# Generalised higher-order Freud weights

Peter A. Clarkson1, Kerstin Jordaan2 and Ana Loureiro1
1 School of Mathematics, Statistics and Actuarial Science,
University of Kent, Canterbury, CT2 7FS, UK
2 Department of Decision Sciences,
University of South Africa, Pretoria, 0003, South Africa
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We discuss polynomials orthogonal with respect to a semi-classical generalised
higher order Freud weight

$\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(tx^{2}-x^{2m}\right),\qquad
x\in\mathbb{R},$

with parameters $\lambda>-1$, $t\in\mathbb{R}$ and $m=2,3,\dots$ . The
sequence of generalised higher order Freud weights for $m=2,3,\dots$, forms a
hierarchy of weights, with associated hierarchies for the first moment and the
recurrence coefficient. We prove that the first moment can be written as a
finite partition sum of generalised hypergeometric ${}_{1}F_{m}$ functions and
show that the recurrence coefficients satisfy difference equations which are
members of the first discrete Painlevé hierarchy. We analyse the asymptotic
behaviour of the recurrence coefficients and the limiting distribution of the
zeros as $n\to\infty$. We also investigate structure and other mixed
recurrence relations satisfied by the polynomials and related properties.

##  1 Introduction

In this paper we consider polynomials orthogonal with respect to the
generalised higher order Freud weight

$\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(tx^{2}-x^{2m}\right),\qquad
x,t\in\mathbb{R},\qquad m=2,3,\dots,$ (1.1)

with $\lambda>-1$ a parameter.

Let $\omega(x)$ be a positive weight function defined on the real line
$\mathbb{R}$ for which all the moments

$\mu_{k}=\int_{-\infty}^{\infty}x^{k}\omega(x)\,{\rm d}x,\qquad k=0,1,\dots,$
(1.2)

exist. Then the sequence of monic orthogonal polynomials
$\big{\\{}P_{n}(x)\big{\\}}_{n\in\mathbb{N}}$, where $P_{n}(x)$ is a
polynomial of degree $n$ in $x$, is given by

$\int_{-\infty}^{\infty}P_{m}(x)P_{n}(x)\,\omega(x)\,{\rm
d}x=h_{n}\delta_{m,n},\qquad h_{n}>0,$

where $\delta_{m,n}$ denotes the Kronekar delta.

A fundamental property of orthogonal polynomials is that they satisfy a three-
term recurrence relation of the form

$P_{n+1}(x)=(x-\alpha_{n})P_{n}(x)-\beta_{n}P_{n-1}(x),$ (1.3)

with $\beta_{n}>0$ and initial values $P_{-1}(x)=0$ and $P_{0}(x)=1$. The
recurrence coefficients $\alpha_{n}$ and $\beta_{n}$ are given by the
integrals

$\alpha_{n}=\frac{1}{h_{n}}\int_{-\infty}^{\infty}xP_{n}^{2}(x)\,\omega(x)\,{\rm
d}x,\qquad\beta_{n}=\frac{1}{h_{n-1}}\int_{-\infty}^{\infty}xP_{n-1}(x)P_{n}(x)\,\omega(x)\,{\rm
d}x.$

The coefficient $\beta_{n}$ in the recurrence relation (1.3) can also be
expressed in terms of the Hankel determinant

$\Delta_{n}=\mathop{\rm
det}\nolimits\big{[}\mu_{j+k}\big{]}_{j,k=0}^{n-1}=\left|\begin{matrix}\mu_{0}&\mu_{1}&\ldots&\mu_{n-1}\\\
\mu_{1}&\mu_{2}&\ldots&\mu_{n}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mu_{n-1}&\mu_{n}&\ldots&\mu_{2n-2}\end{matrix}\right|,\qquad n\geq 1,$ (1.4)

with $\Delta_{0}=1$, $\Delta_{-1}=0$, whose entries are given in terms of the
moments (1.2) associated with the weight $\omega(x)$. Specifically

$\beta_{n}=\frac{\Delta_{n+1}\Delta_{n-1}}{\Delta_{n}^{2}}.$ (1.5)

The monic polynomial $P_{n}(x)$ can be uniquely expressed as the determinant

$P_{n}(x)=\frac{1}{\Delta_{n}}\left|\begin{matrix}\mu_{0}&\mu_{1}&\ldots&\mu_{n}\\\
\mu_{1}&\mu_{2}&\ldots&\mu_{n+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mu_{n-1}&\mu_{n}&\ldots&\mu_{2n-1}\\\ 1&x&\ldots&x^{n}\end{matrix}\right|,$

and the normalisation constants as

$h_{n}=\frac{\Delta_{n+1}}{\Delta_{n}},\qquad h_{0}=\Delta_{1}=\mu_{0}.$ (1.6)

Also from (1.5) and (1.6), we see that the relationship between the recurrence
coefficient $\beta_{n}$ and the normalisation constants $h_{n}$ is given by

$h_{n}=\beta_{n}h_{n-1}.$

For symmetric weights, since $\omega(x)=\omega(-x)$, it follows that
$\alpha_{n}=0$ in (1.3). Hence, for symmetric weights, the sequence of monic
orthogonal polynomials $\big{\\{}P_{n}(x)\big{\\}}_{n\in\mathbb{N}}$, satisfy
the three-term recurrence relation

$P_{n+1}(x)=xP_{n}(x)-\beta_{n}P_{n-1}(x).$ (1.7)

The monic orthogonal polynomials $P_{n}(x)$ associated with symmetric weights
are also symmetric, i.e. $P_{n}(-x)=(-1)^{n}P_{n}(x)$. This implies that each
$P_{n}$ contains only even or only odd powers of $x$ and we can write

$\displaystyle P_{2n}(x)$
$\displaystyle=x^{2n}+\sum_{k=1}^{n}c_{2n-2k}^{(2n)}x^{2n-2k}=x^{2n}+c_{2n-2}^{(2n)}x^{2n-2}+\dots+c_{0}^{(2n)},$
$\displaystyle P_{2n+1}(x)$
$\displaystyle=x^{2n+1}+\sum_{k=1}^{n}c_{2n-2k+1}^{(2n+1)}x^{2n-2k+1}=x^{2n+1}+c_{2n-1}^{(2n+1)}x^{2n-1}+\dots+c_{1}^{(2n+1)}x.$

Substituting these expressions into the recurrence relation (1.7) and
comparing the coefficients on each side, we obtain

$\displaystyle\beta_{2n}$
$\displaystyle=c_{2n-2}^{(2n)}-c_{2n-1}^{(2n+1)},\qquad\beta_{2n+1}=-\frac{c_{0}^{(2n+2)}}{c_{0}^{(2n)}}=-\frac{P_{2n+2}(0)}{P_{2n}(0)}.$
(1.8)

It follows from (1.2) that, for symmetric weights, $\mu_{2k-1}=0$,
$k=1,2,\ldots\ $ and hence it is possible to write the Hankel determinant
$\Delta_{n}$ given by (1.4) in terms of the product of two Hankel determinants
obtained by matrix manipulation, interchanging columns and rows. The product
decomposition, depending on $n$ even or odd, is given by

$\Delta_{2n}=\mathcal{A}_{n}\mathcal{B}_{n},\qquad\Delta_{2n+1}=\mathcal{A}_{n+1}\mathcal{B}_{n},$
(1.9)

where $\mathcal{A}_{n}$ and $\mathcal{B}_{n}$ are the Hankel determinants

$\displaystyle\mathcal{A}_{n}$
$\displaystyle{=\left|\begin{matrix}\mu_{0}&\mu_{2}&\ldots&\mu_{2n-2}\\\
\mu_{2}&\mu_{4}&\ldots&\mu_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mu_{2n-2}&\mu_{2n}&\ldots&\mu_{4n-4}\end{matrix}\right|,}\qquad\mathcal{B}_{n}{=\left|\begin{matrix}\mu_{2}&\mu_{4}&\ldots&\mu_{2n}\\\
\mu_{4}&\mu_{6}&\ldots&\mu_{2n+2}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mu_{2n}&\mu_{2n+2}&\ldots&\mu_{4n-2}\end{matrix}\right|},$ (1.10)

with $\mathcal{A}_{0}=\mathcal{B}_{0}=1$. Consequently, for a symmetric
weight, substituting (1.9) into (1.5), the recurrence coefficient $\beta_{n}$
is given by

$\beta_{2n}=\frac{\mathcal{A}_{n+1}\mathcal{B}_{n-1}}{\mathcal{A}_{n}\mathcal{B}_{n}},\qquad\beta_{2n+1}=\frac{\mathcal{A}_{n}\mathcal{B}_{n+1}}{\mathcal{A}_{n+1}\mathcal{B}_{n}}.$

Semiclassical orthogonal polynomials are natural generalisations of classical
orthogonal polynomials and were introduced by Shohat in [30]. Maroni provided
a unified theory for semiclassical orthogonal polynomials (cf. [23, 24]). The
weights of classical orthogonal polynomials satisfy a first-order ordinary
differential equation, the Pearson equation

$\frac{{\rm d}{}}{{{\rm d}{x}}}(\sigma(x)\omega(x))=\tau(x)\omega(x),$

where $\sigma(x)$ is a monic polynomial of degree at most $2$ and $\tau(x)$ is
a polynomial with degree $1$. For semiclassical orthogonal polynomials, the
weight function $\omega(x)$ satisfies a Pearson equation (1) with either
deg$(\sigma(x))>2$ or deg$(\tau(x))\neq 1$ (cf. [14, 22]). The generalised
higher order Freud weight given by (1.1) is a symmetric weight that satisfies

$\frac{{\rm d}{}}{{{\rm
d}{x}}}\left\\{\sigma(x)\omega(x;t,\lambda)\right\\}=\tau(x)\omega(x;t,\lambda),$
(1.11)

with $\sigma(x)=x$ and $\tau(x)=2(tx^{2}-mx^{2m}+\lambda+1)$ and therefore is
a semiclassical weight.

In §2 we consider the moments of the generalised higher order Freud weight,
obtaining a closed form expression for the first moment. The recurrence
coefficients in the three term recurrence relation satisfied by polynomials
orthogonal with respect to generalised higher order Freud weights are
investigated in §3. We prove structure relations and mixed recurrence
relations satisfied by generalised higher order Freud polynomials in §4. The
asymptotic behaviour of the recurrence coefficients proved in §3 determines
the limiting distribution of the zeros and this, as well as other properties
of the zeros, is investigated in §5. We conclude with the quadratic
decomposition of the generalised higher order Freud weight in §6.

##  2 Moments of the generalised higher order Freud weights

The existence of the first moment $\mu_{0}(t;\lambda,m)$ associated with the
generalised higher order Freud weight (1.1) follows from the fact that, at
$\infty$, the integrand behaves like $\exp(-x^{2})$ and, at $x=0$, the
integrand behaves like $x^{\lambda}$ which, for $\lambda>-1$, is integrable.

######  Lemma 2.1.

Let $x\in\mathbb{R},~{}\lambda>-1,~{}t\in\mathbb{R}$ and $m=2,3,\dots$. Then,
for the generalised higher order Freud weight (1.1), the first moment is given
by

$\displaystyle\mu_{0}(t;\lambda,m)$
$\displaystyle=\int_{-\infty}^{\infty}|x|^{2\lambda+1}\exp(tx^{2}-x^{2m})\,{\rm
d}x=\int_{0}^{\infty}s^{\lambda}\exp(ts-s^{m})\,{\rm d}s$
$\displaystyle=\displaystyle\frac{1}{m}\sum_{k=1}^{m}\frac{t^{k-1}}{{(k-1)}!}\Gamma\left(\frac{\lambda+k}{m}\right)\;{}_{2}F_{m}\left(\frac{\lambda+k}{m},1;\frac{k}{m},\frac{k+1}{m},\dots,\frac{m+k-1}{m};\left(\frac{t}{m}\right)^{m}\right)$
$\displaystyle=\displaystyle\frac{1}{m}\Gamma\left(\frac{\lambda+1}{m}\right)\;{}_{1}F_{m-1}\left(\frac{\lambda+1}{m};\frac{1}{m},\dots\frac{m-1}{m};\left(\frac{t}{m}\right)^{m}\right)$
$\displaystyle\qquad+\displaystyle\frac{1}{m}\sum_{k=2}^{m-1}\frac{t^{k-1}}{{(k-1)}!}\Gamma\left(\frac{\lambda+k}{m}\right)$
$\displaystyle\qquad\qquad\times{}_{1}F_{m-1}\left(\frac{\lambda+k}{m};\frac{k}{m},\frac{k+1}{m},\dots,\frac{m-1}{m},\frac{m+1}{m},\dots,\frac{m+k-1}{m};\left(\frac{t}{m}\right)^{m}\right)$
$\displaystyle\qquad+\displaystyle\frac{t^{m-1}}{m!}\Gamma\left(\frac{\lambda}{m}+1\right)\;{}_{1}F_{m}\left(\frac{\lambda}{m}+1;\frac{m+1}{m},\frac{m+2}{m},\dots,\frac{2m-1}{m};\left(\frac{t}{m}\right)^{m}\right),$

where ${}_{p}F_{q}(a;b;z)$ is the generalised hypergeometric function (cf.
[28, (16.2.1)].

