n-1}x_{2}^{\lambda_{2}+n-2}\cdots x_{n}^{\lambda_{n}}$ in
the polynomial

$p_{\mu}(x_{1},\dots,x_{n})W_{0}(x_{1},\dots,x_{n}).$

Related to this we have the Murnaghan–Nakayama rule expressed in terms of
Schur polynomials.

######  Theorem 2.3 ([23, Theorem 7.17.1]).

If $q\geq 1$ and $\lambda$ is a partition then

$p_{q}s_{\lambda}=\sum_{\mu\in\mathcal{R}(\lambda,q)}(-1)^{ht(\mu/\lambda)}s_{\mu}.$

###  2.3 Bounds for the roots of polynomials

To obtain effective bounds on the modulus of the roots, we will use a result
which is essentially due to Turán.

######  Theorem 2.4 ([25]).

Suppose $P(x)=\sum_{k=0}^{n}\alpha_{k}\textrm{{He}}_{k}(x)$ is a polynomial of
degree $n$. If $z$ is a zero of $P(x)$ and $x_{n,n}$ is the largest root of
$\textrm{{He}}_{n}(x)$ then

$|z|\leq
x_{n,n}+\sum_{k=0}^{n-1}\left|\frac{\alpha_{k}}{\alpha_{n}}\right|^{1/(n-k)}.$

However, Turán only proved an inequality for $|\textrm{{Im}}\,z|$ for a
decomposition in the base of classical Hermite polynomials. For convenience,
we explain how to adapt the proof to obtain Theorem 2.4.

###### Proof.

Let $x_{k,1}\leq x_{k,2}\leq\dots\leq x_{k,k}$ be the roots of
$\textrm{{He}}_{k}(x)$.

If $z$ is a complex number, we define $D(z)=\min|z-x_{k,i}|$, where the
minimum is taken over all $1\leq i\leq k\leq n$. The main step of the proof is
showing that

$D(z)\leq\sum_{k=0}^{n-1}\left|\frac{\alpha_{k}}{\alpha_{n}}\right|^{1/(n-k)},$
(2.2)

if $z$ is a root of $P(x)$ and $z$ is not a root of $\textrm{{He}}_{n}(x)$.

The inequality (2.2) follows verbatim from Turán’s proof by replacing $|y|$
with $D(z)$ everywhere (see also [19] for an exposition of the proof).
Therefore we shall not reproduce the proof here.

Using (2.2) we will show the inequality in the theorem.

If $z$ is a root of $\textrm{{He}}_{n}(x)$ then the inequality is trivially
true, as $x_{n,n}\geq 0$.

So we may assume that $z$ is not a root of $\textrm{{He}}_{n}(x)$. As the
roots of Hermite polynomials interlace, we have $x_{n,1}\leq x_{k,i}\leq
x_{n,n}$. The roots of Hermite polynomials are also symmetric around the
origin, i.e., $x_{n,1}=-x_{n,n}$. Therefore the root with largest absolute
value among $x_{k,i}$ is $x_{n,n}$. Then

$|z-x_{k,i}|\geq|z|-|x_{k,i}|\geq|z|-x_{n,n}.$

Therefore $D(z)\geq|z|-x_{n,n}$, and the inequality follows from (2.2). ∎

##  3 Proof of the main result

In this section we prove our main result Theorem 1.3.

For a partition $\lambda\vdash n$, $\textrm{{He}}_{\lambda}(x)$ is an even,
respectively odd, polynomial if the degree is even, respectively odd. This
follows either from [5, Lemma 3.6] or by examining the expression in Theorem
1.2. Therefore we can write it down as
$\textrm{{He}}_{\lambda}(x)=\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}a_{j}x^{n-2j}$.

We start the proof of Theorem 1.3 by writing the base-change formula.

######  Lemma 3.1.

Let $\lambda\vdash n$ and write
$\textrm{{He}}_{\lambda}(x)=\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}a_{j}x^{n-2j}$.
Then
$\textrm{{He}}_{\lambda}(x)=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}b_{k}\textrm{{He}}_{n-2k}(x)$
where

$b_{k}=\sum_{j=0}^{k}\frac{(n-2j)!}{2^{k-j}(k-j)!(n-2k)!}a_{j},\qquad 0\leq
k\leq\left\lfloor\frac{n}{2}\right\rfloor.$

###### Proof.

The Hermite polynomials have the generating function

$\sum_{m=0}^{\infty}\textrm{{He}}_{m}(x)\frac{t^{m}}{m!}=\exp\left(tx-\frac{t^{2}}{2}\right).$

From this we get (see [22, Chapter 11, Section 110])

