# Active Phase for Activated Random Walk on $\mathbb{Z}^{2}$

Yiping Hu University of Washington, Seattle, USA<EMAIL_ADDRESS>
###### Abstract

We show that for small enough sleep rate, the critical density of the
symmetric Activated Random Walk model on $\mathbb{Z}^{2}$ is strictly less
than one.

##  1 Introduction

The Activated Random Walk (ARW) model is a system of interacting particles,
which is believed to exhibit self-organized criticality. In this paper, we
study the dynamics of the simple ARW on the square lattice $\mathbb{Z}^{2}$.
The process starts with a random number of particles at each site
$x\in\mathbb{Z}^{2}$ distributed as a spatially ergodic distribution with
average density $\zeta>0$. Each particle can be in one of two states, either
active or sleepy. Initially all particles are active. Every active particle
performs a continuous-time simple random walk at rate 1 until it falls asleep,
which happens with sleep rate $\lambda$. A sleepy particle stays put until it
becomes active, i.e. it gets reactivated whenever an active particle arrives
at its site.

For every sleep rate $\lambda$, the system has an absorbing-state phase
transition: if the initial density $\zeta$ of particles is below a critical
value $\zeta_{c}(\lambda)$, all particles eventually sleep and the system
fixates; above $\zeta_{c}(\lambda)$, each particle moves indefinitely and the
system stays active. It is intuitively clear and was proved in [4, 9] that
$\zeta_{c}(\lambda)\leq 1$ for all $\lambda$ in all dimensions.

Our main result answers an open question about the existence of a non-trivial
active phase on $\mathbb{Z}^{2}$.

######  Theorem 1.1.

There is a constant $C>0$ such that for all $\lambda<1$,

$\zeta_{c}(\lambda)\leq\frac{C}{\ln(1/\lambda)}.$

In other words, the system stays active a.s. if
$\zeta>\frac{C}{\ln(1/\lambda)}$.

######  Corollary 1.2.

For small enough $\lambda$, the critical density $\zeta_{c}(\lambda)<1$.

######  Corollary 1.3.

The critical density $\zeta_{c}(\lambda)\to 0$ as $\lambda\to 0$.

Though we choose to demonstrate our proof in $\mathbb{Z}^{2}$, the same
argument works in all dimensions.

In one dimension, similar results as Corollary 1.2 and 1.3 were first proved
in the breakthrough work of [2]. The approach from [2] was later reformulated
in [6] as based on two major ingredients: a mass balance equation between
blocks and a single-block estimate. More recently, this reformulation has been
adopted successfully in [1, 5] to find the right order of asymptotics in
Corollary 1.3 and extend Corollary 1.2 for all $\lambda>0$ respectively. All
these works rely crucially on the topological convenience of $\mathbb{Z}$ in
order to maintain independence structures between blocks. However, we no
longer enjoy such property in $\mathbb{Z}^{2}$, making the block arguments
rather unwieldy. Similar results were also obtained in higher dimensions
$d\geq 3$ [10, 12] or for biased ARW [11, 8], but these arguments only work
for transient walks. See [6] for a more complete survey of results.

In this paper, we take a step back and consider the mass balance equations
between sites instead of blocks. A straightforward energy-entropy calculation
is then employed. Our main goal is to show that a similar line of arguments
works in all dimensions and yields non-trivial results. The argument also
provides simpler proofs and a handy framework for further understanding.

We note that another proof of the non-trivial active phase in two dimensions
was obtained independently by Forien and Gaudillière [3]. The asymptotic upper
bound in Theorem 1.1 is comparable to the one from [3, Theorem 2].

In order to establish Theorem 1.1, it suffices to prove the following
quantitative estimate for the finite-volume dynamics. For any positive integer
$N$, let $\mathcal{B}_{N}:=\\{-N,\dots,N\\}^{2}$ be the finite box in
$\mathbb{Z}^{2}$ and let $|\mathcal{B}_{N}|$ be the cardinality of
$\mathcal{B}_{N}$. Consider the ARW dynamics on $\mathcal{B}_{N}$ where the
walks are killed upon leaving $\mathcal{B}_{N}$. Without loss of generality,
we only consider the initial configurations with at most $|\mathcal{B}_{N}|$
particles in $B_{N}$. The finite-volume dynamics eventually stabilizes, that
is when all particles that remain in $\mathcal{B}_{N}$ are sleeping. We call
$S(\mathcal{B}_{N})$ the number of sleeping particles in $\mathcal{B}_{N}$
after stabilization.

######  Theorem 1.4.

There exist positive constants $C$ and $c$, such that for any integer $N$ and
any initial configuration in $\mathcal{B}_{N}$,

$\mathbb{P}\left(S(\mathcal{B}_{N})\geq\frac{C}{\ln(1/\lambda)}|\mathcal{B}_{N}|\right)\leq
e^{-cN^{2}}.$

Theorem 1.4, combined with the criterion [6, Theorem 2.11] and the
universality result [7], gives a proof of Theorem 1.1. The rest of the paper
is devoted to the proof of 1.4.

##  2 Modified site-wise representation

Formally, the configuration of the system at time $t\geq 0$ is given by
$\eta_{t}\in\\{0,\mathfrak{s},1,2,\dots\\}^{\mathbb{Z}^{2}}$, where
$\mathfrak{s}$ represents a single sleeping particle and we have
$0<\mathfrak{s}<1$. Write $x\sim y$ if $x$ is a neighbor of $y$. In the
continuous-time ARW, for $x\in\mathbb{Z}^{2}$ such that $\eta_{t-}(x)\geq 1$,
the system undergoes transitions $\eta_{t}=\mathfrak{t}_{x,y}\eta_{t-}$ and
$\eta_{t}=\mathfrak{t}_{x,\mathfrak{s}}\eta_{t-}$ at rates
$\frac{1}{4}\eta_{t-}(x)$ and $\lambda\eta_{t-}(x)$ respectively. For $x\sim
y$, the movement transition $\mathfrak{t}_{x,y}$ is defined by

$\mathfrak{t}_{x,y}(\eta)(z)=\begin{cases}\eta(y)+1,&z=y,\\\
\eta(x)-1,&z=x,\\\ \eta(z),&\text{otherwise},\end{cases}$

where we use the convention $\mathfrak{s}+1=2$. The sleeping instruction is
given by

$\mathfrak{t}_{x,\mathfrak{s}}(\eta)(z)=\begin{cases}\mathfrak{s},&z=x\text{
and }\eta(x)=1,\\\ \eta(z),&\text{otherwise.}\end{cases}$

It is useful to consider the site-wise representation of ARW. However, we will
use a slight modification of the usual site-wise representation, cf.[6]. In
our setting, each site $x\in\mathbb{Z}^{2}$ is associated with one stack of
i.i.d. movement instructions $\left(\xi_{k}^{x}\right)_{k\in\mathbb{N}^{+}}$,
together with another independent stack of i.i.d. geometric random variables
$\left(g_{k}^{x}\right)_{k\in\mathbb{N}}$. More specifically, each movement
$\xi_{k}^{x}$ has the distribution of $\mathfrak{t}_{x,y_{k}}$, where $y_{k}$
is a uniformly random neighbor of $x$, while each geometric $g_{k}^{x}$
encodes the number of sleep instructions between consecutive movements
$\xi_{k}^{x}$ and $\xi_{k+1}^{x}$, thus taking value in $\\{0,1,2,\dots\\}$
with success probability $1/(1+\lambda)$.

It has been proved [6, Section 11.2] that the stacks of instructions $\xi$ and
$g$, with poisson clocks attached to every site, gives an explicit
construction of the ARW process $(\eta_{t})_{t\geq 0}$ on $\mathbb{Z}^{2}$. We
will not need the Abelian property of the ARW in the remaining sections, but
note that essential inputs of Theorem 1.1, including [6, Theorem 2.11], [7]
and [6, Section 11.2], all use the Abelian property in their proofs.

For $x\sim y$, let $n_{x,y}(m)$ be the number of times that
$\mathfrak{t}_{x,y}$ appears in the first $m$ movement instructions
$\left(\xi_{k}^{x}\right)_{k=1}^{m}$. Also let
$\chi_{x}(m):=1\\{g_{m}^{x}>0\\}$ be the Bernoulli random variable with
parameter $\lambda/(1+\lambda)$.

##  3 Mass balance equation

In this section we focus on the finite-volume dynamics and use the mass
balance equation to prove Theorem 1.4.

We start by introducing three random fields
$\mathbf{M}=(M_{x})\in\mathbb{N}^{\mathcal{B}_{N}}$,
$\mathbf{S}=(S_{x})\in\\{0,1\\}^{\mathcal{B}_{N}}$ and
$\boldsymbol{\Phi}=(\Phi_{x})\in\mathbb{N}^{\partial\mathcal{B}_{N}}$. First,
define the activity odometer field $M_{x}$ which counts the number of movement
instructions used at $x$ until stabilization. Note that in $M_{x}$, we do not
count any sleep instruction. Secondly, let $S_{x}$ be the indicator random
variable of there being a sleeping particle at $x$ after stabilization.
Thirdly, define the exit measure $\Phi_{x}$ to be the number of particles
killed upon entering a site $x\in\partial B_{N}$. Here, for any
$A\subseteq\mathbb{Z}^{2}$, we define the boundary set $\partial A:=\\{x\in
A^{c};\text{there exists }y\in A\text{ such that }y\sim x\\}$.

Now suppose we’re given deterministic fields
$\mathbf{m}=(m_{x})\in\mathbb{N}^{\mathcal{B}_{N}}$,
$\mathbf{s}=(s_{x})\in\\{0,1\\}^{\mathcal{B}_{N}}$ and
$\boldsymbol{\phi}=(\phi_{x})\in\mathbb{N}^{\partial\mathcal{B}_{N}}$. A tuple
of fields $(\mathbf{m},\mathbf{s},\boldsymbol{\phi})$ is said to satisfy the
mass balance equation if

  1. (i)

the mass balance equation holds at every $x\in\mathcal{B}_{N}$:

$\eta_{0}(x)+\sum_{y\sim x}n_{y,x}(m_{y})=m_{x}+s_{x},$ (1)

where $\eta_{0}$ denotes the initial configuration;

  2. (ii)

the boundary condition is met at every $x\in\partial\mathcal{B}_{N}$:

$n_{R(x),x}(m_{R(x)})=\phi_{x},$ (2)

where $R(x)$ is the unique neighbor of $x$ contained in $\mathcal{B}_{N}$.

Note that the tuple of random fields
$(\mathbf{M},\mathbf{S},\boldsymbol{\Phi})$ defined above always satisfies the
mass balance equation.

To prove Theorem 1.4, we rely on the following entropy bound on all possible
field configurations satisfying the mass balance equation. Denote by
$\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})$ the event where
equation (1) holds at $x\in\mathcal{B}_{N}$; for
$x\in\partial\mathcal{B}_{N}$, we use the same notation for the event where
equation (2) holds at $x$. Also note that
$\left\|\boldsymbol{\Phi}\right\|_{\infty}\leq|\eta_{0}|$, where
$|\eta_{0}|:=\sum_{x\in\mathcal{B}_{N}}\eta_{0}(x)$.

######  Lemma 3.1.

There exists $c_{1}>0$ such that for any $N\in\mathbb{N}^{+}$ and initial
configuration $\eta_{0}$ in $\mathcal{B}_{N}$ with $|\eta_{0}|\leq|B_{N}|$,

$\sum_{\mathbf{m},\mathbf{s},\boldsymbol{\phi}}\mathbb{P}\Bigg{(}\bigcap\limits_{x\in\mathcal{B}_{N}\cup\partial\mathcal{B}_{N}}\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})\Bigg{)}\leq
e^{c_{1}|\mathcal{B}_{N}|},$ (3)

where the summation is over
$\left\|\boldsymbol{\phi}\right\|_{\infty}\leq|\eta_{0}|$ and arbitrary
$\mathbf{m},\mathbf{s}$.

The proof of Lemma 3.1 will be given in the next section.