###### Proof.

Using the power series expansion of the exponential function, we obtain

$\begin{split}\mu_{0}(t;\lambda,m)&=\int_{-\infty}^{\infty}|x|^{2\lambda+1}\exp(tx^{2}-x^{2m})\,{\rm
d}x=\int_{0}^{\infty}s^{\lambda}\exp(ts-s^{m})\,{\rm d}s\\\
&=\displaystyle\int_{0}^{\infty}s^{\lambda}\exp(-s^{m})\sum_{n=0}^{\infty}\frac{(ts)^{n}}{n!}\,{\rm
d}s\\\
&=\displaystyle\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\int_{0}^{\infty}s^{n+\lambda}\exp(-s^{m})\,{\rm
d}s\\\
&=\displaystyle\frac{1}{m}\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\int_{0}^{\infty}y^{(n+\lambda-m+1)/m}\exp(-y)\,{\rm
d}y\\\
&=\displaystyle\frac{1}{m}\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\,\Gamma\left(\frac{\lambda+n+1}{m}\right),\end{split}$

where $\Gamma(x)$ denotes the Gamma function defined in [28, (5.2.1)] and the
fourth equal sign follows from the Lebesgue’s Dominated Convergence Theorem.
Letting $n=mk+j$ for $j=0,1,\dots,m-1$, we can write

$\mu_{0}(t;\lambda,m)=\displaystyle\frac{1}{m}\sum_{k=0}^{\infty}\sum_{j=0}^{m-1}\Gamma\left(\frac{\lambda+j+1}{m}+k\right)\;\frac{t^{mk+j}}{(mk+j)!}.$

Using the Gauss multiplication formula [28, (5.5.6)] yields

$(mk+j)!=j!\,m^{mk}\prod_{\ell=1}^{m}\left(\frac{j+\ell}{m}\right)_{k}$

where $(a)_{k}$ denotes the Pochhammer symbol (cf. [28, §5.2(iii)], while it
follows from [28, (5.5.1)] that

$\Gamma\left(\frac{\lambda+j+1}{m}+k\right)=\left(\frac{\lambda+j+1}{m}\right)_{k}\Gamma\left(\frac{\lambda+j+1}{m}\right),$

and hence we have

$\begin{split}\mu_{0}(t;\lambda,m)&=\displaystyle\frac{1}{m}\sum_{k=0}^{\infty}\sum_{j=0}^{m-1}\frac{\left(\frac{\lambda+j+1}{m}\right)_{k}\Gamma\left(\frac{\lambda+j+1}{m}\right)}{m^{mk}\left(\frac{j+1}{m}\right)_{k}\left(\frac{j+2}{m}\right)_{k}...\left(\frac{j+m}{m}\right)_{k}}\,\frac{t^{mk+j}}{j!}\\\
&=\displaystyle\frac{1}{m}\sum_{j=0}^{m-1}\Gamma\left(\frac{\lambda+j+1}{m}\right)\frac{t^{j}}{j!}\sum_{k=0}^{\infty}\frac{\left(\frac{\lambda+j+1}{m}\right)_{k}}{\left(\frac{j+1}{m}\right)_{k}\left(\frac{j+2}{m}\right)_{k}...\left(\frac{j+m}{m}\right)_{k}}\left(\frac{t}{m}\right)^{mk}\\\
&=\displaystyle\frac{1}{m}\sum_{j=0}^{m-1}\Gamma\left(\frac{\lambda+j+1}{m}\right)\frac{t^{j}}{j!}\;{}_{2}F_{m}\left(\frac{\lambda+j+1}{m},1;\frac{j+1}{m},\frac{j+2}{m},\dots,\frac{m+j}{m};\left(\frac{t}{m}\right)^{m}\right).\end{split}$

∎

######  Remark 2.2.

In our earlier studies of semi-classical orthogonal polynomials, we proved
special cases of Lemma 2.3 and Lemma 2.1, namely for $m=2$ in [6] and for
$m=3,4,5$ in [5].

######  Lemma 2.3.

Let $x\in\mathbb{R},~{}\lambda>-1,~{}t\in\mathbb{R}$ and $m=2,3,\dots$. Then,
for the generalised higher order Freud weight (1.1), the first moment

$\mu_{0}(t;\lambda,m)=\int_{-\infty}^{\infty}|x|^{2\lambda+1}\exp(tx^{2}-x^{2m})\,{\rm
d}x=\int_{0}^{\infty}s^{\lambda}\exp(ts-s^{m})\,{\rm d}s,$

satisfies the ordinary differential equation

$m\frac{{\rm d}^{m}{\varphi}}{{{\rm d}{t}}^{m}}-t\frac{{\rm d}{\varphi}}{{{\rm
d}{t}}}-(\lambda+1)\,\varphi=0.$ (2.1)

###### Proof.

Following [26] and [5], we look for a solution of (2.1) in the form

$\varphi(t)=\int_{0}^{\infty}{\rm e}^{st}\,v(s)\,{\rm d}s.$ (2.2)

In order for (2.2) to satisfy (2.1), it is necessary that

$\frac{{\rm d}^{m}{\varphi}}{{{\rm d}{t}}^{m}}-\frac{1}{m}\frac{{\rm
d}{\varphi}}{{{\rm d}{t}}}-\frac{\lambda+1}{m}\varphi=\int_{0}^{\infty}{\rm
e}^{st}\left(s^{m}-\frac{s}{m}-\frac{\lambda+1}{m}\right)v(s)\,{\rm d}s=0.$

Using integration by parts, this is equivalent to

$\int_{0}^{\infty}{\rm
e}^{st}\left\\{s^{m}v(s)+\frac{1}{m}v(s)+\frac{s}{m}\frac{{\rm d}{v}}{{{\rm
d}{s}}}-\frac{\lambda+1}{m}v(s)\right\\}{\rm d}s=0,$

under the assumption that $\lim_{s\to\infty}sv(s){\rm e}^{st}=0$. Hence, for
$\varphi(t)$ to be a solution of (2.1), we need to choose $v(s)$ so that

$(ms^{m}-\lambda)v(s)+s\frac{{\rm d}{v}}{{{\rm d}{s}}}=0.$

One solution of this equation is $v(s)=s^{\lambda}\exp(-s^{m})$.∎

For the generalised higher order Freud weight (1.1), the even moments can be
written in terms of derivatives of the first moment, as follows

$\displaystyle\mu_{2k}(t;\lambda,m)$
$\displaystyle=\int_{-\infty}^{\infty}x^{2k}|x|^{2\lambda+1}\exp(tx^{2}-x^{2m})\,{\rm
d}x$ $\displaystyle=\frac{{\rm d}^{k}{}}{{{\rm
d}{t}}^{k}}\int_{-\infty}^{\infty}|x|^{2\lambda+1}\exp(tx^{2}-x^{2m})\,{\rm
d}x$ $\displaystyle=\frac{{\rm d}^{k}{}}{{{\rm
d}{t}}^{k}}\mu_{0}(t;\lambda,m),\qquad k=0,1,2,\ldots\ ,$ (2.3)

where the interchange of integration and differentiation is justified by
Lebesgue’s Dominated Convergence Theorem. Furthermore, from the definition we
have

$\displaystyle\mu_{2k+2}(t;\lambda,m)$
$\displaystyle=\mu_{2k}(t;\lambda+1,m),\qquad k=0,1,2,\ldots\ .$ (2.4)

##  3 Recurrence coefficients for generalised higher order Freud weights

######  Lemma 3.1.

For the generalised higher order Freud weight (1.1), the recurrence
coefficient $\beta_{n}$ is given by

$\beta_{2n}=\frac{{\rm d}{}}{{{\rm
d}{t}}}\ln\frac{\mathcal{B}_{n}}{\mathcal{A}_{n}},\qquad\beta_{2n+1}=\frac{{\rm
d}{}}{{{\rm d}{t}}}\ln\frac{\mathcal{A}_{n+1}}{\mathcal{B}_{n}}.$ (3.1)

with $A_{0}=B_{0}=1$ and

$\mathcal{A}_{n}=\operatorname{Wr}\left(\mu_{0},\frac{{\rm d}{\mu_{0}}}{{{\rm
d}{t}}},\ldots,\frac{{\rm d}^{n-1}{\mu_{0}}}{{{\rm
d}{t}}^{n-1}}\right),\qquad\mathcal{B}_{n}=\operatorname{Wr}\left(\frac{{\rm
d}{\mu_{0}}}{{{\rm d}{t}}},\frac{{\rm d}^{2}{\mu_{0}}}{{{\rm
d}{t}}^{2}},\ldots,\frac{{\rm d}^{n}{\mu_{0}}}{{{\rm d}{t}}^{n}}\right),$
(3.2)

where

$\mu_{0}=\mu_{0}(t;\lambda,m)=\int_{0}^{\infty}x^{\lambda}\exp(ts-s^{m})\,{\rm
d}{x},$

and $\operatorname{Wr}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ denotes
the Wronskian given by

$\operatorname{Wr}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})=\left|\begin{matrix}\varphi_{1}&\varphi_{2}&\ldots&\varphi_{n}\\\
\varphi_{1}^{(1)}&\varphi_{2}^{(1)}&\ldots&\varphi_{n}^{(1)}\\\
\vdots&\vdots&\ddots&\vdots\\\
\varphi_{1}^{(n-1)}&\varphi_{2}^{(n-1)}&\ldots&\varphi_{n}^{(n-1)}\end{matrix}\right|,\qquad\varphi_{j}^{(k)}=\frac{{\rm
d}^{k}{\varphi_{j}}}{{{\rm d}{t}}^{k}}.$

###### Proof.

It follows from (1.10) and (2) that $\mathcal{A}_{n}$ and $\mathcal{B}_{n}$
can be written in terms of the Wronskians given by (3.2). Furthermore,

$\displaystyle\mathcal{A}_{n}\frac{{\rm d}{\mathcal{B}_{n}}}{{{\rm
d}{t}}}-\mathcal{B}_{n}\frac{{\rm d}{\mathcal{A}_{n}}}{{{\rm
d}{t}}}=\mathcal{A}_{n+1}\mathcal{B}_{n-1},\qquad\mathcal{B}_{n}\frac{{\rm
d}{\mathcal{A}_{n+1}}}{{{\rm d}{t}}}-\mathcal{A}_{n+1}\frac{{\rm
d}{\mathcal{B}_{n}}}{{{\rm d}{t}}}=\mathcal{A}_{n+1}\mathcal{B}_{n}$ (3.3)

(cf. [34, §6.5.1]) and (3.3), together with (1) yields (3.1). ∎

######  Lemma 3.2.

Let $\omega_{0}(x)$ be a symmetric positive weight on the real line for which
all the moments exist and let $\omega(x;t)=\exp(tx^{2})\,\omega_{0}(x)$, with
$t\in\mathbb{R}$, is a weight such that all the moments of exist. Then the
recurrence coefficient $\beta_{n}(t)$ satisfies the Volterra, or the Langmuir
lattice, equation

$\frac{{\rm d}{\beta_{n}}}{{{\rm d}{t}}}=\beta_{n}(\beta_{n+1}-\beta_{n-1}).$

###### Proof.

See, for example, Van Assche [32, Theorem 2.4]. ∎

######  Lemma 3.3.

For the generalised higher order Freud weight (1.1), the associated monic
polynomials $P_{n}(x)$ satisfy the recurrence relation

$P_{n+1}(x)=xP_{n}(x)-\beta_{n}(t;\lambda)P_{n-1}(x),\qquad n=0,1,2,\ldots\ ,$
(3.4)

with $P_{-1}(x)=0$ and $P_{0}(x)=1$, where

$\displaystyle\beta_{2n}(t;\lambda)$
$\displaystyle=\frac{\mathcal{A}_{n+1}(t;\lambda)\mathcal{A}_{n-1}(t;\lambda+1)}{\mathcal{A}_{n}(t;\lambda)\mathcal{A}_{n}(t;\lambda+1)}=\frac{{\rm
d}{}}{{{\rm
d}{t}}}\ln\frac{\mathcal{A}_{n}(t;\lambda+1)}{\mathcal{A}_{n}(t;\lambda)},$
$\displaystyle\beta_{2n+1}(t;\lambda)$
$\displaystyle=\frac{\mathcal{A}_{n}(t;\lambda)\mathcal{A}_{n+1}(t;\lambda+1)}{\mathcal{A}_{n+1}(t;\lambda)\mathcal{A}_{n}(t;\lambda+1)}=\frac{{\rm
d}{}}{{{\rm
d}{t}}}\ln\frac{\mathcal{A}_{n+1}(t;\lambda)}{\mathcal{A}_{n}(t;\lambda+1)}.$

where $\mathcal{A}_{n}(t;\lambda)$ is the Wronskian given by (3.2) with

$\displaystyle\mu_{0}(t;\lambda,m)$
$\displaystyle=\displaystyle\frac{1}{m}\sum_{k=1}^{m}\frac{t^{k-1}}{{(k-1)}!}\Gamma\left(\frac{\lambda+k}{m}\right)\;{}_{2}F_{m}\left(\frac{\lambda+k}{m},1;\frac{k}{m},\frac{k+1}{m},\dots,\frac{m+k-1}{m};\left(\frac{t}{m}\right)^{m}\right).$

###### Proof.

It follows from substituting (2.4) into the expression for
$\mathcal{B}_{n}(t;\lambda)$ given in (3.2) that
$\mathcal{B}_{n}=\mathcal{A}_{n}(t;\lambda+1)$ and then the result immediately
follows from (1) and (3.1). ∎

###  3.1 Nonlinear recursive relations

We follow the approach found in [23, §7] whose key results are summarised in
[25, Proposition 3.1].