$x^{m}=\sum_{k=0}^{\lfloor\frac{m}{2}\rfloor}\frac{m!}{2^{k}k!(m-2k)!}\textrm{{He}}_{m-2k}(x).$

Replacing every power of $x$ in the expression of $\textrm{{He}}_{\lambda}(x)$
gives

$\displaystyle\textrm{{He}}_{\lambda}(x)$
$\displaystyle=\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}a_{j}\left(\sum_{k=0}^{\lfloor\frac{n-2j}{2}\rfloor}\frac{(n-2j)!}{2^{k}k!(n-2j-2k)!}\textrm{{He}}_{n-2j-2k}(x)\right)$
$\displaystyle=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}\left(\sum_{j=0}^{k}\frac{(n-2j)!}{2^{k-j}(k-j)!(n-2k)!}a_{j}\right)\textrm{{He}}_{n-2k}(x).$
∎

We know that
$a_{j}=(-1)^{j}\frac{H(\lambda)}{2^{j}j!(n-2j)!}\chi^{\lambda}(2^{j}1^{n-2j})$
from Theorem 1.2. Replacing $a_{j}$ in Lemma 3.1 gives

$\displaystyle b_{k}$
$\displaystyle=\sum_{j=0}^{k}\frac{(n-2j)!}{2^{k-j}(k-j)!(n-2k)!}(-1)^{j}\frac{H(\lambda)}{2^{j}j!(n-2j)!}\chi^{\lambda}\big{(}2^{j}1^{n-2j}\big{)}$
$\displaystyle=\sum_{j=0}^{k}(-1)^{j}\frac{H(\lambda)}{2^{k}k!(n-2k)!}\binom{k}{j}\chi^{\lambda}\big{(}2^{j}1^{n-2j}\big{)}$
$\displaystyle=\frac{H(\lambda)}{2^{k}k!(n-2k)!}\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\chi^{\lambda}\big{(}2^{j}1^{n-2j}\big{)}.$

Let $S^{\lambda}_{k}$ be the sum
$\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\chi^{\lambda}\big{(}2^{j}1^{n-2j}\big{)}$.
The somewhat surprising fact is that this sum can be evaluated.

######  Lemma 3.2.

Let $\lambda\vdash n$ and $0\leq k\leq\big{\lfloor}\frac{n}{2}\big{\rfloor}$.
Set $\mu^{(k)}:=\big{(}2^{k}1^{n-2k}\big{)}$. Then
$S^{\lambda}_{k}=2^{k}K_{\lambda^{\prime}\mu^{(k)}}$.

###### Proof.

By adding parts of size $0$, we may assume without lack of generality that
$\lambda$ has length $n$, and that we have $n$ variables $x_{1},\dots,x_{n}$.

Set $\mu^{(j)}:=\big{(}2^{j}1^{n-2j}\big{)}$, $0\leq j\leq k$, and define
$M_{\lambda}$ as the monomial
$x_{1}^{\lambda_{1}+n-1}x_{2}^{\lambda_{2}+n-2}\cdots x_{n}^{\lambda_{n}}$.

From Theorem 2.2, we know that $\chi^{\lambda}\big{(}\mu^{(j)}\big{)}$ is the
coefficient of $M_{\lambda}$ in the polynomial

$p_{\mu^{(j)}}(x_{1},\dots,x_{n})W_{0}(x_{1},\dots,x_{n}).$

Therefore $S^{\lambda}_{k}$ must be the coefficient of $M_{\lambda}$ in the
polynomial

$\displaystyle\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}p_{\mu^{(j)}}W_{0}$
$\displaystyle=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}p_{2}^{j}p_{1}^{n-2j}W_{0}=W_{0}p_{1}^{n-2k}\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}p_{2}^{j}p_{1}^{2(k-j)}$
$\displaystyle=W_{0}p_{1}^{n-2k}\big{(}p_{1}^{2}-p_{2}\big{)}^{k},$

where the last equality follows from the binomial theorem.

However, $p_{1}^{2}-p_{2}=2e_{2}$. Furthermore, $p_{1}=e_{1}$. Hence the above
is the polynomial

$\displaystyle 2^{k}e_{1}^{n-2k}e_{2}^{k}W_{0}$
$\displaystyle=2^{k}e_{\mu^{(k)}}W_{0}=\sum_{\rho}2^{k}K_{\rho^{\prime}\mu^{(k)}}s_{\rho}W_{0}\quad(\textrm{
by Theorem~{}\ref{thm:pieri}})$
$\displaystyle=\sum_{\begin{subarray}{c}\rho\\\
\ell(\rho)=n\end{subarray}}2^{k}K_{\rho^{\prime}\mu^{(k)}}W_{\rho},$

as $s_{\rho}=\frac{W_{\rho}}{W_{0}}$ when $\ell(\rho)\leq n$, and $s_{\rho}=0$
otherwise. Furthermore in the last sum the appropriate number of zero parts
were added to $\rho$ such that $W_{\rho}$ is defined.