###### Proof of Theorem 1.4.

Recall from Theorem 1.4 that $S(\mathcal{B}_{N})$ represents the number of
sleeping particles in $\mathcal{B}_{N}$ after stabilization, so we have
$S(\mathcal{B}_{N})=\sum_{x\in\mathcal{B}_{N}}S_{x}$. Note also that

$S_{x}\leq\chi_{x}(M_{x}),$

where $\chi_{x}(m)$ is defined in Section 2. Thus in order to prove Theorem
1.4, it suffices to upper bound $\mathbb{P}(\mathcal{A}(\mathbf{M}))$, where
$\mathbf{M}$ is the odometer field, and for a fixed $\mathbf{m}$, we use
$\mathcal{A}(\mathbf{m})$ to denote the event that

$\sum_{x\in\mathcal{B}_{N}}\chi_{x}(m_{x})\geq\frac{C}{\ln(1/\lambda)}|\mathcal{B}_{N}|.$

By decomposing over the fields we get $\mathbb{P}(\mathcal{A}(\mathbf{M}))$ is
equal to

$\displaystyle\sum_{\mathbf{m},\mathbf{s},\boldsymbol{\phi}}\mathbb{P}\Bigg{(}\mathcal{A}(\mathbf{m}){\,\textstyle\bigcap\,}\\{(\mathbf{m},\mathbf{s},\boldsymbol{\phi})=(\mathbf{M},\mathbf{S},\boldsymbol{\Phi})\\}{\,\textstyle\bigcap\,}\bigcap\limits_{x\in\mathcal{B}_{N}\cup\partial\mathcal{B}_{N}}\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})\Bigg{)}$
$\displaystyle\leq\sum_{\mathbf{m},\mathbf{s},\boldsymbol{\phi}}\mathbb{P}\Bigg{(}\mathcal{A}(\mathbf{m}){\,\textstyle\bigcap\,}\bigcap\limits_{x\in\mathcal{B}_{N}\cup\partial\mathcal{B}_{N}}\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})\Bigg{)}$
$\displaystyle\leq\sum_{\mathbf{m},\mathbf{s},\boldsymbol{\phi}}\mathbb{P}\left(\mathcal{A}(\mathbf{m})\right)\mathbb{P}\Bigg{(}\bigcap\limits_{x\in\mathcal{B}_{N}\cup\partial\mathcal{B}_{N}}\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})\Bigg{)},$
(4)

where in the last inequality we use the independence between $\chi_{x}(m)$’s
and $n_{x,y}(m)$’s. For any fixed $\mathbf{m}$, the probability of
$\mathcal{A}(\mathbf{m})$ is controlled via Chernoff bound by

$e^{-D_{\text{KL}}\left(\frac{C}{\ln(1/\lambda)}\big{\|}\frac{\lambda}{1+\lambda}\right)\,|\mathcal{B}_{N}|}$
(5)

where

$D_{\text{KL}}(p_{1}\|\,p_{2})=p_{1}\ln\frac{p_{1}}{p_{2}}+(1-p_{1})\ln\left(\frac{1-p_{1}}{1-p_{2}}\right)$

is the Kullback-Leibler divergence between Bernoulli distributed random
variables with parameters $p_{1}$ and $p_{2}$ respectively. By combining (4),
(5) and Lemma 3.1 and picking a large enough $C$, we complete the proof of
Theorem 1.4. ∎

##  4 Entropy bound

In this section we prove the entropy bound in Lemma 3.1. Partition
$\mathcal{B}_{N}$ into a set of cycles
$\mathcal{B}_{N}=\coprod_{n=0}^{N}C_{n}$, where each
$C_{n}:=\\{x\in\mathbb{Z}^{2};\left\|x\right\|_{\infty}=N-n\\}$. Set
$C_{-1}:=\partial\mathcal{B}_{N}$. Recall from Section 2 that there is a stack
of movement instruction $(\xi_{k}^{x})_{k\in\mathbb{N}^{+}}$ at every
$x\in\mathcal{B}_{N}$. Consider the filtration $(\mathcal{F}_{n})_{n=-1}^{N}$,
where $\mathcal{F}_{-1}$ is the trivial $\sigma$-field and for $n\geq 0$, each
$\mathcal{F}_{n}$ is the $\sigma$-field generated by the instructions
$\\{\xi_{k}^{x};k\in\mathbb{N}^{+},x\in C_{i},0\leq i\leq n\\}$ in the
outermost $n+1$ cycles.

######  Lemma 4.1.

Fix $N,\eta_{0},\mathbf{s}$ and $\boldsymbol{\phi}$. For every $n=0,\dots,N$,
a.s.,

$\mathbb{E}\bigg{[}\,\sum_{\mathbf{m}_{n}}\prod_{x\in
C_{n-1}}1_{\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\,\bigg{|}\,\mathcal{F}_{n-1}\bigg{]}\leq
4^{|C_{n}|},$ (6)

where $\mathbf{m}_{n}:=(m_{x})_{x\in C_{n}}$ is the restriction of
$\mathbf{m}$ to $C_{n}$, and the summation is over all
$\mathbf{m}_{n}\in\mathbb{N}^{C_{n}}$.

###### Proof.

Define the backward neighbor function $B$ that maps each $y\in C_{n}$ to a
neighbor of $y$ in $C_{n-1}$; whenever there are multiple such neighbors, just
pick one arbitrarily. Note that $B$ is a bijection from $C_{n}$ to $B(C_{n})$.
Rather than working with (6), it suffices to give an upper bound of a slightly
larger expression

$\mathbb{E}\bigg{[}\,\sum_{\mathbf{m}_{n}}\prod_{y\in
C_{n}}1_{\mathcal{M}(B(y),\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\,\bigg{|}\,\mathcal{F}_{n-1}\bigg{]}.$
(7)

By either the mass balance equation (1) when $n\geq 1$ or the boundary
condition (2) when $n=0$, the event
$\mathcal{M}(B(y),\mathbf{m},\mathbf{s},\boldsymbol{\phi})$ above depends on
$\mathbf{m}_{n}$ only through $m_{y}$. Thus we may rewrite (7) as

$\mathbb{E}\bigg{[}\,\prod_{y\in
C_{n}}\sum_{m_{y}}1_{\mathcal{M}(B(y),\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\,\bigg{|}\,\mathcal{F}_{n-1}\bigg{]}.$
(8)

Now note that all randomness defining
$\mathcal{M}(B(y),\mathbf{m},\mathbf{s},\boldsymbol{\phi})$, except for
$n_{y,B(y)}(m_{y})$, are measurable with respect to $\mathcal{F}_{n-1}$.
Therefore, the sums over $m_{y}$ in (8), conditioned on $\mathcal{F}_{n-1}$,
are stochastically dominated by i.i.d. geometric random variables with
parameter $1/4$. This implies that both (8) and (7) are controlled by
$4^{|C_{n}|}$, thus proving (6). ∎

###### Proof of Lemma 3.1.

We will inductively show that for $0\leq n\leq N$,

$\mathbb{E}\bigg{[}\,\sum_{\mathbf{m}_{0}}\sum_{\mathbf{m}_{1}}\cdots\sum_{\mathbf{m}_{n}}\prod_{x\in
A_{n-1}}1_{\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\bigg{]}\leq
4^{|A_{n}\setminus A_{-1}|},$ (9)

where $A_{n}:=\bigcup_{i=-1}^{n}C_{i}$. Assuming (9) for $n=N$, it follows
that the left-hand side of (3) is bounded above by $4^{|\mathcal{B}_{N}|}\cdot
2^{|\mathcal{B}_{N}|}\cdot|\mathcal{B}_{N}|^{|\partial\mathcal{B}_{N}|}$. This
would imply Lemma 3.1 by choosing $c_{1}$ large enough.

It remains to prove (9) by induction. The base case of $n=0$ is nothing but
Lemma 4.1 when $n=0$. Suppose the inequality is true for $n-1$. By
conditioning on $\mathcal{F}_{n-1}$ and using Lemma 4.1, we get

$\displaystyle\mathbb{E}\bigg{[}\,\sum_{\mathbf{m}_{0}}$
$\displaystyle\sum_{\mathbf{m}_{1}}\cdots\sum_{\mathbf{m}_{n}}\prod_{x\in
A_{n-1}}1_{\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\,\sum_{\mathbf{m}_{0}}\sum_{\mathbf{m}_{1}}\cdots\sum_{\mathbf{m}_{n-1}}\prod_{x\in
A_{n-2}}1_{\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\times$
$\displaystyle\hskip
50.58878pt\mathbb{E}\bigg{[}\,\sum_{\mathbf{m}_{n}}\prod_{x\in
C_{n-1}}1_{\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\,\bigg{|}\,\mathcal{F}_{n-1}\bigg{]}\bigg{]}$
$\displaystyle\leq
4^{|C_{n}|}\cdot\mathbb{E}\bigg{[}\,\sum_{\mathbf{m}_{0}}\sum_{\mathbf{m}_{1}}\cdots\sum_{\mathbf{m}_{n-1}}\prod_{x\in
A_{n-2}}1_{\mathcal{M}(x,\mathbf{m},\mathbf{s},\boldsymbol{\phi})}\bigg{]}$
$\displaystyle\leq 4^{|C_{n}|}\cdot 4^{|A_{n-1}\setminus
A_{-1}|}=4^{|A_{n}\setminus A_{-1}|}.$

∎

### Acknowledgement

The author would like to thank Christopher Hoffman for many useful comments
and helpful conversations. Y.H. was supported by NSF grant DMS-1444084.

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# Are fast radio bursts generated by cosmic string cusps?

Renato Costa<EMAIL_ADDRESS>Jake E. B. Gordin<EMAIL_ADDRESS>Amanda Weltman<EMAIL_ADDRESS>The Cosmology and Gravity Group,
Department of Mathematics and Applied Mathematics, University of Cape Town,
Private Bag, Rondebosch, 7700, South Africa

###### Abstract

We revisit the idea that cosmic strings could source fast radio bursts by
taking into account Lorentz boosts and a weaker assumption about the scaling
law for the energy of particle decay. We show that the distance relation and
time scale, for a specific value of the scaling of energy, are still
compatible with observations. However, the event rate predicted by the model
is too high when compared to the data. We additionally show that a more
realistic string, with a finite thickness, further compounds the problem by
prohibiting cusp formation and point out how a superconducting wiggly string
could circumvent this issue.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Fast radio bursts (FRBs) are millisecond-duration radio emissions of unknown
physical origin (first reported in Lorimer:2007qn ; for a review, see
Petroff:2017pny ; Katz:2016dti ). Among the many attempts to explain FRBs, one
hypothesis is of interest in this paper: cusp decay from cosmic strings as an
FRB mechanism. Electromagnetic signatures from cosmic string cusps at high-
energy regimes have been considered before Brandenberger:1986vj ;
Bhattacharjee:1989vu ; MacGibbon:1989kk ; MacGibbon:1992ug ;
Brandenberger:1993hw . Recently the idea has been applied to radio frequencies
and their cosmological implications Cai:2011bi ; Cai:2012zd ;
Brandenberger:2017uwo .

Standard cosmic strings are described by the Nambu-Goto action and decay
primarily through gravitational radiation Brandenberger:1986vj  (for a review
see Vilenkin:278400 ; Brandenberger:1993by ). It was previously thought that
another signature of cosmic strings would come from non-negligible
electromagnetic radiation emitted by cusps Spergel:1986uu  \- portions of the
string which double back on themselves - due to its considerable size,
$l_{c}\sim r^{1/3}R^{2/3}$, where $r$ is the thickness of the string and $R$
is the radius of the string loop. Given that cosmic strings have yet to be
observed, and assuming cusp decay produces radiation across all frequencies,
FRBs could, in principle, provide a possible observational testing ground for
cosmic strings’ signatures, as explored in Brandenberger:2017uwo .

There were, however, two assumptions made by the authors in
Brandenberger:2017uwo  which, if dispensed with, alter their conclusions. The
first assumption is that the length of the cusp region is unaffected by
relativity. Specifically, Lorentz contractions in the rest frame of the
string, when taken into account, give a cusp length estimation significantly
smaller than the one used in Brandenberger:2017uwo . In fact, it was shown in
BlancoPillado:1998bv  that the cusp length actually scales as
$l_{c}\sim\sqrt{rR}$.

The second assumption is that, when a cusp decays, the number density of
particle emission scales as $N(E)\propto E^{-3/2}$. This relation comes from a
simply QCD multiplicity function MacGibbon:1989kk  and is shown to hold well
in high energy experiments Brandenberger:2017uwo . It is not clear, however,
if this holds all the way down to radio energies. As such, we also consider a
more general scaling law, $N(E)\propto E^{-m}$, to check whether different
values of $m$ can compensate for the much smaller size of the cusp length and
still favour the model proposed in Brandenberger:2017uwo .

There is a third assumption made in Brandenberger:2017uwo  and indeed in
almost all analyses of electromagnetic radiation from cusps
Brandenberger:1986vj ; Bhattacharjee:1989vu ; MacGibbon:1989kk ;
MacGibbon:1992ug ; Brandenberger:1993hw ; back-reaction when cusps form is
nominally omitted. In Nambu-Goto strings this can be ignored, since with a
zero thickness approximation there is no back-reaction. However, from
simulations we know that strings in a distribution of string loops form small-
scale structure (commonly called wiggles) Martins:1999su . These wiggles are
dynamical fields coupled to the strings, and so affect the formation of cusps.
We show explicitly here that cusp formation is unnatural when a cosmic string
is wiggly.

The paper is divided as follows: in section II, we present our analysis
assuming the different values for the cusp length and scaling for the energy
mentioned above and check how the distance relation, time scale and emission
rate change. In section III we consider a more realistic string with a finite
thickness given by a mass current and check if cusps can form. We conclude in
section IV.