Note that for a given $m\geq 1$, we can write

$x^{2m}P_{n}(x)=\sum_{\ell=-m}^{m}C^{(2m)}_{n,n+2\ell}P_{n+2\ell},$ (3.5)

where

$C^{(2m)}_{n,n+2\ell}=\frac{1}{h_{n+2\ell}}\int_{-\infty}^{\infty}x^{2m}P_{n+2\ell}(x)P_{n}(x)\,\omega(x)\,{\rm
d}x\quad\text{for}\quad\ell=-m,\ldots,m.$

Observe that $C^{(2m)}_{n,n+k}=C^{(2m)}_{n+k,n}=0$ for $|k|\geq 2m+1$ and

$C^{(2m)}_{n,n+2\ell}=\frac{h_{n}}{h_{n+2\ell}}C^{(2m)}_{n+2\ell,n}=\frac{1}{\beta_{n+1}\cdots\beta_{n+2\ell}}C^{(2m)}_{n+2\ell,n}\,\quad\text{for}\quad\ell=1,\ldots,m.$

From the recurrence relation (1.7) it follows

$x^{2}P_{n}(x)=P_{n+2}+\left(\beta_{n}+\beta_{n+1}\right)P_{n}+\beta_{n-1}\beta_{n}P_{n-2},\quad
n\geq 0.$ (3.6)

In particular, one has $C_{n,n}^{(2)}=\beta_{n}+\beta_{n+1}$,
$C_{n,n-2}^{(2)}=\beta_{n-1}\beta_{n}$ and $C_{n,n+2}^{(2)}=1$. The
computation of the coefficients
$\big{\\{}C_{n-2\ell,n}^{(2m+2)}\big{\\}}_{\ell=0}^{m+1}$ can be derived from
the coefficients $\big{\\{}C_{n-2\ell,n}^{(2m)}\big{\\}}_{\ell=0}^{m}$ as
follows

$C_{n-2\ell,n}^{(2m+2)}=\beta_{n+2}\beta_{n+1}C_{n-2\ell,n+2}^{(2m)}+(\beta_{n}+\beta_{n+1})C_{n-2\ell,n}^{(2m)}+C_{n-2\ell,n-2}^{(2m)},\quad\ell=0,\ldots,m,$
(3.7)

which is a direct consequence of (3.5) multiplied by $x^{2}$ and (3.6).

######  Proposition 3.4.

The recurrence coefficient $\beta_{n}$ for the generalised higher-order Freud
weight (1.1) satisfies the discrete equation

$2mV_{n\phantom{1}}^{(2m)}-2t\beta_{n}=n+(\lambda+\tfrac{1}{2})[1-(-1)^{n}].$
(3.8)

where

$V_{n\phantom{1}}^{(2m)}=C_{n,n-2}^{(2m-2)}+\beta_{n}C_{n,n}^{(2m-2)}.$ (3.9)

Alternatively, (3.9) can be written as

$V_{n\phantom{1}}^{(2m)}=\frac{1}{h_{n-2}}\int_{-\infty}^{\infty}x^{2m-2}P_{n-2}(x)P_{n}(x)\,\omega(x)\,{\rm
d}x+\frac{\beta_{n}}{h_{n}}\int_{-\infty}^{\infty}x^{2m-2}P^{2}_{n}(x)\,\omega(x)\,{\rm
d}x.$

###### Proof.

For any monic polynomial sequence $\big{\\{}P_{n}(x)\big{\\}}_{n\geq 0}$, one
can always write

$x\frac{{\rm d}{P_{n}}}{{{\rm
d}{x}}}(x)=\sum_{j=0}^{n}\rho_{n,j}P_{n-j}(x),\quad\text{for}\quad n\geq 1,$

with $\rho_{n,0}=n$. The assumption that $\big{\\{}P_{n}(x)\big{\\}}_{n\geq
0}$ is orthogonal with respect to the semiclassical weight $w(x)$ satisfying
the differential equation (1.11) with $\sigma(x)=x$ and
$\tau(x)=2(tx^{2}-mx^{2m}+\lambda+1)$ gives, using integration by parts,

$\displaystyle\rho_{n,j}h_{n-j}$
$\displaystyle=\int_{-\infty}^{\infty}x\frac{{\rm d}{P_{n}}}{{{\rm
d}{x}}}(x)P_{n-j}(x)\,\omega(x)\,{\rm d}x$
$\displaystyle=-\int_{-\infty}^{\infty}\left\\{\tau(x)P_{n-j}(x)+x\frac{{\rm
d}{P_{n-j}}}{{{\rm d}{x}}}(x)\right\\}P_{n}(x)\omega(x)\,{\rm d}x,$

where $\displaystyle
h_{k}=\int_{-\infty}^{\infty}P_{k}^{2}(x)\,\omega(x)\,{\rm d}x>0.$ Therefore
$\rho_{n,j}=0$ for any $j\geq 2m+1$ and the symmetry of the weight implies
$\rho_{n,j}=0$ for any $j$ odd. Therefore we have

$x\frac{{\rm d}{P_{n}}}{{{\rm
d}{x}}}(x)=\sum_{\ell=0}^{m}\rho_{n,2\ell}\,P_{n-2\ell}(x),\quad\text{for}\quad
n\geq 0.$ (3.10)

Recall (3.6) to write

$\frac{1}{h_{n}}\int_{-\infty}^{\infty}x^{2}P^{2}_{n}(x)\,\omega(x)\,{\rm
d}x=(\beta_{n}+\beta_{n+1})\quad\text{and}\quad\frac{1}{h_{n-2}}\int_{-\infty}^{\infty}x^{2}P_{n-2}(x)P_{n}(x)\,\omega(x)\,{\rm
d}x=\beta_{n}\beta_{n-1},$

and hence

$\rho_{n,2\ell}=\begin{cases}\displaystyle\frac{2m}{h_{n}}\int_{-\infty}^{\infty}x^{2m}P^{2}_{n}(x)\,\omega(x)\,{\rm
d}x-2t(\beta_{n}+\beta_{n+1})-\left(2\lambda+2+n\right),&\text{if}\quad\ell=0,\\\\[7.11317pt]
\displaystyle\frac{2m}{h_{n-2}}\int_{-\infty}^{\infty}x^{2m}P_{n-2}(x)P_{n}(x)\,\omega(x)\,{\rm
d}x-2t\,\beta_{n}\beta_{n-1},&\text{if}\quad\ell=1,\\\\[7.11317pt]
\displaystyle\frac{2m}{h_{n-2\ell}}\displaystyle\int_{-\infty}^{\infty}x^{2m}P_{n-2\ell}(x)P_{n}(x)\,\omega(x)\,{\rm
d}x,&\text{if}\quad 2\leq\ell\leq m-1,\\\\[7.11317pt]
\displaystyle{2m}\,\beta_{n}\cdots\beta_{n-2m+1},&\text{if}\quad\ell=m,\\\\[7.11317pt]
0,&\text{otherwise}.\end{cases}$ (3.11)

Take $\ell=0$ in (3.7) and note that
$C_{n,n-2}^{(2m-2)}=C_{n-2,n}^{(2m-2)}\beta_{n}\beta_{n-1}$ to get

$C_{n,n}^{(2m)}=\beta_{n+1}\left(C_{n,n+2}^{(2m-2)}\beta_{n+2}+C_{n,n}^{(2m-2)}\right)+\beta_{n}\left(C_{n-2,n}^{(2m-2)}\beta_{n-1}+C_{n,n}^{(2m-2)}\right).$

The symmetric orthogonality recurrence relation (1.7) implies that

$P_{n+2}(x)P_{n}(x)=P_{n+1}^{2}(x)+\beta_{n}P_{n-1}(x)P_{n+1}(x)-\beta_{n+1}P_{n}^{2}(x),$

which gives the relation

$C_{n,n+2}^{(2m-2)}\beta_{n+2}+C_{n,n}^{(2m-2)}=C_{n-1,n+1}^{(2m-2)}\beta_{n}+C_{n+1,n+1}^{(2m-2)},$
(3.12)

and consequently we have

$C_{n,n}^{(2m)}=V_{n+1}^{(2m)}+V_{n\phantom{1}}^{(2m)}\quad\text{where}\quad
V_{n\phantom{1}}^{(2m)}=\beta_{n}\left(\beta_{n-1}C_{n-2,n}^{(2m-2)}+C_{n,n}^{(2m-2)}\right).$
(3.13)

On the other hand, expressions for the coefficients $\rho_{n,2j}$ can be
obtained through a purely algebraic way, and therefore expressed recursively.
For that, we differentiate with respect to $x$ the recurrence relation (1.7)
and use the structure relation (3.10) to get

$P_{n}(x)+\sum_{\ell=0}^{m}\rho_{n,2\ell}P_{n-2\ell}(x)=\frac{{\rm
d}{P_{n+1}}}{{{\rm d}{x}}}(x)+\beta_{n}\frac{{\rm d}{P_{n-1}}}{{{\rm
d}{x}}}(x).$

We multiply the latter by $x$ and use again (3.10) and then (1.7) to obtain a
linear combination of terms of $\big{\\{}P_{n}(x)\big{\\}}_{n\geq 0}$ and this
gives

$\displaystyle P_{n+1}(x)$
$\displaystyle+\beta_{n}P_{n-1}(x)=\sum_{\ell=0}^{m+1}\left(\rho_{n+1,2\ell}-\rho_{n,2\ell}+\beta_{n}\,\rho_{n-1,2\ell-2}-\beta_{n-2\ell+2}\,\rho_{n,2\ell-2}\right)P_{n-2\ell+1}(x).$

Since the terms are linearly independent, we equate the coefficients of
$P_{n+1},P_{n},\ldots,P_{n-2m-1}$ to get

$\begin{cases}\rho_{n,0}=n,\\\ \rho_{n+1,2}-\rho_{n,2}=2\beta_{n},\\\
\rho_{n+1,2\ell}-\rho_{n,2\ell}=\beta_{n-2\ell+2}\,\rho_{n,2\ell-2}-\beta_{n}\,\rho_{n-1,2\ell-2},&\text{for}\quad\ell=2,\ldots,m-1,\\\
\beta_{n-2m}\,\rho_{n,2m}=\beta_{n}\,\rho_{n-1,2m},&\text{for}\quad
j=m-1.\end{cases}$ (3.14)

We combine (3.11) with (3.14) to conclude that the first equation (when
$\ell=0$) gives

$mV_{n+1}^{(2m)}+mV_{n\phantom{1}}^{(2m)}-t(\beta_{n}+\beta_{n+1})=n+\left(\lambda+1\right),$

which implies (3.8). ∎

The expressions for $V_{n\phantom{1}}^{(2m)}$ can be then obtained recursively
using (3.9), (3.7) and (3.13) to write

$\displaystyle V_{n\phantom{1}}^{(2m)}$
$\displaystyle=\beta_{n}\left(V^{(2m-2)}_{n+1}+V^{(2m-2)}_{n}\right)+\left(\beta_{n}+\beta_{n+1}\right)V^{(2m-2)}_{n}$
$\displaystyle\qquad-\beta_{n}\left(\beta_{n}+\beta_{n+1}\right)\left(V_{n\phantom{1}}^{(2m-4)}+V_{n+1}^{(2m-4)}\right)+\beta_{n}\beta_{n-1}\left(V_{n-2}^{(2m-4)}+V_{n-1}^{(2m-4)}\right)$
$\displaystyle\qquad+\beta_{n}\beta_{n-1}\beta_{n+1}\beta_{n+2}C^{(2m-4)}_{n-2,n+2}.$
(3.15)

Combining (3.12) with (3.13) gives

$V_{n}^{(2m)}-V_{n-1\phantom{1}}^{(2m)}=\beta_{n}\left(V_{n+1}^{(2m-2)}+V_{n}^{(2m-2)}\right)-\beta_{n-1}\left(V_{n-1\phantom{1}}^{(2m-2)}+V_{n-2}^{(2m-2)}\right).$

Using the latter relation, we replace the term
$\left(\beta_{n}+\beta_{n+1}\right)V^{(2m-2)}_{n}$ in (3.15) to get

$\displaystyle V_{n\phantom{1}}^{(2m)}$
$\displaystyle=\beta_{n}\left(V_{n+1}^{(2m-2)}+V_{n\phantom{1}}^{(2m-2)}+V_{n-1}^{(2m-2)}\right)+\beta_{n+1}V_{n-1\phantom{1}}^{(2m-2)}$
$\displaystyle\qquad-\beta_{n+1}\beta_{n-1}\left(V_{n-1}^{(2m-4)}+V_{n-2\phantom{1}}^{(2m-4)}\right)+\beta_{n+2}\beta_{n+1}\beta_{n}\beta_{n-1}C^{(2m-4)}_{n-2,n+2}.$
(3.16)