Recall that we need the coefficient of $M_{\lambda}$. We argue that this can
only come from $\rho=\lambda$.

Let $\rho$ be a partition such that the determinant
$W_{\rho}=\det\big{[}x_{i}^{\rho_{j}+n-j}\big{]}$ contains the monomial
$M_{\lambda}$. Assume for a contradiction that $\rho\neq\lambda$. By comparing
powers of each $x_{i}$, there must exist a permutation $\sigma$ such that
$\rho_{\sigma(i)}+n-\sigma(i)=\lambda_{i}+n-i$. Hence
$\rho_{\sigma(i)}=\lambda_{i}+\sigma(i)-i$ for $1\leq i\leq n$.

Let $i$ be minimal such that $\sigma(i)\neq i$. Then $\sigma(i)>i$. Let $j>i$
such that $\sigma(j)=i$. As $\rho_{i}\geq\rho_{\sigma(i)}$ we have

$\rho_{i}=\lambda_{j}+i-j\geq\rho_{\sigma(i)}=\lambda_{i}+\sigma(i)-i.$

So
$\lambda_{j}\geq\lambda_{i}+\sigma(i)-i+(j-i)>\lambda_{i}+\sigma(i)-i>\lambda_{i}$,
a contradiction to the fact that $\lambda_{i}\geq\lambda_{j}$ for $j\geq i$.

Therefore the coefficient of $M_{\lambda}$ comes from $\rho=\lambda$ only, and
so it is $2^{k}K_{\lambda^{\prime}\mu^{(k)}}$. This finishes the proof. ∎

Theorem 1.3 now follows directly by replacing $S_{k}^{\lambda}$ in the formula
for $b_{k}$:

$b_{k}=\frac{H(\lambda)}{2^{k}k!(n-2k)!}S_{k}^{\lambda}=\frac{H(\lambda)}{k!(n-2k)!}K_{\lambda^{\prime}\mu^{(k)}}.$

##  4 An upper bound for the modulus of the roots

In this section we derive the bounds on the absolute value of the roots. We
again write
$\textrm{{He}}_{\lambda}(x)=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}b_{k}\textrm{{He}}_{n-2k}(x)$.

###### Proof of Corollary 1.5.

The plan is to the compute the coefficients $b_{k}$ exactly and then use
Theorem 2.4.

Let $\lambda=(\lambda_{1},\lambda_{2})\vdash n$. The conjugate of the
partition $\lambda$ is
$\lambda^{\prime}=\big{(}2^{\lambda_{2}}1^{\lambda_{1}-\lambda_{2}}\big{)}$.
We will determine $K_{\lambda^{\prime}\mu^{(k)}}$, where recall that
$\mu^{(k)}=\big{(}2^{k}1^{n-2k}\big{)}$.

If $k>\lambda_{2}$ then $K_{\lambda^{\prime}\mu^{(k)}}=0$, as
$\lambda^{\prime}$ does not dominate $\mu^{(k)}$. So $b_{k}=0$.

If $k\leq\lambda_{2}$ then $K_{\lambda^{\prime}\mu^{(k)}}=F_{\rho}$, where
$\rho=\big{(}2^{\lambda_{2}-k}1^{\lambda_{1}-\lambda_{2}}\big{)}$. Using the
fact that $F_{\rho}=F_{\rho^{\prime}}$ we get
$K_{\lambda^{\prime}\mu^{(k)}}=F_{(\lambda_{1}-k,\lambda_{2}-k)}$. So

$\displaystyle b_{k}$
$\displaystyle=\frac{H(\lambda)}{k!(n-2k)!}F_{(\lambda_{1}-k,\lambda_{2}-k)}$
$\displaystyle\overset{\mbox{\tiny{\eqref{eq:hnfactorial}}}}{=}\frac{(\lambda_{1}+1)!\lambda_{2}!}{(\lambda_{1}-\lambda_{2}+1)k!(n-2k)!}\frac{(n-2k)!}{(\lambda_{1}-k+1)!(\lambda_{2}-k)!}(\lambda_{1}-\lambda_{2}+1)$
$\displaystyle=\frac{(\lambda_{1}+1)!\lambda_{2}!}{(\lambda_{1}-k+1)!(\lambda_{2}-k)!k!}.$