##  II Cusp length with Lorentz Contractions

We start by considering the relativistic effects on the cusp length and a more
general scaling for the energy of decay particles. The number density of
photons from the cusp region, in the limit $E\ll E_{f}$, is given by

$N(E)\sim\frac{1}{\Theta^{2}d^{2}}\frac{\mu
l_{c}}{{E}^{2}_{f}}\left(\frac{E_{f}}{E}\right)^{m},$ (1)

where $E_{f}$ is the fragmentation energy of the particles produced in the
decay process, $d$ is the distance from the cusp at which an observer would
receive such a number density of photons and $\Theta$ is the beaming angle. We
take $E_{f}$ to be on the order of the symmetry breaking scale associated with
the formation of the cosmic string, i.e. $E_{f}\sim\eta$. $\mu$ is the mass
per unit length of the string, and scales as $\mu\sim\eta^{2}$. The beaming
angle is given by MacGibbon:1992ug

$\Theta\sim\frac{1}{\ln(E_{f}/\textnormal{GeV})}.$ (2)

We now follow the same procedure as Brandenberger:2017uwo . The flux of
particles is defined by

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int^{E}_{0}dE^{\prime}\left(N(E^{\prime})E^{\prime}\right)$ (3)
$\displaystyle\sim$ $\displaystyle\frac{\mu
l_{c}}{(2-m)\Theta^{2}d^{2}}\left(\frac{E_{f}}{E}\right)^{m-2}.$

Notice that we must have $m<2$ in order to have a non-divergent flux.
Assuming, $E_{f}\sim\eta$ for the fragmentation energy scale, $l_{c}\sim
r^{1/2}R^{1/2}$ for the cusp length including relativistic corrections
BlancoPillado:1998bv , and $r\sim\eta^{-1}$ the width of the cosmic string, we
have

$S\sim\frac{(R\eta)^{-3/2}}{(2-m)\Theta^{2}}\left(\frac{E}{\eta}\right)^{2-m}\left(\frac{R}{d}\right)^{2}\eta^{3}.$
(4)

It is useful to express $\eta$ in terms of $m_{p}$ and $t_{0}$ (the Planck
mass and present time), which are fixed scales. In general, we can write

$\displaystyle\eta$ $\displaystyle\sim$
$\displaystyle(t_{0}m_{p})\frac{\eta}{t_{0}m_{p}}\sim
10^{60}\frac{1}{t_{0}}(G\mu)^{1/2}$ (5) $\displaystyle\sim$ $\displaystyle
10^{15}\text{GeV},$

where in the last line we used $G\mu\sim 10^{-7}$, $t_{0}\sim
10^{42}\text{GeV}^{-1}$. This value for $G\mu$ is the upper bound on the
tension of cosmic strings given by CMB measurements of the angular power
spectrum Dvorkin:2011aj ; Ade:2013xla ; Charnock:2016nzm . One can, of course,
choose small values for the string tension as we will do later.

The energy flux observed is of order $S_{0}\sim 10^{-48}(\text{GeV})^{3}$, so
we must have

$\displaystyle S(E,R,d(R))$ $\displaystyle=$ $\displaystyle S_{0}.$ (6)

Using the above relation we can estimate the distance of the source from the
detector

$\displaystyle d(R)$ $\displaystyle\sim$
$\displaystyle\frac{R^{1/4}}{\sqrt{(2-m)S_{0}}\Theta}\bigg{(}\frac{E}{\eta}\bigg{)}^{\frac{2-m}{2}}\eta^{3/4}.$
(7)

###  II.1 Distance criterion

The distance (7) must be bigger than the expected distance of a cosmic string
from us. That has to be so, since the expected distance defined by

$\displaystyle n(R,t)Rd_{R}^{3}=1,$ (8)

where $n(R,t)$ is the number density of strings per unit of radius, gives the
minimum distance at which one can expect to find at least one cosmic string.
It is also the expected mean separation distance of a string loop in a string
loop distribution Brandenberger:2017uwo . Using the number density

$n(R,t)=\beta R^{-5/2}t^{-2}t_{eq}^{1/2},\qquad\gamma G\mu t<R<t_{eq}$ (9)

obtained from the VOS scale model Kibble:1984hp ; Vilenkin:1981kz  in a matter
dominated universe111The number density for a radiation dominated universe is
different Brandenberger:2017uwo , however we will not consider it here since
the signal from strings closer to us will dominate., where $\beta$ is a
constant of order 1, we get

$\displaystyle d_{R}\sim 10R^{1/2}t_{0}^{1/2}.$ (10)

The condition $d_{R}<d(R)$ implies

$\displaystyle R^{1/2}$ $\displaystyle<$ $\displaystyle\frac{1\times
10^{-2}}{t_{0}(2-m)\Theta^{2}S_{0}}\bigg{(}\frac{E}{\eta}\bigg{)}^{2-m}\hskip
1.42262pt\eta^{3/2}.$ (11)

The fast radio burst energy peak is of order $E\sim 10^{-15}\text{GeV}$. For
such an $E$, the values for $R$ at different $m$ and $\eta$ are shown in Table
1. According to the scaling model for the distribution of cosmic string loops
Vilenkin:1981kz ; Kibble:1984hp  the radius, in a matter dominated Universe,
lies in the interval $\gamma G\mu t_{0}<R<t_{eq}$. Strings with a radius
smaller than the lower bound decay in less than one Hubble time. From Table 1
we see that, considering the same scaling for the energy as
Brandenberger:2017uwo , i.e. $m=3/2$, Lorentz contraction implies a smaller
radius for the string loop, which lies outside the allowed region222In
Brandenberger:2017uwo , one gets $R\sim 10^{51}\text{(GeV)}^{-1}$ by
considering $G\mu\sim 10^{-7}$, which would be inside the allowed region.. The
only way to have a consistent string loop size is if the scaling of the energy
is given by $m\sim 2$. A scaling below $m=3/2$ makes the situation even worse.
We also compute $R$ for $G\mu\sim 10^{-12}$ and for $G\mu\sim 10^{-18}$, which
implies $\eta\sim 10^{12}$ and $\eta\sim 10^{9}$ respectively, to check the
impact of string tension on the radius of the loop. Changing the string
tension will also change the time scale for the emitted radio burst and the
event rate as we will see in the following sections.

$\eta$ $m$ |  $\sim$ 2 | 3/2 | 1 | 1/2
---|---|---|---|---
$10^{15}$ GeV | $\frac{10^{59}}{(2-m)^{2}}$ | $10^{29}$ | $10^{-1}$ | $10^{-31}$
$10^{12}$ GeV | $\frac{10^{50}}{(2-m)^{2}}$ | $10^{23}$ | $10^{-4}$ | $10^{-31}$
$10^{9}$ GeV | $\frac{10^{42}}{(2-m)^{2}}$ | $10^{18}$ | $10^{-8}$ | $10^{-32}$
Table 1: Values of $R$ in $(\text{GeV})^{-1}$ for different $\eta$ and $m$.
These values have to be compared to $R_{min}\sim\gamma G\mu t_{0}$ for
differing values of $\eta$. For $\eta\sim 10^{15}\text{GeV}$, $R_{min}\sim
10^{36}\text{GeV}^{-1}$; for $\eta\sim 10^{12}\text{GeV}$, $R_{min}\sim
10^{31}\text{GeV}^{-1}$ and for $\eta\sim 10^{9}\text{GeV}$, $R_{min}\sim
10^{25}\text{GeV}^{-1}$.

###  II.2 Time scale

The power of cusp decay is given in Brandenberger:1986vj

$\displaystyle P_{c}\sim\frac{\mu l_{c}}{R}.$ (12)

The time scale for a burst can be estimated as the time taken to go from one
‘side’ of the cusp to the other, i.e. when two strands of the string at the
cusp have moved apart by a distance more than the string width $r$. Since
power is energy per unit time, and the string width is inverse of the energy,
the time scale for the cusp decay is

$\displaystyle T=\frac{1}{P_{c}\hskip 1.42262ptr}\sim R^{1/2}r^{1/2}.$ (13)

This is different to the time scale in Brandenberger:2017uwo  by a factor of
$(R/r)^{1/6}$. Since the radius of the string loop $R$ varies with the scaling
of the energy we will also have different values for the time scale. These are
shown in Table 2. FRBs are observed as millisecond scale pulses at observation
with possibly up to microsecond pulses at emission. To be consistent with the
distance criterion and time scale of the burst, we must tune $m\sim 2$ and
choose $\eta$ of order $10^{12}$ GeV or smaller, as one can check from Table 1
and Table 2. Note that consistency with FRBs data could set another upper
bound for the string tension. From the numbers analysed here it would be of
order $G\mu\sim 10^{-12}$, which implies an energy scale for the formation of
cosmic strings of $\eta\sim 10^{12}\text{GeV}$. Of course a more precise
estimation for the string tension would require a more detailed study. We
continue with our final consistency check, the event rate.

$\eta$ $m$ |  $\sim$ 2 | 3/2 | 1
---|---|---|---
$10^{15}$ GeV | $\frac{10^{-2}}{(2-m)}$ | $10^{-17}$ | $10^{-32}$
$10^{12}$ GeV | $\frac{10^{-5}}{(2-m)}$ | $10^{-18}$ | $10^{-32}$
$10^{9}$ GeV | $\frac{10^{-8}}{(2-m)}$ | $10^{-20}$ | $10^{-32}$
Table 2: Time scale of a burst, in seconds, for different values of $m$ and
$\eta$.

###  II.3 Event rate

The event rate of FRBs is given by

$\displaystyle\mathcal{R}\sim\int^{t_{0}}_{\gamma G\mu
t_{0}}dR\left(d^{3}(R)\hskip 1.42262ptn(R,t_{0})\frac{1}{R}P(R)\right),$ (14)

where $P(R)$ is the probability of having a beamed jet of photons coming from
the cusp in the line of sight. Since we expect one cusp per oscillation, we
divide the integrand by $R$. Assuming $P(R)=P_{0}\sim\Theta$ as approximately
constant and plugging (9) and (7) into (14), we get

$\displaystyle\mathcal{R}\sim\frac{P_{0}(\gamma
G\mu)^{-7/4}}{(2-m)^{3/2}\Theta^{3}}\frac{\beta}{S_{0}^{3/2}}\bigg{(}\frac{E}{\eta}\bigg{)}^{\frac{3(2-m)}{2}}\eta^{9/4}t_{0}^{-13/4}10^{-3}.$

The values of $\mathcal{R}$, for different values of $\eta$ and $m$, are shown
in Table 3. Those numbers should be compared to the estimated number of FRBs
per year333Although few FRBs have been detected so far, the expected rate is
relevant here., which is of order $10^{6}$. We can see that if we tune $m$ in
order to get a distance relation and time scale compatible to the data, the
number of expected FRBs emitted would be much bigger than the estimated rate.
On the other hand, if the scaling for the energy is of order $3/2$ or smaller,
the distance relation and event rate are much smaller than the consistency
checks require. This analysis explicitly shows that, unfortunately, FRBs
cannot be interpreted as a signature of structureless cosmic strings.

$\eta$ $m$ |  $\sim$ 2 | 3/2 | 1
---|---|---|---
$10^{15}$ GeV | $\frac{10^{12}}{(2-m)^{3/2}}$ | $10^{-11}$ | $10^{-33}$
$10^{12}$ GeV | $\frac{10^{14}}{(2-m)^{3/2}}$ | $10^{-7}$ | $10^{-27}$
$10^{9}$ GeV | $\frac{10^{17}}{(2-m)^{3/2}}$ | $10^{-1}$ | $10^{-19}$
Table 3: Number of expected FRBs per year for different values of $m$ and
$\eta$.

##  III Wiggly strings and cusp formation

In this section we consider a more realistic string where it could have some
small-scale structure by adding wiggles to it. Intuitively, the presence of a
mass current could make the situation for FRBs even worse, given that cusps
move at the speed of light and wiggles would prevent their formation. We
explicitly show that it is the case.

###  III.1 Equations of motion

We start from the Polyakov version of the wiggly string action introduced by
Carter:2003fb :

$S_{wP}=-\frac{M^{2}}{2}\int
d^{2}\sigma\sqrt{-h}h^{ab}G_{AB}\partial_{a}X^{A}\partial_{b}X^{B}$ (16)

where

$G_{AB}=\begin{bmatrix}g_{\mu\nu}&0\\\ 0&\psi^{2}/M^{2}\end{bmatrix},\qquad
X^{A}=\begin{bmatrix}x^{\mu}\\\ \phi\end{bmatrix}$ (17)

and $h^{ab}$ is an auxiliary metric, $h=\det(h_{ij})$, $\phi(\tau,\sigma)$ are
the wiggles, $M^{2}$ is the Kibble mass scale and $\psi$ is a normalization
constant. This action essentially is a NG string coupled dynamically to a
scalar field, interpretable as neutral charge carriers (or “mass” currents)
Spergel:1986uu ; Martins:2014kda .