Consider $n\rightarrow n-1$ and $m\rightarrow m-1$ in the latter expression,
and replace it in (3.16) and this yields

$\displaystyle V_{n\phantom{1}}^{(2m)}$
$\displaystyle=\beta_{n}\left(V_{n+1}^{(2m-2)}+V_{n\phantom{1}}^{(2m-2)}+V_{n-1}^{(2m-2)}\right)+\beta_{n-1}\beta_{n+1}V_{n}^{(2m-4)}+\beta_{n}\beta_{n+1}V_{n-2\phantom{1}}^{(2m-4)}$
$\displaystyle\qquad-\beta_{n+1}\beta_{n}\beta_{n-2}\left(V_{n-2}^{(2m-6)}+V_{n-3\phantom{1}}^{(2m-6)}\right)+\beta_{n+1}\beta_{n}\beta_{n-1}\left(\beta_{n+1}\beta_{n-2}C^{(2m-6)}_{n-3,n+1}+\beta_{n+2}C^{(2m-4)}_{n-2,n+2}\right).$

If we replace the term $V_{n-2\phantom{1}}^{(2m-4)}$ by the corresponding
expression given by the latter relation and successively continuing the
process then one can deduce the following expressions for
$V_{n\phantom{1}}^{(2m)}$ as follows

$\displaystyle V_{n\phantom{1}}^{(2)}$ $\displaystyle=\beta_{n},$
$\displaystyle V_{n\phantom{1}}^{(4)}$
$\displaystyle=V_{n\phantom{1}}^{(2)}\left(V_{n+1}^{(2)}+V_{n\phantom{1}}^{(2)}+V_{n-1}^{(2)}\right),$
$\displaystyle V_{n\phantom{1}}^{(6)}$
$\displaystyle=V_{n\phantom{1}}^{(2)}\left(V_{n+1}^{(4)}+V_{n\phantom{1}}^{(4)}+V_{n-1}^{(4)}+V_{n+1}^{(2)}V_{n-1}^{(2)}\right).$

For higher orders we compute the coefficients $V_{n\phantom{1}}^{(2m)}$
recursively as stated below. We opted for not giving the expressions in terms
of $\beta_{n}$ since those are rather long. For $m=4,5$, we have

$\displaystyle V_{n\phantom{1}}^{(8)}$
$\displaystyle=V_{n\phantom{1}}^{(2)}\left(V_{n+1}^{(6)}+V_{n\phantom{1}}^{(6)}+V_{n-1}^{(6)}\right)+V_{n\phantom{1}}^{(4)}V_{n+1}^{(2)}V_{n-1}^{(2)}+V_{n+1}^{(2)}V_{n\phantom{1}}^{(2)}V_{n-1}^{(2)}\left(V_{n+2}^{(2)}+V_{n-2}^{(2)}\right),$
$\displaystyle V_{n\phantom{1}}^{(10)}$
$\displaystyle=V_{n\phantom{1}}^{(2)}\left(V^{(8)}_{n+1}+V_{n\phantom{1}}^{(8)}+V^{(8)}_{n-1}\right)+V_{n\phantom{1}}^{(6)}V^{(2)}_{n+1}V^{(2)}_{n-1}+V^{(2)}_{n+1}V_{n\phantom{1}}^{(2)}V^{(2)}_{n-1}\left(V^{(4)}_{n+2}+V^{(4)}_{n-2}\right)$
$\displaystyle\qquad+V^{(2)}_{n+1}V^{(2)}_{n\phantom{1}}V^{(2)}_{n-1}\left\\{\left(V^{(2)}_{n\phantom{1}}+V^{(2)}_{n-1}\right)V^{(2)}_{n+2}+\left(V^{(2)}_{n+1}+V^{(2)}_{n\phantom{1}}\right)V^{(2)}_{n-2}+V^{(2)}_{n+2}V^{(2)}_{n-2}\right\\}.$

######  Remark 3.5.

For the case when $\lambda=-\tfrac{1}{2}$, Proposition 3.4 was proved by
Benassia and Moro [1], using a result in [2]. Although it is straightforward
to modify the proof presented therein for the case when
$\lambda\not=-\tfrac{1}{2}$, we hereby present an alternative approach purely
depending on the structure relation of the semiclassical polynomials.

When $m=2$ the discrete equation is

$4\beta_{n}\big{(}\beta_{n-1}+\beta_{n}+\beta_{n+1}\big{)}-2t\beta_{n}=n+(\lambda+\tfrac{1}{2})[1-(-1)^{n})],$
(3.17)

which is dPI and when $m=3$ the discrete equation is

$\displaystyle 6\beta_{n}\big{(}\beta_{n-2}\beta_{n-1}$
$\displaystyle+\beta_{n-1}^{2}+2\beta_{n-1}\beta_{n}+\beta_{n-1}\beta_{n+1}+\beta_{n}^{2}+2\beta_{n}\beta_{n+1}+\beta_{n+1}^{2}+\beta_{n+1}\beta_{n+2}\big{)}$
$\displaystyle-2t\beta_{n}=n+(\lambda+\tfrac{1}{2})[1-(-1)^{n})],$ (3.18)

which is a special case of dP${}_{\rm I}^{(2)}$, the second member of the
discrete Painlevé I hierarchy. For further information about the discrete
Painlevé I hierarchy, see [7, 8]. Equations (3.17) and (3.18) with $t=0$ were
derived by Freud [13]; see also [20, 32]. Further equation (3.17) and (3.18)
with $\lambda=-\tfrac{1}{2}$ are also known as “string equations” and arise in
important physical applications such as two-dimensional quantum gravity, cf.
[9, 10, 11, 12, 29].

###  3.2 Asymptotics for the recurrence coefficients as $n\to\infty$

In 1976, Freud [13] conjectured that the asymptotic behavior of recurrence
coefficients $\beta_{n}$ in the recurrence relation (1.7) satisfied by monic
polynomials $\\{p_{n}(x)\\}_{n=0}^{\infty}$ orthogonal with respect to the
weight

$\omega(x)=|x|^{\rho}\exp(-|x|^{m}),$

with $x\in\mathbb{R}$, $\rho>-1$, $m>0$ could be described by

$\displaystyle\lim_{n\rightarrow\infty}\frac{\beta_{n}}{n^{2/m}}=\left[\frac{\Gamma(\tfrac{1}{2}m)\,\Gamma(1+\tfrac{1}{2}m)}{\Gamma(m+1)}\right]^{2/m}.$
(3.19)

Freud stated the conjecture for orthonormal polynomials, proved it for
$m=2,4,6$ and also showed that (3.19) is valid whenever the limit on the left-
hand side exists. Magnus [20] proved Freud’s conjecture for the case when $m$
is an even positive integer and also for weights

$w(x)=\exp\\{-Q(x)\\},$

where $Q(x)$ is an even degree polynomial with positive leading coefficient.
We refer the reader to [27, §4.18] for a detailed history of solutions to
Freud’s conjecture up to that point. The conjecture was settled by Lubinsky,
Mhaskar and Saff in [19] as a special case of a more general result for
recursion coefficients of exponential weights, see also [18]. In [17],
Lubinsky and Saff introduced the class of very smooth Freud weights of order
$\alpha$ with conditions on $Q$ that are satisfied when $Q$ is of the form
$x^{\alpha},$ $\alpha>0$. Associated with each weight in this class, one can
define $a_{n}$ as the unique, positive root of the equation (cf. [19, p. 67]
and references therein)

$n=\frac{2}{\pi}\displaystyle\int_{0}^{1}\frac{a_{n}s\,Q^{\prime}(a_{n}s)}{\sqrt{1-s^{2}}}\,{\rm
d}{s}.$ (3.20)

######  Theorem 3.6.

Consider the generalised higher order Freud weight (1.1). Then the recurrence
coefficients $\beta_{n}$ associated with this weight satisfy

$\lim_{n\to\infty}\frac{\beta_{n}(t;\lambda)}{n^{1/m}}=\frac{1}{4}\left(\frac{(m-1)!}{(\tfrac{1}{2})_{m}}\right)^{\\!1/m}.$

###### Proof.

Let $Q(x)=\tfrac{1}{2}x^{2m}$, then evaluating (3.20) yields

$\displaystyle n$
$\displaystyle=\frac{2ma_{n}^{2m}}{\pi}\displaystyle\int_{0}^{1}\frac{s^{2m}}{\sqrt{1-s^{2}}}\,{\rm
d}{s}=\frac{ma_{n}^{2m}}{\pi}\displaystyle\int_{0}^{1}\frac{t^{m-1/2}}{\sqrt{1-t}}\,{\rm
d}{t}$
$\displaystyle=\frac{ma_{n}^{2m}}{\pi}B(m+\tfrac{1}{2},\tfrac{1}{2})=\frac{ma_{n}^{2m}}{\pi}\frac{\Gamma(m+\tfrac{1}{2})\,\Gamma{(\tfrac{1}{2})}}{\Gamma(m+1)}=\displaystyle\frac{a_{n}^{2m}(\tfrac{1}{2})_{m}}{(m-1)!},$

where $B(p,q)$ denotes the Beta function (cf. [28, (5.12.1)]). Hence
$\displaystyle
a_{n}^{2}=\left(\frac{(m-1)!}{(\tfrac{1}{2})_{m}}\,n\right)^{\\!1/m}$ and the
result follows from [19, Theorem 2.3] taking $W(x)=\exp\\{-Q(x)\\}$,
$w=|x|^{\lambda+1/2}$, $P(x)=\tfrac{1}{2}tx^{2}$ and $\Psi(x)=1$. ∎

######  Remark 3.7.

Taking $m=2$ in Theorem 3.6, we recover [3, Corollary 4.2 (ii)] for the
recurrence coefficients associated with the generalised quartic Freud weight
$|x|^{2\lambda+1}\exp(tx^{2}-x^{4})$ which satisfy

$\lim_{n\to\infty}\frac{\beta_{n}(t;\lambda)}{\sqrt{n}}=\frac{1}{\sqrt{12}},$

while, for $m=3$, the recurrence coefficients associated with the generalised
sextic Freud weight $|x|^{2\lambda+1}\exp(tx^{2}-x^{6})$ satisfy

$\lim_{n\to\infty}\frac{\beta_{n}(t;\lambda)}{\sqrt[3]{n}}=\frac{1}{\sqrt[3]{60}},$

as shown in [5, Corollary 4.8].

##  4 Generalised higher order Freud polynomials

###  4.1 Differential equations

The second order differential equations satisfied by generalised higher order
Freud polynomials can be obtained by using ladder operators as was done for
the special cases $m=2$ and $m=3$ in [6, Theorem 6] and [4, Theorem 4.3],
respectively. An alternative approach is given by Maroni in [22] and [24].

######  Proposition 4.1.

The polynomial sequence $\big{\\{}P_{n}(x)\big{\\}}_{n\geq 0}$ orthogonal with
respect to the generalised higher order Freud weight (1.1) is a solution to
the differential equation

$J(x;n)\frac{{\rm d}^{2}{P_{n+1}}}{{{\rm d}{x}}^{2}}(x)+K(x;n)\frac{{\rm
d}{P_{n+1}}}{{{\rm d}{x}}}(x)+L(x;n)P_{n+1}(x)=0,$

where

$\displaystyle J(x;n)=xD_{n+1}(x),$ $\displaystyle
K(x;n)=C_{0}(x)D_{n+1}(x)-x\frac{{\rm d}{D_{n+1}}}{{{\rm
d}{x}}}(x)+D_{n+1}(x),$ $\displaystyle
L(x;n)=\mathcal{W}\left(\tfrac{1}{2}(C_{n+1}(x)-C_{0}(x)),D_{n+1}(x)\right)-D_{n+1}(x)\sum_{j=0}^{n}\frac{1}{\beta_{j}}D_{j}(x),$

with

$\displaystyle C_{n+1}(x)=-C_{n}(x)+\frac{2x}{\beta_{n}}D_{n}(x),\qquad
D_{n+1}(x)=-x+\frac{\beta_{n}}{\beta_{n-1}}D_{n-1}(x)+\frac{x^{2}}{\beta_{n}}D_{n}(x)-xC_{n}(x),$

subject to the initial conditions $C_{0}(x)=-1+2(tx^{2}-mx^{2m}+\lambda+1)$,
$D_{-1}(x)=0$ and

$D_{0}(x)=2x\left\\{m\sum_{j=1}^{m}\mu_{2j-2}(t,\lambda)x^{2m-2j}-t\mu_{0}(t,\lambda)\right\\}.$

###  4.2 Mixed recurrence relations

######  Lemma 4.2.

Let $\big{\\{}P_{n}(x;\lambda)\big{\\}}_{n=0}^{\infty}$ be the sequence of
monic generalised higher order Freud polynomials orthogonal with respect to
the weight (1.1), then, for $m,n$ fixed,

$\displaystyle xP_{2n}(x;\lambda+1)$ $\displaystyle=P_{2n+1}(x;\lambda),$
(4.1a) $\displaystyle x^{2}P_{2n-1}(x;\lambda+1)$
$\displaystyle=xP_{2n}(x;\lambda)-\left\\{\beta_{2n}(\lambda)+\displaystyle\frac{P_{2n+1}^{\prime}(0;\lambda)}{P_{2n-1}^{\prime}(0;\lambda)}\right\\}P_{2n-1}(x;\lambda).$
(4.1b)