From Stirling’s approximation it follows that
$k!\geq\sqrt{2\pi}k^{k+\frac{1}{2}}{\rm e}^{-k}$ and so we get

$b_{k}\leq\frac{(\lambda_{1}+1)^{k}\lambda_{2}^{k}}{\sqrt{2\pi}k^{k+\frac{1}{2}}{\rm
e}^{-k}}.$

Let $z$ be a root of $\textrm{{He}}_{\lambda}(x)$. From Theorem 2.4 and the
fact that $b_{0}=1$, we obtain $|z|\leq
x_{n,n}+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\sqrt[2k]{b_{k}}$, where
$x_{n,n}$ is the largest root of $\textrm{{He}}_{n}(x)$. But for
$k>\lambda_{2}$ we have $b_{k}=0$, as shown above. Hence

$\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\sqrt[2k]{b_{k}}=\sum_{k=1}^{\lambda_{2}}\sqrt[2k]{b_{k}}\leq\sqrt{(\lambda_{1}+1)\lambda_{2}{\rm
e}}\sum_{k=1}^{\lambda_{2}}\frac{1}{\sqrt{k}\sqrt[4k]{2\pi
k}}<\sqrt{(\lambda_{1}+1)\lambda_{2}{\rm
e}}\sum_{k=1}^{\lambda_{2}}\frac{1}{\sqrt{k}}.$

Using the inequality
$\sum_{k=1}^{\lambda_{2}}\frac{1}{\sqrt{k}}<2\sqrt{\lambda_{2}}$ we get

$\sum_{k=1}^{\lambda_{2}}\sqrt[2k]{b_{k}}<\sqrt{(\lambda_{1}+1)\lambda_{2}{\rm
e}}\big{(}2\sqrt{\lambda_{2}}\big{)}=2\sqrt{{\rm
e}}\lambda_{2}\sqrt{\lambda_{1}+1}.$

Furthermore, Theorem 1.4 shows that $x_{n,n}\leq\frac{2(n-1)}{\sqrt{n+2}}\leq
2\sqrt{n-1}$ (the constant $2$ comes from the rescaling
$\textrm{{He}}_{n}(x)=2^{-\frac{n}{2}}H_{n}\big{(}\frac{x}{\sqrt{2}}\big{)}$).
Hence

$|z|\leq 2\sqrt{n-1}+2\sqrt{{\rm e}}\lambda_{2}\sqrt{\lambda_{1}+1}.$

Now $n-1=\lambda_{1}+\lambda_{2}-1<2(\lambda_{1}+1)$, so

$|z|<2\sqrt{2(\lambda_{1}+1)}+2\sqrt{{\rm
e}}\lambda_{2}\sqrt{\lambda_{1}+1}=2\big{(}\sqrt{{\rm
e}}\lambda_{2}+\sqrt{2}\big{)}\sqrt{\lambda_{1}+1}.$

This finishes the proof. ∎

Turán’s theorem is modelled after Walsh’s theorem [26] (see also [19]), which
gives a similar bound, but in terms of the expansion in the usual basis
$\\{x^{n}\\}_{n\geq 0}$. However, applying Walsh’s theorem to the coefficients
in Theorem 1.2 only gives a bound of the order $O\big{(}n\sqrt{n}\big{)}$. The
bound does not change for partitions of length $2$: for example, for
$\textrm{{He}}_{n,2}(x)$ it is possible to compute the character values
exactly and show that the upper bound in Walsh’s theorem is at least
$\Omega\big{(}n\sqrt{n}\big{)}$. Hence changing to the Hermite basis and
relying on Theorem 1.3 is necessary.

Let us now look at the real and purely imaginary roots of
$\textrm{{He}}_{\lambda}(x)$.

###### Proof of Corollary 1.6.

Let $x_{k,1}\leq x_{k,2}\leq\dots\leq x_{k,k}$ be the roots of
$\textrm{{He}}_{k}(x)$.

We show that $\textrm{{He}}_{\lambda}(x)$ has no real root in the interval
$(x_{n,n},+\infty)$. The roots of Hermite polynomials interlace, so
$x_{n,1}\leq x_{k,i}\leq x_{n,n}$ for all $1\leq i\leq k\leq n$. Therefore if
$z\in\mathbb{R}$ is greater than $x_{n,n}$, then $\textrm{{He}}_{k}(z)>0$ for
all $k\leq n$. Furthermore, $b_{k}\geq 0$ for all $k$ and $b_{0}=1$. Hence
$\textrm{{He}}_{\lambda}(z)>0$, so $z$ is not a root.