The equation of motion for the above action is obtained by varying it with
respect to $X^{A}$

$\displaystyle h^{ab}\partial_{a}\partial_{b}X^{A}$ $\displaystyle=$
$\displaystyle 0.$ (18)

Variation with respect to the auxiliary field $h^{ab}$ gives a set of
constraints

$\displaystyle
T_{ab}=G_{AB}\partial_{a}X^{A}\partial_{b}X^{B}-\frac{1}{2}h_{ab}h^{ij}G_{AB}\partial_{i}X^{A}\partial_{j}X^{B}$
$\displaystyle=$ $\displaystyle 0.$

###  III.2 Constraints and the cusp

We fix world-sheet diffeomorphisms and Weyl invariance by choosing the
conformal gauge $h_{ij}=\eta_{ij}$444Here $\eta_{ij}=diag(-1,1)$ and $c=1$.
and the extra redundancy using the static gauge $x^{0}=\tau$. With these
choices of gauge, the constraints (III.1) become

$\displaystyle-1+\dot{\textbf{x}}\cdot\dot{\textbf{x}}+\textbf{x}^{\prime}\cdot\textbf{x}^{\prime}$
$\displaystyle=$
$\displaystyle-\frac{\psi^{2}}{M^{2}}(\dot{\phi}^{2}+\phi^{\prime 2}),$ (20)
$\displaystyle\dot{\textbf{x}}\cdot\textbf{x}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\psi^{2}}{2M^{2}}\dot{\phi}\phi^{\prime},$ (21)

where $x^{\mu}=(\tau,\textbf{x})$ and $\dot{A}\equiv\partial_{\tau}A$,
$A^{\prime}\equiv\partial_{\sigma}A$ for a general field $A$. The cusp
condition

$\displaystyle\dot{x}^{\mu}\dot{x}_{\mu}=0\rightarrow\dot{\textbf{x}}\cdot\dot{\textbf{x}}=1,$
(22)

while Lorentz contraction sets $\textbf{x}^{\prime}=0$ Spergel:1986uu . Taking
into account the cusp conditions, both constraints can only be satisfied at
the same time if, and only if

$\displaystyle\dot{\phi}(\tau_{c},\sigma_{c})=\phi^{\prime}(\tau_{c},\sigma_{c})=0.$
(23)

This condition was also derived in Spergel:1986uu , but using a completely
different approach.

The general solution of the equation of motion for $\phi$ have the form

$\displaystyle\phi(\tau,\sigma)=f(\sigma-\tau)+g(\sigma+\tau).$ (24)

At the cusp, the constraints require

$\displaystyle\partial_{-}f(\sigma_{-})+\partial_{+}g(\sigma_{+})=0,$ (25)

and

$\displaystyle\partial_{-}f(\sigma_{-})-\partial_{+}g(\sigma_{+})=0.$ (26)

where $\sigma_{\pm}=\sigma\pm\tau$ and
$\partial_{\pm}=\frac{\partial_{\sigma}\pm\partial_{\tau}}{2}$. It implies
$f(\sigma_{-})=c_{1}$ and $g(\sigma_{+})=c_{2}$, where $c_{1}$ and $c_{2}$ are
constants. Note that this is not a requirement neither from the equation of
motion, nor from the constraints. So demanding a cusp implies a fine-tuning on
the initial conditions. In the words of Spergel:1986uu , formation of cusps in
this scenario would demand a ‘conspiracy’.

On the other hand, the situation for a superconducting cosmic string can be
different. The presence of an electromagnetic current would add an extra term
to the constraints (20) and (21) making cusp formation more natural in this
set up or act as a source term in the equations of motion. To explicitly show
that in this set up, one has to generalize the wiggly Polyakov action by
including interactions to the electromagnetic field. We leave this for a
future work. The constraints coming from FRBs data on superconducting strings
parameters, was studied in Ye:2017lqn .

##  IV Discussion and Conclusion

We have computed three consistency conditions, to wit, the distance relation,
time scale, and emission rate that a cosmic string should pass in order to be
a solid candidate as a source of FRBs. We have done so by taking into account
the effect that Lorentz boosts have on the cusp length and assuming a weaker
assumption for the scaling of the energy. It was shown that even though it is
possible to tune the scaling of the energy decay of particles to make the
distance relation and time scale compatible with observations, the emission
rate is too high in that case.

A new derivation for the cusp formation condition using the wiggly version of
the Polyakov action was also provided. It is clear from this condition that
the cusp formation is very unlikely in a string with a finite thickness,
demanding a fine tune of the initial conditions. We also pointed out that, for
a superconducting string, the presence of a charge would weaken the constraint
condition for cusp formation, making it more likely in this set up.

## Acknowledgements

We thank Julien Larena and Bryan Gaensler for valuable discussions, as well as
Bryce Cyr for detailed explanation of his work and careful reading of this
manuscript and Robert Brandenberger and Guilherme Franzmann for their
comments. RC, JEBG and AW acknowledge financial support from the South African
Research Chairs Initiative of the NRF and the DST. JEBG is funded by the
National Astrophysics $\&$ Space Sciences Programme (NASSP). Any opinion,
finding and conclusion or recommendation expressed in this material is that of
the authors and the NRF does not accept any liability in this regard.

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# Application of the Cartier Operator in Coding Theory

Vahid Nourozi  The Klipsch School of Electrical and Computer Engineering, New
Mexico State University, Las Cruces, NM 88003 USA<EMAIL_ADDRESS>
###### Abstract.

The $a$-number is an invariant of the isomorphism class of the $p$-torsion
group scheme. We use the Cartier operator on
$H^{0}(\mathcal{A}_{2},\Omega^{1})$ to find a closed formula for the
$a$-number of the form
$\mathcal{A}_{2}=v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{2}})$ where $q=p^{s}$
over the finite field $\mathbb{F}_{q^{2}}$. The application of the computed
$a$-number in coding theory is illustrated by the relationship between the
algebraic properties of the curve and the parameters of codes that are
supported by it.

###### Key words and phrases:

$a$-number; Cartier operator; Super-singular Curves; Maximal Curves.

##  1\. Introduction

Let $\mathcal{C}$ be a geometrically irreducible, projective, and non-singular
algebraic curve defined over the finite field $\mathbb{F}_{\ell}$ of order
$\ell$. Let $\mathcal{C}(\mathbb{F}_{\ell})$ denote the set of
$\mathbb{F}_{\ell}$-rational points of $\mathcal{C}$. In the study of curves
over finite fields, a fundamental problem is the size of
$\mathcal{C}(\mathbb{F}_{\ell})$. The very basic result here is the Hasse-Weil
bound, which asserts that

$\mid\\#\mathcal{C}(\mathbb{F}_{\ell})-(\ell+1)\mid\leq 2g\sqrt{\ell},$

where $g=g(\mathcal{C})$ is the genus of $\mathcal{C}$.

The curve $\mathcal{C}$ is called maximal over $\mathbb{F}_{\ell}$ if the
number of elements of $\mathcal{C}(\mathbb{F}_{\ell})$ satisfies

$\\#\mathcal{C}(\mathbb{F}_{\ell})=\ell+1+2g\sqrt{\ell}.$

We only consider maximal curves of positive genus, and hence $\ell$ will
always be a square, say $\ell=q^{2}$.

In [10], Ihara showed that if a curve $\mathcal{C}$ is maximal over
$\mathbb{F}_{q^{2}}$, then

$g\leq g_{1}:=\dfrac{q(q-1)}{2}.$

In [5] authors showed that

$\qquad\mbox{either}\qquad g\leq
g_{2}:=\lfloor\frac{(q-1)^{2}}{4}\rfloor\qquad\mbox{or}\qquad
g_{1}=\frac{q(q-1)}{2}.$

Ruck and Stichtenoth [22] showed that, up to $\mathbb{F}_{q^{2}}$-isomorphism,
there is just one maximal curve over $\mathbb{F}_{q^{2}}$ of genus
$\frac{q(q-1)}{2}$, namely the so-called Hermitian curve over
$\mathbb{F}_{q^{2}}$, which can be defined by the affine equation

$y^{q}+y=x^{q+1}.$ (1.1)

From [9] there is a unique maximal curve over $\mathbb{F}_{q^{2}}$ of genus
$g=\frac{(\sqrt{q}-1)(m-1)}{2}$ with $m=\frac{\sqrt{q}+1}{t}$ and
$\sqrt{q}\equiv-1$ mod $t$, which can be defined by the affine equation

$\mathcal{A}_{t}=v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{t}}).$ (1.2)

The curve $\mathcal{A}_{t}$ is an Artin-Schreier curve. Also,
$\mathcal{A}_{1}$ is the affine form of the Hermitian curve (1.1).

In this paper we consider the $\mathcal{A}_{2}$ of genus
$g=\frac{(\sqrt{q}-1)^{2}}{2}$ given by the affine equation

$\mathcal{A}_{2}=v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{2}}).$ (1.3)

Provided that $q/2$ is a Weierstrass non-gap at some point of the curve. It is
easy to see that a maximal curve $\mathcal{C}$ is supersingular since all
slopes of its Newton polygon are equal $1/2$. This fact implies that the
Jacobin $X:=\mbox{Jac}(\mathcal{C})$ has no $p$-torsion points over
$\bar{\mathbb{F}}_{p}$. A relevant invariant of the $p$-torsion group scheme
of the Jacobian of the curve is the $a$-number.

Consider the multiplication by $p$-morphism $[p]:X\rightarrow X$, which is a
finite flat morphism of degree $p^{2g}$. It factors as $[p]=V\circ F$. Here,
$F:X\rightarrow X^{(p)}$ is the relative Frobenius morphism coming from the
$p$-power map on the structure sheaf, and the Verschiebung morphism
$V:X^{(p)}\rightarrow X$ is the dual of $F$. The kernel of multiplication-
by-$p$ on $X$ is defined by the group of $X[p]$. The important invariant is
the $a$-number $a(\mathcal{C})$ of curve $\mathcal{C}$ defined by

$a(\mathcal{C})=\mbox{dim}_{\mathbb{\bar{F}}_{p}}\mbox{Hom}(\alpha_{p},X[p]),$

where $\alpha_{p}$ is the kernel of the Frobenius endomorphism on the group
scheme $\mbox{Spec}(k[X]/(X^{p}))$. Another definition for the $a$-number is

$a(\mathcal{C})=\mbox{dim}_{\mathbb{F}_{p}}(\mbox{Ker}(F)\cap\mbox{Ker}(V)).$

A few results on the rank of the Cartier operator (especially $a$-number) of
curves are introduced by Kodama and Washio [11], González [6], Pries and Weir
[14], Yui [26] and Montanucci and Speziali [13]. Also, I introduced the rank
of the Cartier of the maximal curves in [20], maximal curves with the third
largest genus in [18], the hyperelliptic curve in [17], $\mathbb{F}_{q^{2}}$
maximal function fields in [16], Picard Curve in [15], and explained them in
my PhD dissertation in [19].

In this paper, we determine the $a$-number of certain maximal curves. In the
case $g=g_{1}$, the $a$-number of the Hermitian curves is computed by Gross in
[8]. Here, we compute the $a$-number of maximal curves over
$\mathbb{F}_{q^{2}}$ with genus $g=g_{2}$ for infinitely many values of $q$.

In Section 3, we prove that the $a$-number of the curve $\mathcal{A}_{2}$ with
Equation (1.3) is
$a(\mathcal{A}_{2})=\frac{p-1}{8}(\sqrt{p^{s-2}}+1)(\sqrt{q}-1)$, see Theorem
3.2. Finally, we provided an example of the $a$-number of $\mathcal{A}_{2}$
with the MAGMA code. The proofs use directly the action of the Cartier
operator on $H^{0}(\mathcal{A}_{2},\Omega^{1})$.

##  2\. The Cartier operator

Let $k$ be an algebraically closed field of characteristic $p>0$. Let
$\mathcal{C}$ be a curve defined over $k$. The Cartier operator is a
$1/p$-linear operator acting on the sheaf
$\Omega^{1}:=\Omega^{1}_{\mathcal{C}}$ of differential forms on $\mathcal{C}$
in positive characteristic $p>0$.

Let $K=k(\mathcal{C})$ be the function field of the curve $\mathcal{C}$ of
genus $g$ defined over $k$. A separating variable for $K$ is an element $x\in
K\setminus K^{p}$.

######  Definition 2.1.

(The Cartier operator). Let $\omega\in\Omega_{K/K_{q}}$. There exist
$f_{0},\cdots,f_{p-1}$ such that
$\omega=(f^{p}_{0}+f^{p}_{1}x+\cdots+f^{p}_{p-1}x^{p-1})dx$. The Cartier
operator $\mathfrak{C}$ is defined by

$\mathfrak{C}(\omega):=f_{p-1}dx.$

The definition does not depend on the choice of $x$ (see [23, Proposition 1]).

We refer the reader to [23, 2, 3, 24] for the proofs of the following
statements.

######  Proposition 2.2.

(Global Properties of $\mathfrak{C}$). For all $\omega\in\Omega_{K/K_{q}}$ and
all $f\in K$,

  * •

$\mathfrak{C}(f^{p}\omega)=f\mathfrak{C}(\omega)$;

  * •

$\mathfrak{C}(\omega)=0\Leftrightarrow\exists h\in K,\omega=dh$;

  * •

$\mathfrak{C}(\omega)=\omega\Leftrightarrow\exists h\in K,\omega=dh/h$.