###### Proof.

Let $P_{n}(x;\lambda+1)$ be the polynomials associated with the even weight
function

$\displaystyle\omega(x;\lambda+1)$
$\displaystyle=|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}=x^{2}\omega(x;\lambda),\quad
m=2,3,\dots\ .$

The factor $x^{2}$ by which the weight $\omega(x;\lambda)$ is modified has a
double zero at the origin and therefore Christoffel’s formula (cf. [31,
Theorem 2.5]), applied to the monic polynomials $P_{n}(x;\lambda+1)$, is

$x^{2}P_{n}(x;\lambda+1)=\frac{1}{P_{n}(0;\lambda)P_{n+1}^{\prime}(0;\lambda)-P_{n}^{\prime}(0;\lambda)P_{n+1}(0;\lambda)}\left|\begin{matrix}P_{n}(x;\lambda)&P_{n+1}(x;\lambda)&P_{n+2}(x;\lambda)\\\
P_{n}(0;\lambda)&P_{n+1}(0;\lambda)&P_{n+2}(0;\lambda)\\\
P_{n}^{\prime}(0;\lambda)&P_{n+1}^{\prime}(0;\lambda)&P_{n+2}^{\prime}(0;\lambda)\\\
\end{matrix}\right|.$

Since the weight $\omega(x;\lambda)$ is even, we have that
$P_{2n+1}(0;\lambda)=P_{2n}^{\prime}(0;\lambda)$ while $P_{2n}(0;\lambda)\neq
0$ and $P_{2n+1}^{\prime}(0;\lambda)\neq 0$, hence

$x^{2}P_{n}(x;\lambda+1)=\frac{-1}{P_{n}^{\prime}(0;\lambda)P_{n+1}(0;\lambda)}\left|\begin{matrix}P_{n}(x;\lambda)&P_{n+1}(x;\lambda)&P_{n+2}(x;\lambda)\\\
0&P_{n+1}(0;\lambda)&0\\\
P_{n}^{\prime}(0;\lambda)&0&P_{n+2}^{\prime}(0;\lambda)\\\
\end{matrix}\right|,$

for $n$ odd, while, for $n$ even,

$x^{2}P_{n}(x;\lambda+1)=\frac{1}{P_{n}(0;\lambda)P_{n+1}^{\prime}(0;\lambda)}\left|\begin{matrix}P_{n}(x;\lambda)&P_{n+1}(x;\lambda)&P_{n+2}(x;\lambda)\\\
P_{n}(0;\lambda)&0&P_{n+2}(0;\lambda)\\\ 0&P_{n+1}^{\prime}(0;\lambda)&0\\\
\end{matrix}\right|.$

This yields

$x^{2}P_{n}(x;\lambda+1)=P_{n+2}(x;\lambda)-a_{n}P_{n}(x;\lambda),$ (4.2)

where

$a_{n}=\begin{cases}\displaystyle\frac{P_{n+2}(0;\lambda)}{P_{n}(0;\lambda)},\quad&\text{for}\quad
n\quad\text{even},\\\\[6.0pt]
\displaystyle\frac{P_{n+2}^{\prime}(0;\lambda)}{P_{n}^{\prime}(0;\lambda)},\quad&\text{for}\quad
n\quad\text{odd}.\end{cases}$

Using the three-term recurrence relation (1.7) to eliminate
$P_{n+2}(x;\lambda)$ in (4.2), we obtain

$\displaystyle
x^{2}P_{n}(x;\lambda+1)=xP_{n+1}(x;\lambda)-(\beta_{n+1}(\lambda)+a_{n})P_{n}(x;\lambda).$

It follows from (1.8) that, for $n$ even, $\beta_{n+1}(\lambda)+a_{n}=0$ and
the result follows. ∎

######  Lemma 4.3.

For a fixed $m=2,3,\dots$, let
$\big{\\{}P_{n}(x;\lambda)\big{\\}}_{n=0}^{\infty}$ be the sequence of monic
generalised higher order Freud polynomials orthogonal with respect to the
weight (1.1). Then, for $n$ fixed,

$\displaystyle P_{2n+1}(x;\lambda)$
$\displaystyle=P_{2n+1}(x;\lambda+1)+\beta_{2n}(\lambda+1)P_{2n-1}(x;\lambda+1),$
(4.3a) $\displaystyle P_{2n}(x;\lambda)$
$\displaystyle=P_{2n}(x;\lambda+1)-\frac{\beta_{2n}(\lambda)\beta_{2n-1}(\lambda+1)P_{2n-1}^{\prime}(0;\lambda)}{P_{2n+1}^{\prime}(0;\lambda)}P_{2n-2}(x;\lambda+1).$
(4.3b)

###### Proof.

Substitute (4.1a) into the three term recurrence relation

$P_{2n+1}(x;\lambda)=xP_{2n}(x;\lambda)-\beta_{2n}(\lambda)P_{2n-1}(x;\lambda),$
(4.4)

to eliminate $P_{2n+1}(x;\lambda)$ and obtain

$xP_{2n}(x;\lambda+1)=xP_{2n}(x;\lambda)-\beta_{2n}(\lambda)P_{2n-1}(x;\lambda).$

Let $\displaystyle
a_{2n}=\frac{P_{2n+1}^{\prime}(0;\lambda)}{P_{2n-1}^{\prime}(0;\lambda)}$.
Substitute (4.1b) into (4.4) to eliminate $P_{2n-1}(x;\lambda)$ and obtain

$\displaystyle xP_{2n}(x;\lambda+1)$
$\displaystyle=xP_{2n}(x;\lambda)-\frac{\beta_{2n}(\lambda)}{\beta_{2n}(\lambda)+a_{2n}}\left(xP_{2n}(x;\lambda)-x^{2}P_{2n-1}(x;\lambda+1)\right).$
(4.5)

Simplification and rearrangement of terms in (4.5) yields

$\left(1-\frac{\beta_{2n}(\lambda)}{\beta_{2n}(\lambda)+a_{2n}}\right)P_{2n}(x;\lambda)=P_{2n}(x;\lambda+1)-\frac{\beta_{2n}(\lambda)}{\beta_{2n}(\lambda)+a_{2n}}xP_{2n-1}(x;\lambda+1),$

then, using the three term recurrence relation to eliminate
$xP_{2n-1}(x;\lambda+1)$, we obtain

$\displaystyle\left(1-\frac{\beta_{2n}(\lambda)}{\beta_{2n}(\lambda)+a_{2n}}\right)P_{2n}(x;\lambda)$
$\displaystyle=\left(1-\frac{\beta_{2n}(\lambda)}{\beta_{2n}(\lambda)+a_{2n}}\right)P_{2n}(x;\lambda+1)$
$\displaystyle\qquad+\frac{\beta_{2n}(\lambda)}{\beta_{2n}(\lambda)+a_{2n}}\beta_{2n-1}(\lambda+1)P_{2n-2}(x;\lambda+1),$

which simplifies to (4.3b). Substituting (4.1a) into the three term recurrence
relation

$P_{2n+1}(x;\lambda+1)=xP_{2n}(x;\lambda+1)-\beta_{2n}(\lambda+1)P_{2n-1}(x;\lambda+1),$

yields (4.3a). ∎

###  4.3 Quasi-orthogonality for $\lambda\in(-2,-1)$

Lemma 4.3 yields the quasi-orthogonality of generalised higher order Freud
polynomials for $-2<\lambda<-1$.

######  Theorem 4.4.

Suppose $-2<\lambda<-1$. For each fixed $m=2,3,\dots$, the generalised higher
order Freud polynomial $P_{n}(x;\lambda)$ is quasi-orthogonal of order $2$ on
$\mathbb{R}$ with respect to the weight

$|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)},\qquad t\in\mathbb{R}.$

###### Proof.

Suppose $-2<\lambda<-1$, then $\lambda+1>-1$. When $n$ is even, we have from
(4.3b) that

$\displaystyle\int_{-\infty}^{\infty}$ $\displaystyle
x^{k}P_{n}(x;\lambda)\,\,|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}\,{\rm
d}x$
$\displaystyle=\int_{-\infty}^{\infty}x^{k}P_{n}(x;\lambda+1)\,\,|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}\,{\rm
d}x$ (4.6)
$\displaystyle\qquad-\frac{\beta_{n}(\lambda)\beta_{n-1}(\lambda+1)P_{n-1}^{\prime}(0;\lambda)}{P_{n+1}^{\prime}(0;\lambda)}\int_{-\infty}^{\infty}x^{k}P_{n-2}(x;\lambda+1)\,\,|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}\,{\rm
d}x,$

while, for $n$ is odd, it follows from (4.3a) that

$\displaystyle\int_{-\infty}^{\infty}$ $\displaystyle
x^{k}P_{n}(x;\lambda)\,\,|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}\,{\rm
d}x$
$\displaystyle=\int_{-\infty}^{\infty}x^{k}P_{n}(x;\lambda+1)\,\,|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}\,{\rm
d}x$ (4.7)
$\displaystyle\qquad+\beta_{n}(\lambda+1)\int_{-\infty}^{\infty}x^{k}P_{n-2}(x;\lambda+1)\,\,|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}\,{\rm
d}x.$

Since $\lambda+1>-1$, it follows from the orthogonality of the generalised
higher order Freud polynomials that

$\int_{-\infty}^{\infty}x^{k}P_{n}(x;\lambda+1)\,\,|x|^{2\lambda+3}\exp\big{(}tx^{2}-x^{2m}\big{)}=0\qquad\text{for}\quad
k=0,\dots,n-1,$

and we see that all the integrals on the righthand side of (4.6) and (4.7) are
equal to zero for $k=0,\dots,n-3$. ∎

##  5 Zeros of generalised higher order Freud polynomials

###  5.1 Asymptotic zero distribution

The asymptotic behaviour of the recurrence coefficients of generalised higher
order Freud polynomials orthogonal with respect to (1.1), satisfying Freud’s
conjecture, given by (3.19), is independent of the values of $t$ and
$\lambda$. The asymptotic behaviour implies that the recurrence coefficients
are regularly varying, irrespective of $t$ and $\lambda$. To consider the
asymptotic distribution of the zeros of generalised higher order Freud
polynomials orthogonal with respect to the weight (1.1) as $n\to\infty$, we
use an appropriate scaling and apply the property of regular variation as
detailed in [16].

######  Theorem 5.1.

Let $\phi(n)=n^{1/(2m)}$ and assume that $n,N$ tend to infinity in such a way
that the ratio $n/N\to\ell$. Then, for the sequence of scaled monic
polynomials $P_{n,N}(x)=(\phi(N))^{-n}P_{n}(\phi(N)x)$ associated with the
generalised higher order Freud weight (1.1), the asymptotic zero distribution,
as $n\to\infty$, has density

$a_{m}(\ell)=\frac{2m}{c\pi(2m-1)}\left(1-{x^{2}}/{c^{2}}\right)^{\\!1/2}{}_{2}F_{1}\left(1,1-m;\tfrac{3}{2}-m;{x^{2}}/{c^{2}}\right),$
(5.1)

where

$c=2a\ell^{1/(2m)}\quad\text{with}\quad
a=\tfrac{1}{2}\left(\frac{(m-1)!}{\\!(\tfrac{1}{2})_{m}}\right)^{\\!1/(2m)},$

defined on the interval $(-2a\ell^{1/(2m)},2a\ell^{1/(2m)})$.