On the other hand,
$\textrm{{He}}_{\lambda}(-z)=(-1)^{|\lambda|}\textrm{{He}}_{\lambda}(z)$ (see
[5, Lemma 3.6]). Therefore $\textrm{{He}}_{\lambda}(x)$ has no roots in the
interval $(-\infty,-x_{n,n})$. This shows that any real root $z$ of
$\textrm{{He}}_{\lambda}(x)$ satisfies $|z|\leq x_{n,n}$.

Now let $z\in{\rm i}\mathbb{R}$ be a purely imaginary root of
$\textrm{{He}}_{\lambda}(x)$. Then $\textrm{{He}}_{\lambda}(z)={\rm
i}^{|\lambda|}\textrm{{He}}_{\lambda^{\prime}}(-{\rm i}z)$ (see [5,
Proposition 3.8]). Hence ${\rm i}z$ is a real root of
$\textrm{{He}}_{\lambda^{\prime}}(x)$. But $|\lambda^{\prime}|=n$, so
$|z|=|{\rm i}z|\leq x_{n,n}$ by the above argument. This finishes the proof. ∎

##  5 Proof of Proposition 1.7

In this section we will work with the unnormalized Hermite functions. For any
$n\geq 0$, define

$\varphi_{n}(x):={\rm e}^{-\frac{x^{2}}{2}}H_{n}(x).$

The functions $\varphi_{n}(x)$ have a nicer analytic behavior than $H_{n}(x)$
or $\textrm{{He}}_{n}(x)$ because they approach $0$ at infinity. Furthermore,
many previous results are stated in terms of the unnormalized Hermite
functions. For instance this allows us to apply results from [11].

The functions $\varphi_{n}(x)$ satisfy the Schrödinger equation

$-\varphi_{n}^{\prime\prime}(x)+x^{2}\varphi_{n}(x)=(2n+1)\varphi_{n}(x).$

Furthermore, for any non-negative integers $n_{1},n_{2},\dots,n_{r}$ we have
(see [11, Proposition 3.1]):

$\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r}}(x)]={\rm
e}^{-\frac{rx^{2}}{2}}\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)].$
(5.1)

######  Proposition 5.1.

Let $n_{1},n_{2},\dots,n_{r}$ be non-negative integers. Suppose $R\geq 0$ is
such that any root $x_{0}$ of $\varphi_{n_{i}}(x)$ verifies $|x_{0}|\leq R$,
for all $1\leq i\leq r$. If $z$ is a real root of
$\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\allowbreak\varphi_{n_{r}}(x)]$
then $|z|\leq R$.

Proposition 5.1 is equivalent to Proposition 1.7. This follows from Definition
1.1, the fact that $\textrm{{He}}_{n}(x)$ is a rescaling of $H_{n}(x)$, and
the fact that the roots of Hermite polynomials interlace.

###### Proof of Proposition 5.1.

Define $I_{R}:=(-\infty,-R)\cup(R,+\infty)$. We may assume without lack of
generality that the numbers $n_{1},\dots,n_{r}$ are distinct, otherwise the
determinant vanishes and the claim is trivially true. We will show that for
$x\in I_{R}$,
$\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r}}(x)]\neq
0$. We can order the numbers in increasing order, i.e.,
$n_{1}<n_{2}<\dots<n_{r}$.

The proof relies on the following simple observation.

######  Observation 5.2.

Suppose $\psi_{1}(x)$ and $\psi_{2}(x)$ verify the Schrödinger equation

$-\psi_{i}^{\prime\prime}(x)+V(x)\psi_{i}(x)=E_{i}\psi(x)$ (5.2)

on $I_{R}$, and do not vanish in $I_{R}$. Let
$w(x):=\textrm{{Wr}}[\psi_{1}(x),\psi_{2}(x)]$. If $E_{1}\neq E_{2}$ and
$\lim\limits_{x\rightarrow\pm\infty}w(x)=0$ then $w(x)$ has no zeros in
$I_{R}$.

###### Proof.

Note that

$\displaystyle
w(x)=\psi_{1}(x)\psi_{2}^{\prime}(x)-\psi_{1}^{\prime}(x)\psi_{2}(x),$
$\displaystyle
w^{\prime}(x)=\psi_{1}(x)\psi_{2}^{\prime\prime}(x)-\psi_{1}^{\prime\prime}(x)\psi_{2}(x)\overset{\mbox{\tiny{\eqref{eq:schrodinger}}}}{=}(E_{1}-E_{2})\psi_{1}(x)\psi_{2}(x).$

As $E_{1}\neq E_{2}$ and $\psi_{1}(x)\psi_{2}(x)\neq 0$ on $I_{R}$,
$w^{\prime}(x)$ has constant sign on each interval $(R,+\infty)$ and
$(-\infty,-R)$. As $\lim\limits_{x\rightarrow\pm\infty}w(x)=0$, it follows
that on each interval $(R,+\infty)$ and $(-R,-\infty)$, either $w(x)$ strictly
decreases from a positive value to $0$, or strictly increases from a negative
value to $0$. Hence $w(x)$ is never $0$ in these intervals. ∎

We now prove the statement by induction on $r\geq 1$.