######  Remark 2.3.

Moreover, one can easily show that

$\mathfrak{C}(x^{j}\,dx)=\begin{cases}0&\text{if }\hskip 11.38092ptp\nmid
j+1,\\\ x^{s-1}\,dx&\text{if }\hskip 11.38092ptj+1=ps.\end{cases}$

If $\mbox{div}(\omega)$ is effective, then differential $\omega$ is
holomorphic. The holomorphic differentials set $H^{0}(\mathcal{C},\Omega^{1})$
is a $g$-dimensional $k$-vector subspace of $\Omega^{1}$ such that
$\mathfrak{C}(H^{0}(\mathcal{C},\Omega^{1}))$ is the same as
$H^{0}(\mathcal{C},\Omega^{1})$. Any curve $\mathcal{C}$ has a $a$-number that
is equal to the size of the Cartier operator’s kernel
$H^{0}(\mathcal{C},\Omega^{1})$, or to put it another way, it is the size of
the space of exact holomorphic differentials on $\mathcal{C}$ (see [12,
5.2.8]).

The following theorem is due to Gorenstein; see [7, Theorem 12].

######  Theorem 2.4.

A differential $\omega\in\Omega^{1}$ is holomorphic if and only if it is of
the form $(h(x,y)/F_{y})dx$, where $H:h(X,Y)=0$ is a canonical adjoint.

######  Theorem 2.5.

[13] With the above assumptions,

$\mathfrak{C}(h\dfrac{dx}{F_{y}})=(\dfrac{\partial^{2p-2}}{\partial
x^{p-1}\partial y^{p-1}}(F^{p-1}h))^{\frac{1}{p}}\dfrac{dx}{F_{y}}$

for any $h\in K(\mathcal{X})$.

The differential operator $\nabla$ is defined by

$\nabla=\dfrac{\partial^{2p-2}}{\partial x^{p-1}\partial y^{p-1}},$

has the property

$\nabla(\sum_{i,j}c_{i,j}X^{i}Y^{j})=\sum_{i,j}c_{ip+p-1,jp+p-1}X^{ip}Y^{jp}.$
(2.1)

##  3\. The $a$-number of Curve $\mathcal{A}_{2}$

In this section, we consider the curve $\mathcal{A}_{2}$ is given by the
equation $y^{\sqrt{q}}+y=x^{\frac{\sqrt{q}+1}{2}}$ of genus
$g(\mathcal{A}_{2})=\frac{(\sqrt{q}-1)^{2}}{4}$, with $q=p^{s}$ and $p>2$ over
$\mathbb{F}_{q^{2}}$, where $\sqrt{q}\equiv-1$ mod $2$ . From Theorem 2.4, one
can find a basis for the space $H^{0}(\mathcal{A}_{2},\Omega^{1})$ of
holomorphic differentials on $\mathcal{A}_{2}$, namely

$\mathcal{B}=\\{x^{i}y^{j}dx\mid\frac{\sqrt{q}+1}{2}i+\sqrt{q}j\leq 2g-2\\}.$

######  Proposition 3.1.

The rank of the Cartier operator $\mathfrak{C}$ on the curve $\mathcal{A}_{2}$
equals the number of pairs $(i,j)$ with $\frac{\sqrt{q}+1}{2}i+\sqrt{q}j\leq
2g-2$ such that the system of congruences mod $p$

$\Bigg{\\{}\begin{array}[]{c}k\sqrt{q}+h-k+j\equiv 0,\\\
(p-1-h)(\frac{(\sqrt{q}+1)}{2})+i\equiv p-1,\\\ \end{array}$ (3.1)

has a solution $(h,k)$ for $0\leq h\leq\frac{p-1}{2},0\leq k\leq h$.

###### Proof.

By Theorem 2.5,
$\mathfrak{C}((x^{i}y^{j}/F_{y})dx)=(\nabla(F^{p-1}x^{i}y^{j}))^{1/p}dx/F_{y}$.
So, we apply the differential operator $\nabla$ to

$(y^{\sqrt{q}}+y-x^{\frac{\sqrt{q}+1}{2}})^{p-1}x^{i}y^{j}=\sum_{h=0}^{p-1}\sum_{k=0}^{h}(^{p-1}_{h})(^{h}_{k})(-1)^{h-k}x^{(p-1-h)(\frac{(\sqrt{q}+1)}{2})+i}y^{k\sqrt{q}+h-k+j}$
(3.2)

for each $i,j$ such that $\frac{\sqrt{q}+1}{2}i+\sqrt{q}j\leq 2g-2$.

From the Formula (2.1),
$\nabla(y^{\sqrt{q}}+y+x^{\frac{\sqrt{q}+1}{2}})^{p-1}x^{i}y^{j}\neq 0$ if and
only if for some $(h,k)$, with $0\leq h\leq\frac{p-1}{2}$ and $0\leq k\leq h$,
satisfies both the following congruences mod $p$:

$\Bigg{\\{}\begin{array}[]{c}k\sqrt{q}+h-k+j\equiv 0,\\\
(p-1-h)(\frac{(\sqrt{q}+1)}{2})+i\equiv p-1.\\\ \end{array}$ (3.3)

Take $(i,j)\neq(i_{0},j_{0})$ in this situation both
$\nabla(y^{\sqrt{q}}+y+x^{\frac{\sqrt{q}+1}{2}})^{p-1}x^{i}y^{j}$ and
$\nabla(y^{\sqrt{q}}+y+x^{\frac{\sqrt{q}+1}{2}})^{p-1}x^{i_{0}}y^{j_{0}}$ are
nonzero. We claim that they are linearly independent over $k$. To show
independence, we prove that, for each $(h,k)$ with $0\leq h\leq p-1$ and
$0\leq k\leq h$ there is no $(h_{0},k_{0})$ with $0\leq h_{0}\leq p-1$ and
$0\leq k_{0}\leq h_{0}$ such that

$\Bigg{\\{}\begin{array}[]{c}k\sqrt{q}+h-k+j=k_{0}\sqrt{q}+h_{0}-k_{0}+j_{0},\\\
(p-1-h)(\frac{(\sqrt{q}+1)}{2})+i=(p-1-h_{0})(\frac{(\sqrt{q}+1)}{2})+i_{0}.\\\
\end{array}$ (3.4)

If $h=h_{0}$, then $j\neq j_{0}$ by $i=i_{0}$ from the second equation,
therefore $k\neq k_{0}$. We may assume $k>k_{0}$. Then
$j-j_{0}=(\sqrt{q}-1)(k-k_{0})>\sqrt{q}-1$, a contradiction as
$j-j_{0}\leq\frac{(\sqrt{q}-1)^{2}-4}{2\sqrt{q}}$. Similarly, if $k=k_{0}$,
then $h\neq h_{0}$ by $(i,j)\neq(i_{0},j_{0})$. We assume that $h>h_{0}$. Then
$i-i_{0}=\frac{\sqrt{q}+1}{2}(h-h_{0})>\frac{\sqrt{q}+1}{2}$, a contradiction
as $i-i_{0}\leq\frac{(\sqrt{q}-1)^{2}-4}{\sqrt{q}+1}$. ∎

For convenience, we signify the matrix representing the $p$-th power of the
Cartier operator $\mathfrak{C}$ with $A_{s}:=A(\mathcal{A}_{2})$ on the curve
$\mathcal{A}_{2}$ for the basis $\mathcal{B}$, where $q=p^{s}$.

######  Theorem 3.2.

If $q=p^{s}$ for $s\geq 2$, $s$ be even and $p>2$, then the $a$-number of the
curve $\mathcal{X}$ equals

$\dfrac{p-1}{8}(\sqrt{p^{s-2}}+1)(\sqrt{q}-1).$

###### Proof.

First we prove that, if $q=p^{s},s\geq 2$ and be even, then
$\mbox{rank}(A_{s})=\dfrac{p+1}{8}(\sqrt{p^{s}}-1)(\sqrt{p^{s-2}}-1)$. In this
case, $1\leq\frac{\sqrt{q}+1}{2}i+\sqrt{q}j\leq g$ and System (3.1) mod $p$
reads

$\Bigg{\\{}\begin{array}[]{c}h-k+j\equiv 0,\\\
-\frac{h}{2}-\frac{1}{2}+i\equiv p-1.\\\ \end{array}$ (3.5)

First assume that $s=2$, for $\sqrt{q}=p$, we have $\frac{p+1}{2}i+pj\leq g$
and System (3.5) becomes

$\Bigg{\\{}\begin{array}[]{c}j=k-h,\\\ i=p+\frac{h}{2}-\frac{1}{2},\\\
\end{array}$

in this case $\frac{p+1}{2}i+pj\leq g$ that is,
$\frac{h(1-3p)}{4}+kp\leq\frac{-p^{2}-3p+2}{4}$ then
$h\geq\frac{-p^{2}-3p+2}{1-3p}$, thus $h\geq\frac{3p+10}{9}$ a contradiction
by Proposition 3.1. As a consequence, there is no pair $(i,j)$ for which the
above system admits a solution $(h,k)$. Thus, $\mbox{rank}(A_{1})=0$.

Let $s=4$, so $\sqrt{q}=p^{2}$. For $\frac{p^{2}+1}{2}i+p^{2}j\leq g$, the
above argument still works. Therefore,
$\frac{(p-1)^{2}}{4}+1\leq\frac{p^{2}+1}{2}i+p^{2}j\leq\frac{(p^{2}-1)^{2}}{4}$
and our goal is to determine for which $(i,j)$ there is a solution $(h,k)$ of
the system mod $p$

$\Bigg{\\{}\begin{array}[]{c}h-k+j\equiv 0,\\\
-\frac{h}{2}-\frac{1}{2}+i\equiv p-1.\\\ \end{array}$

Take $l,m\in Z^{+}_{0}$ so that

$\Bigg{\\{}\begin{array}[]{c}j=lp+k-h,\\\ i=mp+p+\frac{h}{2}-\dfrac{1}{2}.\\\
\end{array}$

In this situation, $i<\frac{2g}{p^{2}+1}=\frac{(p^{2}-1)^{2}}{2(p^{2}+1)}$, so
$mp+p+\frac{h}{2}-\frac{1}{2}\leq\frac{(p^{2}-1)^{2}}{2(p^{2}+1)}$. Then
$m\leq\frac{(p^{2}-1)^{2}}{2(p^{2}+1)}$. And $j<\frac{(p^{2}-1)^{2}}{4p^{2}}$,
so $lp+k-h<\frac{(p^{2}-1)^{2}}{4p^{2}}$, Then
$l<\frac{(p^{2}-1)^{2}}{4p^{2}}$. From this we can say that
$\frac{p^{2}-1}{4}-1\leq l\leq\frac{p^{2}-1}{4}$, and $\frac{p^{2}-1}{2}-1\leq
m\leq\frac{p^{2}-1}{2}$. In this way, $\dfrac{(p^{2}-1)^{2}}{8}$ suitable
values for $(i,j)$ are obtained, whence
$\mbox{rank}(A_{2})=\dfrac{(p^{2}-1)^{2}}{8}$.

For $s\geq 6$, $\mbox{rank}(A_{s})$ equals $\mbox{rank}(A_{s-1})$ plus the
number of pairs $(i,j)$ with
$\frac{(\sqrt{p^{s-2}}-1)^{2}}{4}+1\leq\frac{\sqrt{q}+1}{2}i+\sqrt{q}j\leq\frac{(\sqrt{p^{s}}-1)^{2}}{4}$
such that the system mod $p$

$\Bigg{\\{}\begin{array}[]{c}h-k+j\equiv 0,\\\
-\frac{h}{2}-\frac{1}{2}+i\equiv p-1,\\\ \end{array}$

has a solution. With our usual conventions on $l,m$, a computation shows that
such pairs $(i,j)$ are obtained for $0\leq
l\leq\frac{(\sqrt{p^{s}}-1)^{2}}{4\sqrt{p^{s+2}}}$ from this we have
$\frac{\sqrt{p^{s-4}}(p^{2}-1)}{4}-1\leq
l\leq\frac{\sqrt{p^{s-4}}(p^{2}-1)}{4}$, and $0\leq
m\leq\frac{(\sqrt{p^{s}}-1)^{2}}{2(\sqrt{p^{s}}+1)}$ from this we have
$\frac{(\sqrt{p^{s-2}}-1)(p+1)}{2}-1\leq
m\leq\frac{(\sqrt{p^{s-2}}-1)(p+1)}{2}$. In this case, we have

$\frac{(\sqrt{p^{s-2}}-1)(p+1)\sqrt{p^{s-4}}(p^{2}-1)}{8}$

choices for $(h,k)$. Therefore, we get

$\mbox{rank}(A_{s})=\mbox{rank}(A_{s-1})+\frac{(\sqrt{p^{s-2}}-1)(p+1)\sqrt{p^{s-4}}(p^{2}-1)}{8}.$

Now, our claim on the rank of $A_{s}$ follows by induction on $s$. Hence

$\begin{array}[]{ccccccc}a(\mathcal{X})&=&\dfrac{(\sqrt{p^{s}}-1)^{2}}{4}-\dfrac{(p+1)(\sqrt{p^{s}}-1)(\sqrt{p^{s-2}}-1)}{8}\\\
&=&\dfrac{(\sqrt{p^{s}}-1)}{8}(\sqrt{p^{s}}+p-\sqrt{p^{s-2}}-1)\\\
&=&\dfrac{(\sqrt{p^{s}}-1)}{8}(p(\sqrt{p^{s-2}}+1)-(\sqrt{p^{s-2}}+1))\\\
&=&\dfrac{(\sqrt{p^{s}}-1)}{8}((\sqrt{p^{s-2}}+1)(p-1))\\\
&=&\dfrac{(p-1)}{8}((\sqrt{p^{s-2}}+1)(\sqrt{q}-1)).\\\ \end{array}$

∎

For the finite field $\mathbb{F}_{q^{2}}$, consider the following curve
written in affine form:

$\mathcal{A}_{t}=v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{t}}),$

where $\sqrt{q}\equiv-1$ mod $t$. From this, we are led to the following
problem.