###### Proof.

The scaled monic polynomials $P_{n,N}(x)=(\phi(N))^{-n}P_{n}(\phi(N)x)$
associated with the generalised higher order Freud weight (1.1) have
recurrence coefficient
$\beta_{n,N}(t;\lambda)=\frac{\beta_{n}(t;\lambda)}{\left(\phi(N)\right)^{2}}$.
Since $\phi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ and, for every $\ell>0$, we have

$\lim_{x\to\infty}\frac{\phi(x\ell)}{\phi(x)}=\ell^{1/(2m)},$

$\phi$ is regularly varying at infinity with exponent of variation
$\tfrac{1}{2m}$ (cf. [33]). Since it follows from (3.6) that

$\lim_{n/N\to\ell}\sqrt{\beta_{n,N}(t;\lambda)}=\lim_{n/N\to\ell}\frac{\sqrt{\beta_{n}(t;\lambda)}}{\phi(n)}\frac{\phi(n)}{\phi(N)}=a\ell^{1/(2m)},$

the recurrence coefficients $\beta_{n.N}(t;\lambda)$ are said to be regularly
varying at infinity with index $\tfrac{1}{2m}$ (cf. [16, Section 4.5]). From
the property of regular variation, using [16, Theorem 1.4], it follows that
the asymptotic zero distribution has density

$\displaystyle\frac{1}{\pi\ell}\int_{0}^{\ell}s^{-1/(2m)}$
$\displaystyle\left(2a-xs^{-1/(2m)}\right)^{\\!-1/2}\left(2a+xs^{-1/(2m)}\right)^{\\!-1/2}\,{\rm
d}s$
$\displaystyle=\frac{m}{a\pi\ell}\int_{0}^{\ell^{1/(2m)}}y^{2m-2}\left(1-\left(\frac{x}{2ay}\right)^{\\!2}\right)^{\\!-1/2}\,{\rm
d}y$
$\displaystyle=\frac{m}{a\pi\ell}\int_{0}^{\ell^{1/(2m)}}y^{2m-2}\sum_{k=0}^{\infty}\frac{\left(\tfrac{1}{2}\right)_{k}}{k!}\left(\frac{x}{2ay}\right)^{\\!2k}\,{\rm
d}y$
$\displaystyle=\frac{m}{a\pi\ell}\sum_{k=0}^{\infty}\frac{\left(\tfrac{1}{2}\right)_{k}}{k!}\left(\frac{x}{2a}\right)^{\\!2k}\int_{0}^{\ell^{1/(2m)}}y^{2m-2k-2}\,{\rm
d}y$
$\displaystyle=\frac{m}{a\pi\ell^{1/(2m)}}\sum_{k=0}^{\infty}\frac{\left(\tfrac{1}{2}\right)_{k}}{k!}\frac{1}{2m-2k-1}\left(\frac{x}{2a\ell^{1/(2m)}}\right)^{\\!2k}$
$\displaystyle=\frac{m}{a\pi\ell^{1/(2m)}(2m-1)}\sum_{k=0}^{\infty}\frac{\left(\tfrac{1}{2}\right)_{k}\left(\tfrac{1}{2}-m\right)_{k}}{\left(\tfrac{3}{2}-m\right)_{k}k!}\left(\frac{x}{2a\ell^{1/(2m)}}\right)^{\\!2k}$
$\displaystyle=\frac{m}{a\pi(2m-1)\ell^{1/(2m)}}{}_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1}{2}-m;\tfrac{3}{2}-m;\left(\frac{x}{2a\ell^{1/(2m)}}\right)^{\\!2}\right).$

∎

Figure 5.1 shows the zeros and the asymptotic distribution according to
Theorem 5.1.

Figure 5.1: The zeros of $P_{n,N}(x)$ (red) for $\lambda=0.5$, $t=1$, $m=3$,
$n=N=10$ and $\ell=1$ with the corresponding limiting distribution (5.1)
(blue) and endpoints $(-2a,0)$ and $(2a,0)$ (green).

Figure 5.2 shows the asymptotic distribution of zeros according to Theorem 5.1
for various values of $\ell$.

Figure 5.2: The limiting distribution of the zeros $a_{3}(\ell)$ for
$\ell=0.5$ (green), $\ell=1$ (blue) and $\ell=2$ (red).

######  Remark 5.2.

Note that the formula on [16, p. 189, line 22] should be
$\displaystyle\frac{1}{t}\int_{0}^{t}\frac{1}{s^{\lambda}}\omega^{\prime}_{[b-2a,b+2a]}(xs^{-\lambda})\,{\rm
d}s$.

###  5.2 Bounds for the extreme zeros

From the three-term recurrence relation (3.4), we obtain bounds for the
extreme zeros of monic generalised higher order Freud polynomials.

######  Theorem 5.3.

For each $n=2,3,\dots,$ the largest zero, $x_{1,n}$, of monic generalised
higher order Freud polynomials $P_{n}(x)$ orthogonal with respect to the
weight (1.1), satisfies

$0<x_{1,n}<\max_{1\leq k\leq n-1}\sqrt{c_{n}\beta_{k}(t;\lambda)},$

where $\displaystyle c_{n}=4\cos^{2}\left(\frac{\pi}{n+1}\right)+\varepsilon$,
$\varepsilon>0$.

###### Proof.

The upper bound for the largest zero $x_{1,n}$ follows by applying [15,
Theorem 2 and 3], based on the Wall-Wetzel Theorem to the three-term
recurrence relation (3.4). ∎

###  5.3 Monotonicity of the zeros

######  Theorem 5.4.

Consider $0<x_{\lfloor n/2\rfloor,n}<\dots<x_{2,n}<x_{1,n}$, the positive
zeros of monic orthogonal polynomials $P_{n}(x)$ with respect to the
generalised higher order Freud weight (1.1) where $\lfloor k\rfloor$ denotes
the largest integer less than or equal to $k$. Then, for $\lambda>-1$,
$t\in\mathbb{R}$ and for a fixed value of $\nu$, $\nu\in\\{1,2,\dots,\lfloor
n/2\rfloor\\}$, the $\nu$-th zero $x_{n,\nu}$ increases when (i), $\lambda$
increases; and (ii), $t$ increases.

###### Proof.

This follows from [4, Lemma 4.5], taking $C(x)=x$, $D(x)=x^{2}$,
$\rho=2\lambda+1$ and $\omega_{0}(x)=\exp(-x^{2m})$. ∎

###  5.4 Interlacing of the zeros

Next, for fixed $\lambda>-1$, $t\in\mathbb{R}$ and $k\in(0,1]$, we consider
the relative positioning of the zeros of the monic generalised higher order
Freud polynomials $\big{\\{}P_{n}(x;\lambda)\big{\\}}$ orthogonal with respect
to the weight (1.1), and the zeros of $\\{P_{n}(x;\lambda+k)$, $k\in(0,1]$,
orthogonal with respect to the weight

$\omega(x;t,\lambda)=|x|^{2\lambda+2k+1}\exp\left(tx^{2}-x^{2m}\right),\qquad
m=2,3,\dots\ .$

The zeros of monic generalised higher order Freud polynomials
$\big{\\{}P_{n}(x;\lambda)\big{\\}}$ orthogonal with respect to the symmetric
weight (1.1) are symmetric around the origin. We denote the positive zeros of
$P_{2n}(x;\lambda)$ by

$0<x_{n,2n}^{\lambda}<x_{n-1,2n}^{\lambda}<\dots<x_{2,2n}^{\lambda}<x_{1,2n}^{\lambda},$

and the positive zeros of $P_{2n+1}(x;\lambda)$ by

$0<x_{n,2n+1}^{\lambda}<x_{n-1,2n+1}^{\lambda}<\dots<x_{2,2n+1}^{\lambda}<x_{1,2n+1}^{\lambda}.$

######  Theorem 5.5.

Let $\lambda>-1$ and $t\in\mathbb{R}$. Let
$\big{\\{}P_{n}(x;\lambda)\big{\\}}$ be the monic generalised higher order
Freud polynomials orthogonal with respect to the weight (1.1). Then, for
$\ell\in\\{1,\dots,n-1\\}$ and $k\in(0,1]$, we have

$\displaystyle
x_{\ell+1,2n}^{\lambda}<x_{\ell,2n-1}^{\lambda}<x_{\ell,2n-1}^{\lambda+k}<x_{\ell,2n-1}^{\lambda+1}<x_{\ell,2n}^{\lambda},$
(5.2)

and

$\displaystyle
x_{\ell+1,2n+1}^{\lambda}<x_{\ell,2n}^{\lambda}<x_{\ell,2n}^{\lambda+k}<x_{\ell,2n}^{\lambda+1}=x_{\ell,2n+1}^{\lambda}.$
(5.3)

###### Proof.

The zeros of two consecutive polynomials in the sequence of generalised higher
order Freud orthogonal polynomials are interlacing, that is,

$0<x_{n,2n}^{\lambda}<x_{n-1,2n-1}^{\lambda}<x_{n-1,2n}^{\lambda}<\dots<x_{2,2n}^{\lambda}<x_{1,2n-1}^{\lambda}<x_{1,2n}^{\lambda}$
(5.4)

and

$0<x_{n,2n}^{\lambda}<x_{n,2n+1}^{\lambda}<x_{n-1,2n}^{\lambda}<\dots<x_{2,2n+1}^{\lambda}<x_{1,2n}^{\lambda}<x_{1,2n+1}^{\lambda}.$
(5.5)

On the other hand, we proved in Theorem 5.4 that the positive zeros of
generalised higher order Freud polynomials monotonically increase as the
parameter$\lambda$ increases. This implies that, for each fixed
$\ell\in\\{1,2,\dots,n\\}$ and $k\in(0,1)$,

$x_{\ell,2n}^{\lambda}<x_{\ell,2n}^{\lambda+k}<x_{\ell,2n}^{\lambda+1},$ (5.6)

and

$x_{\ell,2n-1}^{\lambda}<x_{\ell,2n-1}^{\lambda+k}<x_{\ell,2n-1}^{\lambda+1}.$
(5.7)

Next, we prove that the zeros of $P_{2n}(x;\lambda)$ interlace with those of
$P_{2n-1}(x;\lambda+1)$. From (4.1b),

$P_{2n-1}(x;\lambda+1)=\frac{xP_{2n}(x;\lambda)-(\beta_{2n}(\lambda)+P_{2n+1}^{\prime}(0;\lambda)/P_{2n-1}^{\prime}(0;\lambda))P_{2n-1}(x;\lambda)}{x^{2}}.$
(5.8)

Evaluating (5.8) at consecutive zeros $x_{\ell}=x_{\ell,n}^{(\lambda)}$ and
$x_{\ell+1}=x_{\ell+1,n}^{(\lambda)}$, $\ell=1,2,\ldots,n-1$, of
$P_{2n}(x;\lambda)$, we obtain

$\begin{split}P_{2n-1}&(x_{\ell};\lambda+1)P_{2n-1}(x_{\ell+1};\lambda+1)\\\
&=\frac{(\beta_{2n}(\lambda)+P_{2n+1}^{\prime}(0;\lambda)/P_{2n-1}^{\prime}(0;\lambda))^{\\!2}P_{2n-1}(x_{\ell};\lambda)P_{2n-1}(x_{\ell+1};\lambda)}{x_{\ell}^{2}x_{\ell+1}^{2}}<0,\end{split}$

since the zeros of $P_{2n}(x;\lambda)$ and $P_{2n-1}(x;\lambda)$ separate each
other. So there is at least one positive zero of $P_{2n}(x;\lambda+1)$ in the
interval $(x_{\ell},x_{\ell}+1)$ for each $\ell=1,2,\ldots,n-1$ since there
are exactly $n$ positive zeros and and this implies that

$0<x_{n,2n}^{\lambda}<x_{n-1,2n-1}^{\lambda+1}<x_{n-1,2n}^{\lambda}<x_{n-2,2n-1}^{\lambda+1}<\dots<x_{2,2n-1}^{\lambda+1}<x_{2,2n}^{\lambda}<x_{1,2n-1}^{\lambda+1}<x_{1,2n}^{\lambda}.$
(5.9)

Equations (5.4), (5.7) and (5.9) yield (5.2). To prove (5.3), we note that by
(4.1a) the $n$ positive zeros of $P_{2n}(x,\lambda+1)$ and
$P_{2n+1}(x;\lambda)$ coincide, i.e
$x_{\ell,2n}^{\lambda+1}=x_{\ell,2n+1}^{\lambda}$ for
$\ell\in\\{1,2,\dots,n\\}$, and the result follows using (5.5) and (5.6).∎

Figure 5.3 shows the interlacing of the zeros of polynomials orthogonal with
respect to the generalised higher order Freud weight (1.1) for $m=3$ as
described in (5.4) of Theorem 5.5.

Figure 5.3: The zeros of $P_{7}(x;\lambda)$ (green), $P_{7}(x;\lambda+1)$
(red) and $P_{8}(x;\lambda)$ (blue) for $\lambda=0.5$ and $t=1$.

Figure 5.4 illustrates the interlacing of the zeros of polynomials orthogonal
with respect to the generalised higher order Freud weight (1.1) for $m=3$ as
described in (5.5) of Theorem 5.5.

Figure 5.4: The zeros of $P_{8}(x;\lambda)$ (green), $P_{8}(x;\lambda+0.5)$
(red), $xP_{8}(x;\lambda+1)$ (blue) and $P_{9}(x;\lambda)$ (blue) for
$\lambda=1.5$ and $t=2.3$.

##  6 Quadratic decomposition of the generalised higher order Freud weight

Suppose

$P_{2n}(x;t,\lambda)=B_{n}(x^{2};t,\lambda,)\qquad
P_{2n+1}(x;t,\lambda)=xR_{n}(x^{2};t,\lambda),\qquad\text{for all}\quad n\geq
0,$

then from the recurrence relation (1.7) we have

$\displaystyle B_{n+1}(x;t,\lambda)$
$\displaystyle=R_{n+1}(x;t,\lambda)+\beta_{2n+2}R_{n}(x;t,\lambda),$
$\displaystyle xR_{n}(x;t,\lambda)$
$\displaystyle=B_{n+1}(x;t,\lambda)+\beta_{2n+1}B_{n}(x;t,\lambda),$

and this gives second order recurrence relations for both $\\{B_{n}\\}_{n\geq
0}$ and $\\{R_{n}\\}_{n\geq 0}$ as follows

$\displaystyle\begin{cases}B_{n+1}(x;t,\lambda)=\left(x-\beta_{2n}-\beta_{2n+1}\right)B_{n}(x;t,\lambda)-\beta_{2n-1}\beta_{2n}B_{n-1}(x;t,\lambda),\
n\geq 1,\\\ B_{1}(x;t,\lambda)=x-\beta_{1},\ B_{0}(x;t,\lambda)=1,\end{cases}$
$\displaystyle\begin{cases}R_{n+1}(x;t,\lambda)=\left(x-\beta_{2n+2}-\beta_{2n+1}\right)R_{n}(x;t,\lambda)-\beta_{2n+1}\beta_{2n}R_{n-1}(x;t,\lambda),\\\
R_{1}(x;t,\lambda)=x-\beta_{1}-\beta_{2},\ R_{0}(x;t,\lambda)=1.\end{cases}$

Furthermore, $\\{B_{n}\\}_{n\geq 0}$ and $\\{R_{n}\\}_{n\geq 0}$ satisfy the
orthogonality relations

$\displaystyle\int_{0}^{\infty}B_{k}(x;t,\lambda)B_{n}(x;t,\lambda)\,x^{\lambda}\exp(tx-x^{m})\,{\rm
d}x=h_{2n}(t,\lambda)\delta_{n,k},$
$\displaystyle\int_{0}^{\infty}R_{k}(x;t,\lambda)R_{n}(x;t,\lambda)\,x^{\lambda+1}\exp(tx-x^{m})\,{\rm
d}x=h_{2n+1}(t,\lambda)\delta_{n,k},\quad n,k\geq 0.$

## Acknowledgement

PAC and KJ gratefully acknowledge the support of a Royal Society Newton
Advanced Fellowship NAF$\backslash$R2$\backslash$180669.