If $r=1$, the Wronskian is just $\varphi_{n_{1}}(x)$, so the claim is
trivially true.

If $r=2$, notice that

$\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x)]\overset{\mbox{\tiny{\eqref{eq:weighted_wronskian}}}}{=}{\rm
e}^{-x^{2}}\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x)].$

Therefore
$\lim\limits_{x\rightarrow\pm\infty}\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x)]=0$.
By the choice of $R$, $\varphi_{n_{1}}(x)$ and $\varphi_{n_{2}}(x)$ have no
roots in $I_{R}$. Then the statement follows from Observation 5.2.

Now assume $r\geq 3$ and the induction hypothesis holds. Define

$\displaystyle\psi_{n_{r-2}}(x)=\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r-2}}(x)],$
$\displaystyle\psi_{n_{r-1}}(x)=\frac{\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r-2}}(x),\varphi_{n_{r-1}}(x)]}{\psi_{n_{r-2}}(x)},$
$\displaystyle\psi_{n_{r}}(x)=\frac{\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r-2}}(x),\varphi_{n_{r}}(x)]}{\psi_{n_{r-2}}(x)}.$

From the induction hypothesis it follows that
$\psi_{n_{r-2}}(x),\psi_{n_{r-1}}(x)$ and $\psi_{n_{r}}(x)$ do not vanish in
$I_{R}$. Then from the definition they are repeatedly differentiable in
$I_{R}$. In this situation, Crum [8] showed that for
$i\in\\{n_{r-1},n_{r}\\}$, $\psi_{i}$ verifies the Schrödinger equation

$-\psi_{i}^{\prime\prime}(x)+V(x)\psi_{i}(x)=(2i+1)\psi_{i}(x),$ (5.3)

where

$V(x)=x^{2}-2\frac{\partial^{2}}{\partial x^{2}}\log\psi_{n_{r-2}}(x).$

Crum proved this in the case when $n_{1},n_{2},\dots,n_{r-2}$ are consecutive
integers starting from $0$, and only for the interval $(0,1)$ with boundary
conditions. However, the proof of (5.3) remains valid for a sequence
$n_{1}<n_{2}<\dots<n_{r-2}$ of non-consecutive integers, in a neighborhood of
$x$ where the Wronskians do not vanish.

Let $w(x):=\textrm{{Wr}}[\psi_{n_{r-1}}(x),\psi_{n_{r}}(x)]$. Jacobi’s
identity for Wronskians tells us that

$\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r}}(x)]=\psi_{n_{r-2}}(x)w(x).$
(5.4)

Hence

$\displaystyle w(x)$
$\displaystyle=\frac{\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r}}(x)]}{\textrm{{Wr}}[\varphi_{n_{1}}(x),\varphi_{n_{2}}(x),\dots,\varphi_{n_{r-2}}(x)]}\overset{\mbox{\tiny{\eqref{eq:weighted_wronskian}}}}{=}\frac{{\rm
e}^{-\frac{rx^{2}}{2}}\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)]}{{\rm
e}^{-\frac{(r-2)x^{2}}{2}}\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r-2}}(x)]}$
$\displaystyle={\rm
e}^{-x^{2}}\frac{\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r}}(x)]}{\textrm{{Wr}}[H_{n_{1}}(x),H_{n_{2}}(x),\dots,H_{n_{r-2}}(x)]}.$

Then $\lim\limits_{x\rightarrow\pm\infty}w(x)=0$. From this and (5.3), we may
apply Observation 5.2 to deduce that $w(x)$ has no zeros in $I_{R}$. As
$\psi_{n_{r-2}}(x)\neq 0$ on $I_{R}$, the right-hand side of (5.4) does not
vanish in $I_{R}$. Hence the left-hand side Wronskian in (5.4) does not vanish
in $I_{R}$ either. ∎

##  6 Extension of the results to other polynomials

In this section we consider the possibility of extending our results to
polynomials which are obtained from the following generating function:

$\exp\left(xt-\frac{t^{q}}{q}\right)=\sum_{n=0}^{\infty}Q_{n}(x)\frac{t^{n}}{n!},$

where $q$ is a positive integer. For $q=2$ we recover the Hermite polynomials
$\textrm{{He}}_{n}(x)$. Note that we omit the dependence of $Q_{n}(x)$ on $q$
as this will always be clear from the context.