######  Problem 3.3.

What is the dimension of the space of exact holomorphic differentials of
$\mathcal{A}_{t}$?

######  Example 3.4.

Consider the curve $\mathcal{X}$ with function field $K(x,y)$ given by

$y+y^{5}=x^{3},$

where $p=5$ and $s=2$. It is easily seen that a basis for
$H^{0}(\mathcal{X},\Omega^{1})$ is given by

$\mathcal{B}=\\{dx,xdx,x^{2}dx,ydx\\}.$

Let us compute the image of $\mathfrak{C}(\omega)$ for any
$\omega\in\mathcal{B}$. It is straightforward, and from Remark 2.3, we see
that

$\mathfrak{C}(dx)=\mathfrak{C}(xdx)=\mathfrak{C}(x^{2}dx)=0.$

Also,

$\mathfrak{C}(ydx)=\mathfrak{C}((x^{3}-y^{5})dx)=0.$

Hence, $a(\mathcal{X})=4$.

### Certain Generalization of Hermitian Curves.

Let $\mathcal{Y}$ be the non-singular model over $\mathbb{F}_{q}$ of the plane
curve given by the following equation

$y^{m}=x+x^{\ell}+\cdots+x^{\ell^{2r-1}},$

where $q=\ell^{2r}$ and be a square, and $m\geq 2$ be an integer such that
$m\mid(\ell^{r}+1)$.

There is just one point $Q$ over $x=\infty$, since $\mbox{gcd}(m,\ell)=1$. The
curve $\mathcal{Y}$ is the Hermitian curve if $r=1$ and $m=\ell+1$, see [25].

Directly from the Riemann-Hurwitz genus formula, we have genius $\mathcal{Y}$
is $g=\frac{(m-1)(\ell^{2r-1}-1)}{2}$. Also, the number of rational points of
$\mathcal{Y}$ is
$\sharp\mathcal{Y}(\mathbb{F}_{q})=1+\ell^{2r-1}+m(\ell-1)\ell^{2r-1}$. In
particular, $\mathcal{Y}$ is maximal if and only if $r=1$.

### Conjecture.

Let $m=2$ and $r\geq 2$, then the dimension of the space
$H^{0}(\mathcal{Y},\Omega^{1})$, of exact holomorphic differentials on
$\mathcal{Y}$, is

$a(\mathcal{Y})=\dfrac{(\ell^{2r-2}+1)(p-1)}{4}.$

##  4\. Application in coding theory

In a finite field, the $a$-number of a curve $\mathcal{X}$ corresponds to the
size of the space containing exact holomorphic differentials on $\mathcal{X}$.
The Cartier operator is utilized in the construction of algebraic geometry
codes that establish a connection between this invariant and coding theory.

More precisely, the $a$-number establishes a minimum distance requirement for
codes that are constructed in this manner. A greater $a$-number is preferable
because codes with greater minimum distances are capable of rectifying a
greater number of errors. By demonstrating how the $a$-number of the Artin-
Schreier curve $\mathcal{X}$ provides a minimum distance $\delta$ constraint
on codes $C(D,G)$ derived from $\mathcal{X}$ utilizing the Cartier operator
and divisors $D$ and $G$, the following theorem establishes this relationship
explicitly.

The relationship between the $a$-number and minimal code distance is
quantified in the theorem. This illustrates how the $a$-number, which is
derived from algebraic characteristics of the curve, can also impact the
error-correcting capability of codes that are supported by the curve.

Please read my paper on algebraic geometry codes [21] in order to comprehend
the following theorem:

######  Theorem 4.1.

Let $X$ be the Artin-Schreier curve over $\mathbb{F}{q^{2}}$ defined by
$y^{\sqrt{q}}+y=x^{\frac{\sqrt{q}+1}{2}}$ where $q=p^{s}$. Let $G=mP\infty$ be
a divisor on $X$, where $P_{\infty}$ is the point at infinity. Let $C(D,G)$ be
the algebraic geometry code constructed using the Cartier operator, with $D$ a
rational divisor of degree $d$. Then the minimum distance $\delta$ of $C(D,G)$
satisfies:

$\delta\geq d-m-2g+a(X)$

where $a(X)$ is the $a$-number of $X$.

###### Proof.

By the Cartier operator code construction, if $\omega_{1},...,\omega_{n}$ is a
basis for $H^{0}(X,\Omega^{1})$ then $\eta_{i}=C(\omega_{i})$ is a basis for
the exact differentials. Let $\nu_{1},...,\nu_{a(X)}$ be a basis for the
kernel of $C$, which has dimension $a(X)$.

We have $l(G)\geq m+1-g$ from the Riemann-Roch theorem. If $f\in L(G)$, write:

$f=a_{1}\eta_{1}+...+a_{n}\eta_{n}+b_{1}\nu_{1}+...+b_{a(X)}\nu_{a(X)}$

Applying $C$, since it’s $0$ on $\text{ker}(C)$, we get:
$C(f)=a_{1}\eta_{1}+...+a_{n}\eta_{n}$. And,
$\text{deg}(C(f))\leq\text{deg}(D)-m-a(X)$. It’s known
$\delta\geq\text{deg}(D)-\text{deg}(C(f))$. Thus $\delta\geq d-m-2g+a(X)$

So the $a$-number directly gives a lower bound on the minimum distance
$\delta$. A higher $a$-number leads to a larger minimum distance. ∎

### Acknowledgements.

This paper was written while Vahid Nourozi was visiting Unicamp (Universidade
Estadual de Campinas), supported by TWAS/Cnpq (Brazil) with fellowship number
$314966/2018-8$.

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# Asymptotic analysis of a family of Sobolev orthogonal polynomials related to
the generalized Charlier polynomials

Diego Dominici
Johannes Kepler University
Research Institute for Symbolic Computation (RISC)
Altenberger Straße 69, 4040 Linz, Austria e-mail<EMAIL_ADDRESS>Juan
José Moreno Balcázar
Departamento de Matemáticas e
Instituto Carlos I de Física Teórica y Computacional
Universidad de Almería
La Cañada de San Urbano s/n
04120 Almería
Spain e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper we tackle the asymptotic behaviour of a family of orthogonal
polynomials with respect to a nonstandard inner product involving the forward
operator $\Delta$. Concretely, we treat the generalized Charlier weights in
the framework of $\Delta$–Sobolev orthogonality. We obtain an asymptotic
expansion for this orthogonal polynomials where the falling factorial
polynomials play an important role.

##  1 Introduction

Let $\mathbb{N}_{0}$ be the set of nonnegative integers

$\mathbb{N}_{0}=\mathbb{N}\cup\left\\{0\right\\}=\left\\{0,1,2,\ldots\right\\}.$

If $\mathfrak{L}:\mathbb{C}\left[x\right]\rightarrow\mathbb{C}$ is a linear
functional, we say that a sequence $\left\\{p_{n}\right\\}_{n\geq 0},$
$\deg\left(p_{n}\right)=n,$ is an _orthogonal polynomial sequence_ with
respect to $\mathfrak{L}$ if

$\mathfrak{L}\left[p_{k}p_{n}\right]=h_{n}\delta_{k,n},\quad
k,n\in\mathbb{N}_{0},\quad h_{n}\neq 0,$ (1)

where $\delta_{k,n}$ denotes the Kronecker delta. If $h_{n}=1,$ then
$\left\\{p_{n}\right\\}_{n\geq 0}$ is said to be an _orthonormal polynomial
sequence_. We denote by $\\{\mu_{n}\\}_{n}\geq 0$ the _moment sequence_ of the
functional $\mathfrak{L}$ on the monomial basis,

$\mu_{n}=\mathfrak{L}[x^{n}],\quad n\in\mathbb{N}_{0}.$

Let $\left\\{\mathrm{p}_{n}\right\\}_{n\geq 0}$ be the sequence of monic
polynomials, orthogonal with respect to $\mathfrak{L}$. From (1), we see that

$\mathfrak{L}\left[x\mathrm{p}_{n}\mathrm{p}_{k}\right]=0,\quad k\neq n,n\pm
1,$

and therefore the polynomials $\mathrm{p}_{n}\left(x\right)$ satisfy the
_three-term recurrence relation_

$x\mathrm{p}_{n}=\mathrm{p}_{n+1}+\beta_{n}\mathrm{p}_{n}+\gamma_{n}\mathrm{p}_{n-1},\quad
n\in\mathbb{N}_{0},$ (2)

with initial values $\mathrm{p}_{0}(x)=1,$ $\mathrm{p}_{1}(x)=x-\beta_{0}.$
Using (1), the coefficients $\beta_{n},\gamma_{n}$ are given by

$\beta_{n}=\frac{\mathfrak{L}\left[x\mathrm{p}_{n}^{2}\right]}{h_{n}},\quad\gamma_{n}=\frac{\mathfrak{L}\left[x\mathrm{p}_{n}\mathrm{p}_{n-1}\right]}{h_{n-1}},\quad
n\in\mathbb{N},$ (3)

with initial values

$\beta_{0}=\frac{\mu_{1}}{\mu_{0}},\quad\gamma_{0}=0.$ (4)

Note that (again using again (1)), we have

$h_{n}=\mathfrak{L}\left[x^{n}\mathrm{p}_{n}\right]=\mathfrak{L}\left[x\mathrm{p}_{n}\mathrm{p}_{n-1}\right]=\gamma_{n}h_{n-1},\quad
n\in\mathbb{N},$

and therefore

$\gamma_{n}=\frac{h_{n}}{h_{n-1}},\quad n\in\mathbb{N}.$ (5)

The monic _Generalized Charlier polynomials_ , $P_{n}\left(x;z\right)$ are
orthogonal with respect to the linear functional

$L\left[p\right]={\displaystyle\sum\limits_{x=0}^{\infty}}\frac{p\left(x\right)}{\left(b+1\right)_{x}}\frac{z^{x}}{x!},\quad
p\in\mathbb{C}\left[x\right],\quad z>0,$ (6)

where the _Pochhammer symbol_ is defined by [17, 5.2.4]

$\left(c\right)_{n}={\displaystyle\prod\limits_{j=0}^{n-1}}\left(c+j\right),\quad
n\in\mathbb{N},\quad\left(c\right)_{0}=1.$ (7)

In [13] Hounkonnou, Hounga, and Ronveaux studied the orthogonal polynomials
associated with the linear functional

$L_{r}\left[p\right]={\displaystyle\sum\limits_{x=0}^{\infty}}p\left(x\right)\frac{z^{x}}{\left(x!\right)^{r}},\quad
p\in\mathbb{C}\left[x\right],\quad r\in\mathbb{N}.$ (8)

When $r=2,$ they derived nonlinear recurrences (known as the _Laguerre-Freud
equations_) for the recurrence coefficients, and a second-order difference
equation for the orthogonal polynomials associated with $L_{r}$. Note that the
case $r=2$ is a particular example of (6) with $b=0.$

In [19] Van Assche and Foupouagnigni also considered (8) with $r=2.$ They
simplified the Laguerre-Freud equations obtained in [13], and obtained

$u_{n+1}+u_{n-1}=\frac{1}{\sqrt{z}}\frac{nu_{n}}{1-u_{n}^{2}},\quad
v_{n}=\sqrt{z}u_{n+1}u_{n},$

with $\gamma_{n}=z\left(1-u_{n}^{2}\right)$ and $\beta_{n}=v_{n}+n$. They
showed that these equations are related to the discrete Painlevé II equation
dPII. In [18], Smet and Van Assche studied the orthogonal polynomials
associated with (6). They obtained the Laguerre-Freud equations

$\displaystyle\left(\gamma_{n+1}-z\right)\left(\gamma_{n}-z\right)$
$\displaystyle=z\left(\beta_{n}-n\right)\left(\beta_{n}-n+b\right),$ (9)
$\displaystyle\beta_{n}+\beta_{n-1}$
$\displaystyle=n-1-b+\frac{nz}{\gamma_{n}},$

and showed that these equations are a limiting case of the discrete Painlevé
IV equation dPIV.