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# Towards Decentralized Human-Swarm Interaction by Means of Sequential Hand
Gesture Recognition

Zahi Kakish, Sritanay Vedartham, and Spring Berman This work was supported by
the Arizona State University Global Security Initiative.Zahi Kakish and Spring
Berman are with the School for Engineering of Matter, Transport and Energy,
Arizona State University (ASU), Tempe, AZ 85281, USA {zahi.kakish,
<EMAIL_ADDRESS>Vedartham is a high school student at BASIS,
Scottsdale, AZ 85259, USA<EMAIL_ADDRESS>
###### Abstract

In this work, we present preliminary work on a novel method for Human-Swarm
Interaction (HSI) that can be used to change the macroscopic behavior of a
swarm of robots with decentralized sensing and control. By integrating a small
yet capable hand gesture recognition convolutional neural network (CNN) with
the next-generation Robot Operating System _ros2_ , which enables
decentralized implementation of robot software for multi-robot applications,
we demonstrate the feasibility of programming a swarm of robots to recognize
and respond to a sequence of hand gestures that capable of correspond to
different types of swarm behaviors. We test our approach using a sequence of
gestures that modifies the target inter-robot distance in a group of three
Turtlebot3 Burger robots in order to prevent robot collisions with obstacles.
The approach is validated in three different Gazebo simulation environments
and in a physical testbed that reproduces one of the simulated environments.

##  I INTRODUCTION

As the price of low-power computing devices continues to decrease, the
robotics community recognizes the increasing viability of robotic swarms for a
multitude of applications. The emphasis on scalability and hardware redundancy
in robotic swarms, however, makes developing robust and stable management and
control tools difficult. In addition, a human-in-the-loop approaches for
managing a fleet of robots leads to cognitive overload by operators impairing
their ability to continue working [1]. Our work seeks to solve this issue in
Human-Swarm Interfaces (HSI) by attempting to remove the interface. Instead, a
human operator may give commands through a sequence of hand gestures and have
their commands relayed to an entire swarm or team of robots.

Yet before this can happen, further progress is required in Human-Swarm
interfaces and control. Past work in the field has expanded upon control using
a variety of different control interfaces which are often referred to as
Human-Swarm Interfaces. Kolling _et al._ provide an essential survey of the
HSI field as it currently stands [2]. As mentioned earlier in [1], one of the
most difficult aspects of human-swarm control is burdening the user with
cognitive overload. Managing multiple robots is a difficult task, and,
therefore, HSI systems are designed with this premise. Abstracting complex
swarm behavior is essential to discretizing the tasks into manageable and
comprehensible feedback to the human user. Lin _et al._ were able to generate
artificial task functions capable of abstracting a swarm in real-time to
assist users [3]. After abstracting complex swarm behavior, [4] and [5] mapped
the different abstracted behaviors onto single gesture inputs for tablet
interfaces. However, both these methods require a centralized server to
function because of their robot’s low computational power, and they require
the difficult development of an external mobile application.

Various approaches have been attempted to control multi-robot systems by other
physical means, including wearable devices using haptic feedback as seen in
[6][7] and Electroencephalogram brain-computer interfaces (BCI) as reported by
Karavas _et al._ [8]. However, both these approaches are hindered by the
necessity of wearable hardware to function and cumbersome application process
to read noisy EEG signals, respectively. These issues make the their
applicability in everyday usage more difficult.

In this work, we present preliminary research into a human-swarm control
paradigm capable of future use in decentralized control of a robotic swarm
through human input that is realizable with the current generation of computer
hardware. By decentralized, we mean that the robots’ systems are independent
of one another but are capable of inter-communication. With the advent of
accelerated training in machine learning and advances in computer vision, many
complex classification tasks are scalable to less computationally powerful
devices. We seek to utilize this by combining advances in decentralized
robotic systems using _ros2_ [9] with specialized convolutional neural
networks (CNN) capable of recognizing hand gestures. We hypothesize that
concatenating a series of different hand gestures produces more refined
control of a particular swarm task specified by a user.

In Section II, we establish a simplified system composed of a CNN and
simplified swarm behavior to validate our hypothesis. Next, we test our
approach using a simplified swarm behavior on a small swarm size (three
robots) in various simulation testbeds. The system is further validated
through physical experimentation using three robots in a re-creation of the
final simulated testbed. Finally, we expand upon the results of the
experiments and discuss the future applicability of this approach towards a
decentralized swarm controlled by a human-in-the-loop. As of time of writing,
this is the only known work that formulates sequential hand gestures for
controlling multiple robots using CNNs in a semi-decentralized manner.

##  II METHODOLOGY

Our system is primary composed of two parts: (1) a deep neural network
classifying various hand gestures, and (2) a simplified swarm cohesion model
for a small swarm size. This paper will provide a validation of sequential
hand gestures as a means of controlling a swarm. Therefore, we do not
formulate the algorithm as a generalized model for scalable swarm sizes, which
we leave to future work.

###  II-A Deep Neural Network

Since low-powered, small computationally powerful devices, such as the
Raspberry Pi and other _ARM_ devices, are common among swarm robotics hardware
architecture, most conventional and state-of-the-art neural network models are
unusable or difficult to implement. To circumvent this issue, a special neural
network named a SqueezeNet was deployed for hand gesture recognition [10]. The
SqueezeNet model is capable of performing as well, or even surpassing,
classification rates of many the most popular model types with only a fraction
of the parameters, thus reducing the model size to roughly $\sim 5$ MB. This
substantial reduction in model size makes this model ideal for use in hardware
typically found in swarm robotics applications.

####  II-A1 Dataset and Preprocessing

A training set consisting of six different gestures and $2,956$ images was
utilized for training the SqueezeNet model [11]. The images are silhouettes of
the gestures in black and white. A gesture set, $G$, containing all possible
gestures for classification and available in the training data is defined as:

$G=\\{~{}\textit{C},~{}\textit{Fist},~{}\textit{L},~{}\textit{Ok},~{}\textit{Peace},~{}\textit{Palm}~{}\\}$
(1)

Figure 1 contains examples of the gestures found in the dataset. The dataset
was expanded to ensure more robust classification of gestures during real-time
operation. The data augmentation began with inverting the images horizontally
to resemble the gesture but with the opposite hand. The original image and the
inverted image are used to generate three new images, respectively. One
rotated by $90^{\circ}$ clockwise, another $90^{\circ}$ counter-clockwise, and
an image flipped upside-down. Therefore, seven new images are generated for
each image in the original dataset.

To ensure that the model is capable of real-time performance, a seventh
gesture was added to the dataset: ‘None’. Since the user’s hand is not
consistently in the frame, the image is classified as ‘None’ and awaits a
gesture to come into the frame. A dataset containing blank images is generated
during preprocessing consisting of black images and black images containing
Gaussian noise to add to SqueezeNet model’s classification capabilities.

Further preprocessing was performed before training to ensure optimal results.
Images in the dataset were cropped from their original size of $640\times 576$
to $570\times 570$. The images were then scaled to $240\times 240$. This final
image size is big enough to ensure adequate real-time classification of
gestures without increasing computational cost.

Figure 1: Silhouettes of gestures available through the dataset in [11]. In
addition to these gestures, a blank image was used to classify the gesture
_None_. A) a _C_ shape B) a _Fist_ C) an _L_ shape D) the _Okay_ sign E) a
_Palm_ F) the _Peace_ sign.

####  II-A2 Training

The SqueezeNet model was programmed and trained using the _TensorFlow_ API
version $1.14$ [12] running on a development box containing four _Nvidia_ RTX
2080 graphics cards. The model was trained for $10$ epochs with a Stochastic
Gradient Descent optimizer, which had a learning rate $\alpha=0.001$, a
coefficient of momentum $\eta=0.9$, and a learning rate decay
$\lambda=0.0002$. The batch size was set to two images to prevent memory
issues due to the images’ size. The dataset was split into a training and
validation set consisting of $90\%$ and $10\%$ of the images, respectively.

###  II-B Gesture Driven Controller

We first consider a waypoint system in which a finite number of robots, $N$,
plot a straight motion to a desired goal position $(x_{d},y_{d})_{N}$ from an
initial starting position $(x_{i},y_{i})_{N}$. To help validate the
applicability of sequential gesture control, we will only focus on one
direction, $x$, and keep the $y$ values arbitrary. For example, if the initial
position of a robot is $(0,0)$ and the desired position is set to be $(10,0)$,
the desired position is considered achieved if only the robot’s x-coordinate
matches the desired value. A set of waypoints, $\mathcal{W}$, for each robot
$n$ is thus generated from augmenting $x_{i}$ by $x_{\text{offset}}$, which is
calculated from a user-defined number of waypoints, $M$.

$\displaystyle x_{\text{offset}}$ $\displaystyle=\frac{|x_{i}-x_{d}|}{M}$ (2)
$\displaystyle\mathcal{W}_{n}$ $\displaystyle=\\{(x_{i}+k\times
x_{\text{offset}},y):k=0,\ldots,M\\}$ (3)

Each robot, $n$, moves from one waypoint to another in the set $\mathcal{W}$
until it reaches the final desired position $(x_{d},y_{d})_{n}$.

With the waypoints generated, we define
$\mathcal{C}:y_{n}\times\beta\to\mathbb{R}$ to be the cohesion of the swarm,
that is, a metric measuring the inter-robot distance of a swarm, where $y_{n}$
is the robot $n$’s $y$ position and $\beta$ is the cohesion factor. $\beta$ is
a binary variable that represents an increase in cohesion if $\beta=0$ or a
decrease in cohesion if $\beta=1$. To increase the cohesion of a swarm is to
reduce the distance between each robot, and to decrease the cohesion is to
increase the distance between each robot. Additionally, we consider a small
swarm size of three agents in a line separated by an offset in the $y$
direction. The cohesion of this small swarm is best represented by a constant
addition or subtraction of a small, static offset $y_{\text{offset}}$. An
individual robot, $n$, applies a cohesion change by the following piecewise
formula:

$\mathcal{C}(y_{n},\beta)=\begin{cases}y_{n}-y_{\text{offset}}&\quad\text{if
}y_{n}>0\text{ and }\beta=0\\\ y_{n}+y_{\text{offset}}&\quad\text{if
}y_{n}>0\text{ and }\beta=1\\\ y_{n}+y_{\text{offset}}&\quad\text{if
}y_{n}<0\text{ and }\beta=0\\\ y_{n}-y_{\text{offset}}&\quad\text{if
}y_{n}<0\text{ and }\beta=1\\\ \end{cases}$ (4)

Next, we consider a differential drive capable of driving from one waypoint to
the next based on the unicycle model [13] as defined by the following state-
space formulation, where $x$ and $y$ are the robot’s position and $\phi$ is
its angle.

$\dot{x}=\begin{bmatrix}\dot{x}\\\ \dot{y}\\\ \dot{\phi}\\\
\end{bmatrix}=\begin{bmatrix}v_{o}\cos\phi\\\ v_{o}\sin\phi\\\
\omega\end{bmatrix}$ (5)

Here, $v_{o}$ and $\omega$ is the robot’s linear and angular velocity,
respectively. Since we choose to apply a constant linear velocity, the
dynamics of the robot using the unicycle model are represented primarily by
$\omega$. To control the robot’s motion from one point to another reference
point, the difference between a desired angle, $\phi_{d}$, and the robot’s
current angle, $\phi$, is calculated and used as the error

$e=\phi_{d}-\phi.$ (6)

To prevent singularities from arising and restrict the error between $0$ and
$2\pi$, we update the calculated error using $\mathit{atan2}$

$e_{new}=\mathit{atan2}(\sin(e),\cos(e)).$ (7)

Finally, our control input into the robot is applied with a proportional gain
$K_{p}$.

$\omega=K_{p}\dot{e}_{new}$ (8)

This forms the basis for the control of the swarm using sequential hand
gestures.