The polynomials $Q_{n}(x)$ have similar properties as the Hermite polynomials.
They are $(q-1)$-orthogonal and Appell polynomials [3], and satisfy the
recurrence relation:

$Q_{n}(x)=xQ_{n-1}(x)-\frac{(n-1)!}{(n-q)!}Q_{n-q}(x),$

with $Q_{n}(x)=x^{n}$, $0\leq n\leq q-1$.

$Q_{n}(x)$ are $d$-symmetric polynomials with $d:=q-1$:

$Q_{n}(\omega_{q}z)=\omega_{q}^{n}Q_{n}(z),$

where $\omega_{q}:=\exp\big{(}\frac{2\pi{\rm i}}{q}\big{)}$. In particular,
the zeros of $Q_{n}(x)$ lie on the $q$-star

$\bigcup_{\ell=0}^{q}[0,\infty)\times\omega_{q}^{\ell}.$

$Q_{n}(x)$ has exactly $\big{\lfloor}\frac{n}{q}\big{\rfloor}$ positive simple
real zeros [1, Theorem 2.2], and the largest zero in absolute value from
$Q_{1}(x),\dots,Q_{n}(x)$ belongs to $Q_{n}(x)$ [1, equations (2.5)–(2.7)].

For a partition $\lambda$ and $q\geq 3$ arbitrary, one can define the
Wronskian polynomial $Q_{\lambda}(x)$ in analogy with the case $q=2$:

$Q_{\lambda}(x):=\frac{\textrm{{Wr}}[Q_{n_{1}}(x),Q_{n_{2}}(x),\dots,Q_{n_{r}}(x)]}{\Delta(n_{\lambda})},$

where $n_{\lambda}=(n_{1},n_{2},\dots,n_{r})$ is the degree vector of
$\lambda$.

These polynomials were studied in [3, Section 7] and in [2]. For $q=3$,
$Q_{\lambda}(x)$ can be used to define the Yablonskii–Vorobiev polynomials,
which give rational solutions to the second Painlevé equation. Most of the
properties of Wronskian Hermite polynomials generalize to arbitrary $q$. In
particular, $Q_{\lambda}(x)$ are monic integer polynomials with coefficients
determined by the characters of the symmetric group.

######  Theorem 6.1 ([3, Theorem 7]).

Let $\lambda\vdash n$. Then

$Q_{\lambda}(x)=\sum_{k=0}^{\lfloor
n/q\rfloor}(-1)^{k}H(\lambda)\frac{\chi^{\lambda}\big{(}q^{k}1^{n-qk}\big{)}}{q^{k}k!(n-qk)!}x^{n-qk}.$

Therefore it is natural to ask if Theorem 1.3 generalizes, for example with
$K_{\lambda^{\prime}(2^{k}1^{n-2k})}$ replaced by
$K_{\lambda^{\prime}(q^{k}1^{n-qk})}$. We show here that the generalization
only partially holds: enough to imply Corollary 1.6 for any $q$, but the
coefficients do not have a simple form.

As in Lemma 3.1, we start with the change of basis formula:

######  Lemma 6.2.

Let $\lambda\vdash n$ and write
$Q_{\lambda}(x)=\sum_{j=0}^{\lfloor\frac{n}{q}\rfloor}a_{j}x^{n-qj}$. Then
$Q_{\lambda}(x)=\sum_{k=0}^{\lfloor\frac{n}{q}\rfloor}b_{k}Q_{n-qk}(x)$ where

$b_{k}=\sum_{j=0}^{k}\frac{(n-qj)!}{q^{k-j}(k-j)!(n-qk)!}a_{j},\qquad 0\leq
k\leq\left\lfloor\frac{n}{q}\right\rfloor.$

The proof is similar to that of Lemma 3.1 so we omit it.

Now once more we have to evaluate the expression obtained by replacing

$a_{j}=(-1)^{j}\frac{H(\lambda)}{q^{j}j!(n-qj)!}\chi^{\lambda}\big{(}q^{j}1^{n-qj}\big{)}$

in $b_{k}$. Thus we are led to the evaluation of the sum
$S_{q,k}^{\lambda}=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\chi^{\lambda}\big{(}q^{j}1^{n-qj}\big{)}$.
There is no analogue to Lemma 3.2. The most we can say is the following.