We are interested in an inner product in the framework of Sobolev-type
orthogonality. Concretely, a $\Delta$–Sobolev inner product involving the
linear operator $L$ given in (6), i.e.

$\left\langle p,q\right\rangle=L\left[pq\right]+\lambda L\left[\Delta p\Delta
q\right],\quad p,q\in\mathbb{C}\left[x\right],$ (10)

where $\lambda\in\mathbb{C}$ and the _finite difference operators_ (in $x)$
$\Delta,\nabla$ are defined by

$\Delta\left[p\right]=p\left(x+1\right)-p\left(x\right),\quad\nabla\left[p\right]=p(x)-p\left(x-1\right).$

We will denote by $S_{n}\left(x;\lambda,z\right)$ the monic polynomials
orthogonal with respect to the inner product (10).

The study of Sobolev orthogonality, and corresponding orthogonal polynomials,
is a relatively recent topic in the theory of orthogonal polynomials. The
first seminal paper was written by Lewis in 1947 (see [15]) and other
foundational articles were written in the sixties and seventies of the last
century. However, the eclosion of investigations about this topic took place
in the nineties. Sobolev orthogonal polynomials are attractive because they
are not orthogonal in a standard way. For this reason nice properties of
standard orthogonal polynomials such as the three-term recurrence relation,
Christoffel–Darboux formula, etc. are lost. Therefore, it was necessary to
construct a new (unfinished) theory. Originally, the Sobolev inner products
involved the derivative operator. But, there is no reason to consider other
operators. In this paper, as we have mentioned previously, we consider a
Sobolev inner product involving the forward difference operator $\Delta$, the
so called $\Delta$–Sobolev orthogonality in some papers (see, for example,
[1], [2], [3], [16]).

The paper is organized as follows: in Section 2 we introduce some basic facts
which are useful to establish the main result in this paper. The
$\Delta$–Charlier–Sobolev inner product is introduced in Section 3, where we
obtain some properties of the corresponding orthogonal polynomials which allow
us to obtain an asymptotic expansion for $S_{n}\left(x;\lambda,z\right).$

##  2 Preliminary material

In this section, we review some of material that we will need in the rest of
the paper.

###### Lemma 1

If

$\phi\left(x\right)=x\left(x+b\right),\quad\psi\left(x\right)=z,$ (11)

then the functional (6) satisfies the _Pearson equation_

$L\left[\psi\mathfrak{S}p\right]=L\left[\phi p\right],\quad
p\in\mathbb{C}\left[x\right],$ (12)

where $\mathfrak{S}$ denotes the shift operator (in $x)$

$\mathfrak{S}\left[p\right]=p\left(x+1\right).$

Proof. We see from (6) that

$\displaystyle{\displaystyle\sum\limits_{x=0}^{\infty}}\frac{zp\left(x+1\right)}{\left(b+1\right)_{x}}\frac{z^{x}}{x!}={\displaystyle\sum\limits_{x=1}^{\infty}}\frac{p\left(x\right)}{\left(b+1\right)_{x-1}}\frac{z^{x}}{\left(x-1\right)!}$
$\displaystyle={\displaystyle\sum\limits_{x=1}^{\infty}}\frac{x\left(x+b\right)p\left(x\right)}{\left(b+1\right)_{x}}\frac{z^{x}}{x!}={\displaystyle\sum\limits_{x=0}^{\infty}}\frac{x\left(x+b\right)p\left(x\right)}{\left(b+1\right)_{x}}\frac{z^{x}}{x!},$

and (12) follows.

In general, we say that a functional $L$ satisfying the Pearson equation (12)
where $\phi\left(x\right),\psi\left(x\right)$ are fixed polynomials is
_discrete semiclassical_. Note that we can also write (12) as

$L\left[\psi\Delta p\right]=L\left[\left(\phi-\psi\right)p\right],\quad
p\in\mathbb{C}\left[x\right].$

The _class_ of the functional is defined by

$s=\max\left\\{\deg\left(\phi-\psi\right)-1,\deg\left(\phi\right)-2\right\\},$
(13)

and semiclassical functional of class __ $s=0$ are called _classical_ [12].
Note that from (11) and (13) it follows that the generalized Charlier
polynomials are discrete semiclassical of class $s=1.$ In [10], we classified
the discrete semiclassical orthogonal polynomials of class $s\leq 1$ and in
[11] we extended our results to $s\leq 2.$

###### Proposition 2

Let $\mathrm{p}_{n}\left(x\right)$ be the sequence of monic polynomials
orthogonal with respect to a linear functional $L$ satisfying the Pearson
equation (12) with $\deg\left(\phi\right)=r,\deg\left(\psi\right)=t$.

(i) The polynomials $\mathrm{p}_{n}\left(x\right)$ satisfy the structure
equation

$\psi\left(x\right)\mathrm{p}_{n}\left(x+1\right)={\displaystyle\sum\limits_{k=-r}^{t}}A_{k}\left(n\right)\mathrm{p}_{n+k}\left(x\right),$
(14)

where the coefficients $A_{k}\left(n\right)$ are solutions of the recurrence
equation

$\displaystyle\gamma_{n+k+1}A_{k+1}\left(n\right)-\gamma_{n}A_{k+1}\left(n-1\right)+A_{k-1}\left(n\right)-A_{k-1}\left(n+1\right)$
(15) $\displaystyle=\left(\beta_{n}-\beta_{n+k}-1\right)A_{k}\left(n\right),$

with

$A_{t}\left(n\right)=z,\quad
A_{-r}\left(n\right)=\gamma_{n}\gamma_{n-1}\cdots\gamma_{n-r+1}.$ (16)

and $A_{k}\left(n\right)=0,$ $k\notin\left[-r,t\right]$.

(ii) The generalized Charlier polynomials $P_{n}\left(x;z\right)$ satisfy

$\Delta P_{n}=nP_{n-1}+\xi_{n}P_{n-2},\quad n\in\mathbb{N}_{0},$ (17)

where

$\xi_{n}=\frac{\gamma_{n}\gamma_{n-1}}{z},\quad n\in\mathbb{N}_{0}.$ (18)

Proof. (i) See [5].

(ii) If $\phi\left(x\right),\psi\left(x\right)$ are given by (11), then
$t=0,r=2$ and therefore

$A_{0}\left(n\right)=z,\quad A_{-2}\left(n\right)=\gamma_{n}\gamma_{n-1}.$
(19)

Setting $k=0$ in (15), we get

$A_{-1}\left(n+1\right)-A_{-1}\left(n\right)=z,$

and we conclude that

$A_{-1}\left(n\right)=nz.$ (20)

Using (19) and (20) in (14) we obtain

$zP_{n}\left(x+1\right)=zP_{n}\left(x\right)+nzP_{n-1}\left(x\right)+\gamma_{n}\gamma_{n-1}P_{n-2}\left(x\right),$

and (17) follows.

###### Remark 3

If $k=-1,-2$ then (15) gives

$\displaystyle\gamma_{n}\left(\gamma_{n-1}-\gamma_{n+1}\right)$
$\displaystyle=nz\left(\beta_{n}-\beta_{n-1}-1\right),$ $\displaystyle
nz\gamma_{n-1}-\left(n-1\right)z\gamma_{n}$
$\displaystyle=\gamma_{n}\gamma_{n-1}\left(\beta_{n}-\beta_{n-2}-1\right),$

from which the Laguerre-Freud equations (9) can be derived (see [18], equation
2.14 and beyond).

Equation (17) was derived in [18] using the method presented in [14]. For a
different approach using infinite matrices, see [6].

Let $\varphi_{n}\left(x\right)$ denote the _falling factorial polynomials_
defined by $\varphi_{0}\left(x\right)=1$ and

$\varphi_{n}\left(x\right)={\displaystyle\prod\limits_{k=0}^{n-1}}\left(x-k\right),\quad
n\in\mathbb{N}.$ (21)

Note that we can write

$\varphi_{n}\left(x\right)=\frac{\Gamma\left(x+1\right)}{\Gamma\left(x-n+1\right)}=n!\binom{x}{n},\quad
n\in\mathbb{N}_{0},$ (22)

where $\Gamma$ denotes the gamma function [17, 5.2.1].

###### Proposition 4

The moments of the functional $L$ on the basis $\varphi_{n}\left(x\right)$ are
given by

$\nu_{n}\left(z\right)=L\left[\varphi_{n}\right]=\frac{z^{n}}{\left(b+1\right)_{n}}\
_{0}F_{1}\left(\begin{array}[c]{c}-\\\ b+n+1\end{array};z\right),$ (23)

where$\ {}_{p}F_{q}$ is the generalized hypergeometric function [17, 16.2.1].

Proof. Using (6) and (22), we have

$L\left[\varphi_{n}\right]=\
{\displaystyle\sum\limits_{x=n}^{\infty}}\frac{1}{\left(b+1\right)_{x}}\frac{z^{x}}{\left(x-n\right)!}={\displaystyle\sum\limits_{x=0}^{\infty}}\frac{1}{\left(b+1\right)_{x+n}}\frac{z^{x+n}}{x!},$

and since

$\left(c\right)_{n+m}=\left(c\right)_{n}\left(c+n\right)_{m},$

we obtain

$L\left[\varphi_{n}\right]={\displaystyle\sum\limits_{x=0}^{\infty}}\frac{1}{\left(b+1\right)_{n}\left(b+n+1\right)_{x}}\frac{z^{x+n}}{x!},$

and (23) follows.

###### Lemma 5

The polynomials $\varphi_{n}\left(x\right)$ satisfy the connection (or
linearization) formula

$\varphi_{n}\left(x\right)\varphi_{m}\left(x\right)={\displaystyle\sum\limits_{k=0}^{\min\left\\{n,m\right\\}}}\binom{n}{k}\binom{m}{k}k!\varphi_{n+m-k}\left(x\right).$
(24)

Proof. From the definition of $\varphi_{n}\left(x\right)$, we see that

$\varphi_{n+m}\left(x\right)=\varphi_{n}\left(x\right)\varphi_{m}\left(x-n\right).$
(25)

Suppose that $m\leq n.$ Using (25), we have

${\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{m}{k}k!\varphi_{n+m-k}\left(x\right)={\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{m}{k}k!\varphi_{n}\left(x\right)\varphi_{m-k}\left(x-n\right),$

and using (22), we can write

${\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{m}{k}k!\varphi_{m-k}\left(x-n\right)=m!{\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{x-n}{m-k}.$

Using the Chu–Vandermonde identity, we have

${\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{x-n}{m-k}=\binom{x}{m},$

and therefore

${\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{m}{k}k!\varphi_{m-k}\left(x-n\right)=\varphi_{m}\left(x\right).$

###### Corollary 6

For all $m,n\in\mathbb{N}_{0},$ $m\leq n,$ we have

$L\left[\varphi_{n}\varphi_{m}\right]=\binom{n}{m}\frac{m!}{\left(b+1\right)_{n}}z^{n}\left[1+\frac{n+1}{\left(n-m+1\right)\left(n+b+1\right)}z+O\left(z^{2}\right)\right],\quad$
(26)

as $z\rightarrow 0.$

Proof. Using (24), we get

$L\left[\varphi_{n}\varphi_{m}\right]={\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{m}{k}k!\nu_{n+m-k}\left(z\right),$

and (23) gives

$\displaystyle
L\left[\varphi_{n}\varphi_{m}\right]={\displaystyle\sum\limits_{k=0}^{m}}\binom{n}{k}\binom{m}{k}k!\frac{z^{n+m-k}}{\left(b+1\right)_{n+m-k}}\left(1+\frac{z}{n+m-k+b+1}+\cdots\right)$
$\displaystyle=\binom{n}{m}\frac{m!}{\left(b+1\right)_{n}}z^{n}+\binom{n+1}{m}\frac{m!}{\left(b+1\right)_{n+1}}z^{n+1}+O\left(z^{n+2}\right),\quad
z\rightarrow 0.$

##  3 Sobolev polynomials

Let $S_{n}\left(x;\lambda,z\right)$ be the monic polynomials orthogonal with
respect to the inner product (10). Introducing the sequences

$\mu_{i,j}\left(\lambda,z\right)=\left\langle\varphi_{i},\varphi_{j}\right\rangle,\quad\nu_{i,j}\left(z\right)=L\left[\varphi_{i}\varphi_{j}\right],$

and using the identity

$\Delta\varphi_{n}=n\varphi_{n-1},\quad n\in\mathbb{N}_{0},$

we have

$\mu_{i,j}=\left\langle\varphi_{i},\varphi_{j}\right\rangle=L\left[\varphi_{i}\varphi_{j}\right]+\lambda
L\left[i\varphi_{i-1}j\varphi_{j-1}\right]=\nu_{i,j}+\lambda ij\nu_{i-1,j-1},$
(27)

for all $i,j\in\mathbb{N}_{0}.$ Using (26) in (27), we have

$\mu_{i,j}\left(z\right)=\frac{\lambda
j}{\left(i-j\right)!}i!\frac{z^{i-1}}{\left(b+1\right)_{i-1}}+\frac{\left(\lambda
i-1\right)j+i+1}{\left(i+1-j\right)!}i!\frac{z^{i}}{\left(b+1\right)_{i}}+O\left(z^{i+1}\right)$
(28)

as $z\rightarrow 0,$ with $j\leq i.$

In [7], we obtained power series solutions for the determinant of a square
matrix whose entries are power series in $z.$ Using (26) and (28), we see that