##  III SIMULATIONS

A series of 3D simulations were designed to test the viability of sequential
gesture-based control of a swarm. To formalize the experimental simulations,
the following assumptions were made. First, the simulations are limited to
three robots. Though it may seem like a small swarm size, we believe that this
size provides a viable minimal benchmark for how our algorithm will actually
work. Second, odometry is provided by the simulation environment and not by
internal (i.e. encoders) or external (i.e. GPS, cameras, etc.) sensors.
Finally, a black background is used when reading images from the gestures to
streamline classifications.

###  III-A Structure

The simulation was developed using _ros2_ , the next-generation of the Robot
Operating System (ROS). Compared to the original ROS, ros2 provides an
enhanced middleware programming environment, the removal of a ROS Master to
make multi-robot decentralized approaches easier, and expanded platform and
architecture support. _Gazebo_ , a 3D robot simulation environment, is used to
render the experiment. Gazebo is a project developed in tandem with ROS/ros2,
and comes with many ROS drivers to simplify development of robots by providing
a simulation environment that functions similarly to a physical one. The
experiment is run using the Turtlebot3 Burger robot by _ROBOTIS_ [14] who
provide 3D Gazebo models for use in simulation.

Each robot runs two separate nodes: one that contains drivers to connect to
Gazebo to simulate differential drive control, sensor readings, etc., and
another that is the primary motion controller as defined in Section II-B. The
robots do not intercommunicate and only subscribe to messages from one
external node that tells them what gestures were classified. This external
node reads information from a generic webcam running the SqueezeNet model.
Additionally, images captured by the webcam are modified to reflect those used
for training, which were silhouette images of the different hand gestures.

The captured camera images are cropped from the $640\times 480$ resolution to
$480\times 480$ resolution and then scaled to $240\times 240$, which is the
input size for our SqueezeNet model. The images are then converted to
greyscale and a slight Gaussian blur is applied to prevent fine details in the
image from causing artifacts when converted to a binary image. Finally, the
image is converted using a binary filter and inputted into the SqueezeNet
model. The predicted gesture is then published to all the robots. This message
is not unique to each robot.

###  III-B Gestures

For this experiment, the number of gestures used have been reduced to the
following five:

$G_{possible}=\\{\textit{Palm},~{}\textit{Peace},~{}\textit{Fist},~{}\textit{C},~{}\textit{L}\\}$
(9)

These gestures are mapped to the subsequent actions the swarm may undergo:

  * •

Palm: Stop movement of the swarm

  * •

Peace: Resume movement of the swarm

  * •

Fist: Read cohesion action

  * •

C Sign: Increase

  * •

L Sign: Decrease

Additionally, there is a sixth classified gesture is _None_ , which, as stated
in Section II-A1, means that there is no gesture recognized and no swarm
action applicable. _Palm_ (Stop) and _Peace_ (Resume) are the only two
gestures capable of controlling the swarm alone. The rest require the user to
give a sequence of gestures for the swarm to read. One a gesture pertaining to
the swarm behavior (cohesion) the user wishes to modify and the next is a
gesture mapped to a modification variable (increase or decrease).

In the simulation, we will rely on two sequences used to modify the cohesion
of the swarm to help negotiate an obstacle. The first is increasing the
cohesion of the swarm, $\beta=0$, which means that the swarm will group closer
to one another. This is done by giving the following commands in this order:

$\textit{Palm}\rightarrow\textit{Fist}\rightarrow\textit{C}\rightarrow\textit{Peace}$
(10)

This sequence is explained as “stop the swarm, read my cohesion command,
increase cohesion by one step size, and resume moving.” The second would be to
decrease the cohesion of the swarm, $\beta=1$, resulting in a increase in
distance between the robots. This is done using the same hand gestures but
with the decrease cohesion command.

$\textit{Palm}\rightarrow\textit{Fist}\rightarrow\textit{L}\rightarrow\textit{Peace}$
(11)

Just like the increase cohesion sequence, this sequence is similarly explained
as “stop the swarm, read my cohesion command, decrease cohesion by one step
size, and resume moving.” As described earlier, the cohesion of the swarm is
set as steps. Specifically, each call to increase or decrease the swarm
cohesion decreases or increases the distance between the robots by a
calculated $y_{\text{offset}}$, respectively. If an obstacle requires the user
to change the swarms cohesion by multiple steps of $y_{\text{offset}}$, the
user does not need to repeat the whole sequence twice but can concatenate the
swarm command multiple times. For example, if the swarm cohesion is needed to
increase by two steps, a user would provide the following command:

$\textit{Palm}\rightarrow\textit{Fist}\rightarrow\textit{C}\rightarrow\textit{Fist}\rightarrow\textit{C}\rightarrow\textit{Peace}$
(12)

Figure 2: The three testbeds created for simulation. A) An $8m\times 4m$
testbed containing multiple openings for the swarm to traverse through. B) An
$8m\times 4m$ testbed containing only one small opening for the agents to
negotiate. Compared to the previous testbed, this requires the user to string
together multiple sequences of the same gesture to complete. C) A small
$2.5m\times 2.5m$ testbed with one small $1m$ wide opening. This last testbed
is recreated for physical validation in Section IV

###  III-C Simulated Environment

The three robots are placed in a line within three simulated testbeds, shown
in Figure 2. The first contains a series of two types of openings: one small-
sized opening in the middle and two intermediate-sized openings located on
both ends of the testbed. The second testbed has only one small-sized opening.
Each testbed provides a different validation for the capability of our
sequential gesture control scheme. The first demonstrates how individual
commands can be given at the onset of an obstacle, and the other shows how for
more difficult obstacles a user is capable of stringing together multiple
gesture actions into one command. The first two testbeds are $8m\times 4m$ in
size. Each robot is placed $1m$ from the end of the testbed and spread to have
an inter-robot distance of $1.5m$ between one another. To complete each task,
the robots will have to move forward and reach the other end of the testbed.
The swarm of robots will need to negotiate the obstacles before them by
relying on a user’s sequential gesture input.

The third testbed is a recreation of a real testbed used in the physical
experiment section of the paper. Compared to the large surface area of the
other testbeds, this testbed is significantly smaller at $2.5m\times 2.5m$.
Additionally, the initial inter-robot distance is reduced to $0.6m$ and the
robots are placed $0.5m$ from the end. The testbed contains a $1m$ sized
opening a meter from the robots starting position. Like the other testbeds,
the robots will have to negotiate this obstacle using a single sequence of
gestures to increase cohesion. After going through the obstacle, a different
sequence is used to decrease the cohesion. This testbed’s results act as
validation for the success of this experiment using real robots. Figure 3
provides a brief overview of the robots increasing and then decreasing their
cohesion to negotiate a small opening.

Figure 3: An overview of the physical experiment with gesture sequences
required for the swarm to get through the small opening in the middle of the
arena. A) The robot swarm begins on one end of the testbed. The _Peace_
gesture is used as a standalone command to start the experiment. B) Once the
robots get closer to the obstacle, the following sequence
($\textit{Palm}\rightarrow\textit{Fist}\rightarrow\textit{C}\rightarrow\textit{Peace}$)
is given to increase the cohesion of the swarm and resume motion. C) After the
robots have cleared the opening, another gesture sequence
($\textit{Palm}\rightarrow\textit{Fist}\rightarrow\textit{L}\rightarrow\textit{Peace}$)
is used to return the swarm back to their initial cohesion size. D) Finally,
the robots reach the final waypoint and the experiment completes.

##  IV PHYSICAL EXPERIMENTS

Figure 4: The physical testbed.

To validate the use of sequential hand gesture control of a swarm in a
physical setting, an analogous physical testbed to the third simulated testbed
was built as shown in Figure 4. The experiment was set up in the same manner
as the simulated one. For this paper, the individual robot controller and
gesture recognition code was written with the intention of interchangeability
between the simulated and physical experiments. The only difference were
parameter files which provided environmental constraints and individual robot
attributes depending on the environment the test is being run.

However, certain aspects of the simulation were not available for use in our
physical experiments. Odometry within simulation is accurately calculated by
the Gazebo environment, but getting this same information in a physical
experiment required use of an overhead camera and _ArUco_ fiducial markers
[15] [16] to calculate each individual robot’s pose. A ros2 node was developed
to track individual robot position from the overhead camera and calculate
poses from a $0.127\textit{m}\times 0.127\textit{m}$ marker placed atop each
robot that was detected. This odometry information is then published to the
robots for use in their controller node’s.

During simulation, all the ros2 individual robot nodes and gesture node were
run on the same computer. The physical experiment distributes the computation
over a wireless network. Each robot runs their respective controller on their
hardware, but do not communicate to one another. The only communication they
receive are from two external sources: the node calculating individual robot
odometry from a central overhead camera and the node reading and classifying
the sequential gestures from the user. Each of these nodes are also run on
separate computers due to convenience rather than the inability of running
both on the same one. This decentralization comes with a cost, however, in the
form an approximately $0.5s$ delay. Even though the individual robots were
capable of running the gesture detection node, we chose to keep the node
running on a separate computer to keep the test runs between the simulation
and physical experiments similar.

The physical experimental procedure is nearly identical to the simulation
except for the software structure changes presented in this section. There are
a few small changes in comparison to the simulation; however, we do not
believe it reduces the validity of our physical experiment. One additional
change to this experiment was the linear velocity of each robot was reduced
due to the half-second network lag in the system.

##  V DISCUSSION

We have successfully demonstrated our hypothesis by showcasing a simplified
cohesion control model for both simulated and physical testbeds. The
supplemental video attached provides the results of a single run on each
simulated and physical testbed. Each simulated test completed successfully and
the controller responded correctly to the properly classified sequential
gesture commands given from the SqueezeNet model in real-time. As mentioned in
Section III-A, the run corresponding to Testbed $2$ demonstrated the ability
for the systems to read multiple instances of the same _increase cohesion_
gesture sequence in one input. Results from that run show that the provided
input sequence was easily registered and enacted by the robots. Additionally,
the physical experiment was able to finish successfully even with the network
delay present in the system. We believe that re-creations of bigger testbeds,
such as Testbed $1$ and $2$ in our simulations, would yield successful runs.
Although the system is semi-decentralized due to odometry calculations and
having the gesture recognition node running separate from the robots, these
results of these tests show the feasibility of a human operator interacting
with a decentralized robot swarm by showing a robot a sequence of hand
gestures.

Although all the experiments were successful, we did run into a minor
classification issue during test runs. The SqueezeNet CNN would sporadically
misclassify the hand gesture upon the subject’s hand leaving the camera’s
viewing area or when switching between hand gestures. We believe that this
issue is likely due to gestures created during the hand’s motion unable to be
classified. To help reduce this error, publishing of the predicted gesture was
limited to once every half second instead of one every tenth of a second,
which is the refresh rate of the ros2 node that classified the gestures.

##  VI FUTURE WORK

As stated in the beginning of Section II, the experiments shown were not
generalized, but future work hopes to encompass multiple, generalized swarm
behaviors. Research by Harriott _et al._ gives an interesting breakdown of
numerous biologically-inspired swarm behavior metrics [17] as possible future
behaviors employable by our system. A further expanded repertoire of behaviors
would increase the feasibility of this system for everyday work with human
operators.

Although this light-weight SqueezeNet model is capable of being run on low-
powered hardware such as the Raspberry Pi due to its size (approximately $\sim
5$ MB), the model proved very sensitive to lighting conditions and cluttered
backgrounds. Generating the silhouette images is very difficult in more
dynamic backgrounds, which we believe is necessary if a gesture recognition
system were to be placed on each robot. Therefore, all of our tests were run
with a static black background. As of time of writing, the team at _Google AI_
has developed algorithms for more robust hand gesture recognition without the
need for silhouette images capable of running on Android and iOS devices [18].
These algorithms are capable of running on hardware similar to that of the
Turtlebot3 and is able to classify a larger set of hand gestures. Our future
work seeks to explore this and other algorithms capable of recognizing hand
gestures in more dynamic and cluttered backgrounds.

Finally, we intend this work to help us pursue human-swarm interaction capable
of rewarding a swarm for the actions or tasks it accomplishes. Specifically, a
form of dynamic on-line reinforcement learning (RL) on a decentralized swarm.
Work by Nam et al. show the feasibility in applying RL to predict or learn a
human operators “trust” is in the swarm they are operating [19] [20]. We hope
that future work allows more unique and novel methods for human and swarm to
learn from one another in different environments.

##  VII CONCLUSION

Our paper provides preliminary work into using a sequence of hand gestures to
control swarm behavior. This system is a combination of a small sized CNN
model capable of recognizing silhouette images of hand gestures in real-time
and a decentralized robot development environment, _ros2_. We test this
control strategy in a semi-decentralized manner in three simulated testbeds
using three Turtlebot3 Burger robots. The system is further tested on a
physical testbed with the same robots. Both environments yielded successful
runs using a single swarm metric (cohesion of the swarm). In the future, we
intend to expand upon this work and create a fully realized, decentralized
swarm system controllable by a human operator with no external equipment by
giving a sequence of commands to control the whole swarm.

## ACKNOWLEDGMENT

The authors would like to thank Shiba Biswal (ASU), Karthik Elamvazhuthi
(UCLA), and Varun Nalam (ASU) for their assistance in this work.

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