######  Lemma 6.3.

Let $q\geq 2,\lambda\vdash n$ and $0\leq
k\leq\big{\lfloor}\frac{n}{q}\big{\rfloor}$. Then $S_{q,k}^{\lambda}\geq 0$.

###### Proof.

Set $\mu^{(q,j)}:=\big{(}q^{j}1^{n-qj}\big{)}$ for $0\leq j\leq k$.

Proceeding as in the proof of Lemma 3.2, with the same notations, we must
determine the coefficient of the monomial $M_{\lambda}$ in the polynomial

$\displaystyle\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}p_{\mu^{(q,j)}}W_{0}$
$\displaystyle=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}p_{q}^{j}p_{1}^{n-qj}W_{0}$
$\displaystyle=W_{0}p_{1}^{n-qk}\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}p_{q}^{j}p_{1}^{q(k-j)}$
$\displaystyle=W_{0}p_{1}^{n-qk}\big{(}p_{1}^{q}-p_{q}\big{)}^{k}.$

Let us try to interpret $p_{1}^{n-qk}\big{(}p_{1}^{q}-p_{q}\big{)}^{k}$
combinatorially.

Let $\rho$ be any partition. Then from Theorem 2.3,

$p_{1}^{q}s_{\rho}=\sum_{|\gamma/\rho|=q}\alpha_{\gamma}s_{\gamma},$

where $\alpha_{\gamma}$ counts the number of ways one can obtain $\gamma$ from
$\rho$ by adding $q$ labelled squares to the Ferrers diagram $D_{\rho}$. Note
that these squares need not form a hook or be connected.

Similarly,

$p_{q}s_{\rho}=\sum_{\gamma\in\mathcal{R}(\rho,q)}(-1)^{ht(\gamma/\rho)}s_{\gamma}.$

Adding these two, we obtain
$\big{(}p_{1}^{q}-p_{q}\big{)}s_{\rho}=\sum_{|\gamma/\rho|=q}\beta_{\gamma}s_{\gamma}$,
where

$\beta_{\gamma}=\begin{cases}\alpha_{\gamma},&\textrm{if $\gamma/\rho$ is not
a $q$-hook},\\\ \alpha_{\gamma}+1,&\textrm{if $\gamma/\rho$ is a $q$-hook
spanning an even number of rows},\\\ \alpha_{\gamma}-1,&\textrm{if
$\gamma/\rho$ is a $q$-hook spanning an odd number of rows}.\end{cases}$

In particular, $\beta_{\gamma}\geq 0$.

Starting from the Schur polynomial $s_{0}=1$ for the partition with only $0$
parts and length $n$, and applying the above, we obtain:

$p_{1}^{n-qk}\big{(}p_{1}^{q}-p_{q}\big{)}^{k}W_{0}=\sum_{\begin{subarray}{c}\rho\\\
\ell(\rho)=n\end{subarray}}K_{\rho}s_{\rho}W_{0}=\sum_{\begin{subarray}{c}\rho\\\
\ell(\rho)=n\end{subarray}}K_{\rho}W_{\rho},$

where $K_{\rho}\geq 0$. Exactly as in the proof of Lemma 3.2 one can now
conclude that the coefficient of $M_{\lambda}$ is $K_{\lambda}$, and so
$S_{q,k}^{\lambda}=K_{\lambda}$. ∎

However, for $q\geq 3$ the coefficients $K_{\rho}$ that appear in the proof
above are no longer related to the Kostka numbers. In the case $q=2$ there are
only two types of $2$-hooks: horizontal $2$-hooks ($2$ squares in the same
row), and vertical $2$-hooks ($2$ squares in the same column). Then
$\beta_{\gamma}=0$ when $\gamma/\rho$ is an horizontal $2$-hook. This property
explains why $K_{\rho}$ corresponds to a Kostka number.

Nevertheless, knowing that the expansion in the $\\{Q_{n}(x)\\}_{n\geq 0}$
basis has non-negative coefficients implies the analogue of Corollary 1.6.

######  Corollary 6.4.

Let $q\geq 2$ and $\lambda\vdash n$. If $z$ is a root of $Q_{\lambda}(x)$
located on the $2q$-star then $|z|\leq x_{n}$, where $x_{n}$ is the largest
real root of $Q_{n}(x)$.

The proof uses Lemma 6.3 together with the property
$Q_{\lambda}(x)=(-\omega_{2q})^{n}Q_{\lambda^{\prime}}\big{(}{-}\omega_{2q}^{-1}x\big{)}$
(see [4, Section 7.2]), and we omit it.

Note that for $q=2$ we recover Corollary 1.6.

### Acknowledgements

The authors are indebted to the referees for the careful reading and for
suggesting to extend the results to $q\geq 3$.

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