$H_{n}\left(z\right)\sim
z^{\binom{n}{2}}{\displaystyle\prod\limits_{k=1}^{n-1}}\frac{k!}{\left(b+1\right)_{k}},$

and

$\widetilde{H}_{n}\left(\lambda,z\right)\sim\lambda^{n-1}z^{\binom{n-1}{2}}{\displaystyle\prod\limits_{k=1}^{n-2}}\frac{\left(k+1\right)\left(k+1\right)!}{\left(b+1\right)_{k}},$

as $z\rightarrow 0,$ where $H_{0}=\widetilde{H}_{0}=1$ and

$H_{n}\left(z\right)=\underset{0\leq i,j\leq
n-1}{\det}\left(\nu_{i,j}\right),\quad\widetilde{H}_{n}\left(\lambda,z\right)=\underset{0\leq
i,j\leq n-1}{\det}\left(\mu_{i,j}\right),\quad n\in\mathbb{N}.$

Since the determinants $H_{n}\left(z\right)$ and the norms of the polynomials
are related by (see [4], Theorem 3.2)

$h_{n}\left(z\right)=\frac{H_{n+1}\left(z\right)}{H_{n}\left(z\right)},$

we get

$h_{n}\left(z\right)=\frac{n!}{\left(b+1\right)_{n}}z^{n}+O\left(z^{n+1}\right),\quad
n\in\mathbb{N}_{0},$ (29)

as $z\rightarrow 0.$ Similarly, for all $n\in\mathbb{N}$

$\widetilde{h}_{n}\left(\lambda,z\right)=\lambda\frac{nn!z^{n-1}}{\left(b+1\right)_{n-1}}+\frac{n!\left(n+b-1+b\lambda
n\right)}{\left(n+b-1\right)\left(b+1\right)_{n}}z^{n}+O\left(z^{n}\right),$
(30)

as $z\rightarrow 0,$ where

$\left\langle
S_{n},S_{n}\right\rangle=\widetilde{h}_{n}\left(\lambda,z\right)=\frac{\widetilde{H}_{n+1}\left(\lambda,z\right)}{\widetilde{H}_{n}\left(\lambda,z\right)}.$
(31)

The polynomials $S_{n}\left(x;\lambda,z\right)$ and $P_{n}\left(x;z\right)$
are related by the following expression.

###### Theorem 7

We have

$P_{n}\left(x;z\right)=S_{n}\left(x;\lambda,z\right)+a_{n}\left(\lambda,z\right)S_{n-1}\left(x;\lambda,z\right),\quad
n\in\mathbb{N},$ (32)

where

$a_{n}\left(\lambda,z\right)=\frac{\left(n-1\right)\lambda}{z}\frac{h_{n}\left(z\right)}{\widetilde{h}_{n-1}\left(z,\lambda\right)},\quad
n\in\mathbb{N}.$ (33)

Proof. Since the polynomials $S_{n}\left(x;\lambda,z\right)$ are a basis of
$\mathbb{C}\left[x\right]$ and
$P_{n}\left(x;z\right),S_{n}\left(x;\lambda,z\right)$ are monic, it follows
that

$P_{n}=S_{n}+{\displaystyle\sum\limits_{k=0}^{n-1}}c_{n,k}S_{k}.$

Using orthogonality, we have

$c_{n,k}=\frac{\left\langle P_{n},S_{k}\right\rangle}{\widetilde{h}_{k}},$

and using (10) we get

$\widetilde{h}_{k}c_{n,k}=L\left[P_{n}S_{k}\right]+\lambda L\left[\Delta
P_{n}\Delta S_{k}\right].$ (34)

Using (17) in (34), we obtain

$\widetilde{h}_{k}c_{n,k}=L\left[P_{n}S_{k}\right]+\lambda
nL\left[P_{n-1}\Delta S_{k}\right]+\lambda\xi_{n}L\left[P_{n-2}\Delta
S_{k}\right]=0,\quad 0\leq k\leq n-2,$

and therefore the only nonzero coefficient is

$c_{n,n-1}=\lambda\frac{\xi_{n}}{\widetilde{h}_{n-1}}L\left[P_{n-2}\Delta
S_{n-1}\right].$

But since

$\Delta
S_{n-1}=\left(n-1\right)x^{n-2}+O\left(x^{n-3}\right)=\left(n-1\right)P_{n-2}+O\left(x^{n-3}\right),$

we see that

$L\left[P_{n-2}\Delta S_{n-1}\right]=\left(n-1\right)h_{n-2}.$

Finally, we can use (5) and (18) to obtain

$\xi_{n}h_{n-2}=\frac{\gamma_{n}\gamma_{n-1}}{z}h_{n-2}=\frac{h_{n}}{z}.$ (35)

Thus, we conclude that

$c_{n,n-1}=\left(n-1\right)\lambda\frac{\xi_{n}}{\widetilde{h}_{n-1}}h_{n-2}=\frac{\left(n-1\right)\lambda}{z}\frac{h_{n}}{\widetilde{h}_{n-1}}.$

###### Remark 8

If we use (30)-(29) in (33), we get

$a_{n}\left(\lambda,z\right)=\frac{nz}{\left(n+b\right)\left(n+b-1\right)}+O\left(z^{2}\right),\quad
z\rightarrow 0,\quad n\geq 2.$ (36)

Next, we shall find a recurrence for the Sobolev norms
$\widetilde{h}_{n}\left(\lambda,z\right).$

###### Theorem 9

For all $n\in\mathbb{N},$ the functions
$\widetilde{h}_{n}\left(\lambda,z\right)$ defined by (31) satisfy the
nonlinear recurrence

$\widetilde{h}_{n}=\lambda
n^{2}h_{n-1}+\left(1+\lambda\frac{\gamma_{n}\gamma_{n-1}}{z^{2}}\right)h_{n}-\left(n-1\right)^{2}\frac{\lambda^{2}}{z^{2}}\frac{h_{n}^{2}}{\widetilde{h}_{n-1}}.$
(37)

Proof. Using (10) and (31) we get

$\widetilde{h}_{n}=\left\langle
S_{n},P_{n}\right\rangle=L\left[S_{n}P_{n}\right]+\lambda L\left[\Delta
S_{n}\Delta P_{n}\right]=h_{n}+\lambda L\left[\Delta S_{n}\Delta
P_{n}\right].$

But from (32) we have

$L\left[\Delta S_{n}\Delta P_{n}\right]=L\left[\left(\Delta
P_{n}\right)^{2}\right]-a_{n}L\left[\Delta S_{n-1}\Delta P_{n}\right],$

while (17) gives

$L\left[\left(\Delta P_{n}\right)^{2}\right]=n^{2}h_{n-1}+\xi_{n}^{2}h_{n-2},$

and

$\displaystyle L\left[\Delta S_{n-1}\Delta P_{n}\right]$
$\displaystyle=nL\left[\Delta S_{n-1}P_{n-1}\right]+\xi_{n}L\left[\Delta
S_{n-1}P_{n-2}\right]$ $\displaystyle=0+\left(n-1\right)\xi_{n}h_{n-2}.$

Hence,

$\widetilde{h}_{n}=h_{n}+\lambda
n^{2}h_{n-1}+\lambda\left[\xi_{n}^{2}-a_{n}\left(n-1\right)\xi_{n}\right]h_{n-2},$

or using (33) and (35), we conclude that

$\widetilde{h}_{n}=h_{n}+\lambda
n^{2}h_{n-1}+\lambda\frac{\gamma_{n}\gamma_{n-1}}{z^{2}}h_{n}-\left(n-1\right)^{2}\frac{\lambda^{2}}{z^{2}}\frac{h_{n}^{2}}{\widetilde{h}_{n-1}}.$

Since $\widetilde{h}_{0}=h_{0}$ we know from (4) that $\gamma_{0}=0,$ we can
use (37) and obtain

$\displaystyle\widetilde{h}_{1}$ $\displaystyle=h_{1}+\lambda h_{0},$
$\displaystyle\widetilde{h}_{2}$
$\displaystyle=h_{2}+\left(4h_{1}+\frac{\gamma_{1}\gamma_{2}h_{2}}{z^{2}}-\frac{\lambda}{z^{2}}\frac{h_{2}^{2}}{h_{1}+\lambda
h_{0}}\right)\lambda.$

Using (33), it follows that

$a_{1}=0,\quad a_{2}=\frac{\lambda h_{2}}{z\left(h_{1}+\lambda h_{0}\right)}.$

###### Remark 10

Note that using (33), we can rewrite (37) as

$\frac{n\lambda}{z}\frac{h_{n+1}}{a_{n+1}}=\lambda
n^{2}h_{n-1}+\left(1+\lambda\frac{\gamma_{n}\gamma_{n-1}}{z^{2}}-\left(n-1\right)\frac{\lambda}{z}a_{n}\right)h_{n},$

or, using (5)

$\frac{n\gamma_{n+1}}{za_{n+1}}=\frac{n^{2}}{\gamma_{n}}+\frac{1}{\lambda}+\frac{\gamma_{n}\gamma_{n-1}}{z^{2}}-\frac{\left(n-1\right)a_{n}}{z},\quad
n\in\mathbb{N}.$ (38)

##  4 Asymptotic analysis

In [8] we shown that the 3-term recurrence coefficients of the generalized
Charlier polynomials have the asymptotic expansions

$\beta_{n}\left(z\right)=n+\frac{bz}{n^{2}}-\frac{b\left(2b+1\right)z}{n^{3}}+O\left(n^{-4}\right),$

and

$\gamma_{n}\left(z\right)=z-zbn^{-1}+zb^{2}n^{-2}-bz\left(2z+b^{2}\right)n^{-3}+O\left(n^{-4}\right),$
(39)

as $n\rightarrow\infty.$ We continued our work in [9], where we obtained
asymptotic expansions for all discrete semiclassical orthogonal polynomials.

###### Theorem 11

Let the functions $a_{n}\left(\lambda,z\right)$ satisfy the nonlinear
recurrence (41), with $a_{n}\left(\lambda,0\right)=0.$ If we write

$a_{n}\left(\lambda,z\right)\sim z{\displaystyle\sum\limits_{k\geq
1}}\alpha_{k}\left(\lambda,z\right)n^{-k},\quad n\rightarrow\infty,$ (40)

then

$\alpha_{1}=1,\quad\alpha_{2}=1-2b,\quad\alpha_{3}=1+3b\left(b-1\right)-\frac{z}{\lambda}.$

Proof. Let’s start by rewriting (38) as

$\left[n^{2}\frac{z}{\gamma_{n}}+\frac{z}{\lambda}+\frac{\gamma_{n}\gamma_{n-1}}{z}-\left(n-1\right)a_{n}\right]a_{n+1}-n\gamma_{n+1}=0,\quad
n\in\mathbb{N},$ (41)

and suppose that

$a_{n}\left(\lambda,z\right)={\displaystyle\sum\limits_{k=-N}^{N}}u_{k}\left(\lambda,z\right)n^{-k}.$
(42)

Using (39) and (42) in (41), we see that as $n\rightarrow\infty$

$u_{k}=0,\quad k\leq-2,\quad u_{-1}\left(u_{-1}-1\right)=0.$

Thus, there are two solutions of (41), one with asymptotic behavior

$a_{n}=n+b+1+\left(b+1+\frac{z}{\lambda}\right)n^{-1}+O\left(n^{-2}\right),\quad
n\rightarrow\infty$

and the other

$a_{n}=zn^{-1}+\left(1-2b\right)zn^{-2}{+\left(1+3b\left(b-1\right)-\frac{z}{\lambda}\right)zn^{-3}}+O\left(n^{-4}\right),\quad
n\rightarrow\infty.$

Since from (36) we know that $a_{n}\left(\lambda,0\right)=0,$ we must choose
the second solution and (40) follows.

###### Remark 12

Using (33) and (40) we deduce

$\frac{h_{n+1}\left(z\right)}{\widetilde{h}_{n}\left(z,\lambda\right)}\sim\frac{z^{2}}{(n+1)n\lambda}+O(n^{-3}).$

In particular,

$\lim_{n\rightarrow\infty}n^{2}\frac{h_{n+1}\left(z\right)}{\widetilde{h}_{n}\left(z,\lambda\right)}=\frac{z^{2}}{\lambda}.$

From (5) and (39), we obtain

$\lim_{n\rightarrow\infty}\frac{\widetilde{h}_{n}\left(z,\lambda\right)}{n^{2}h_{n}\left(z\right)}=\frac{\lambda}{z}.$

The above asymptotic behaviour of the norms can also be obtained from Theorem
9 via Poincaré’s Theorem. That technique has given fruitful results to obtain
asymptotic properties in the context of Sobolev orthogonality.

In [9] we studied the asymptotic behavior of the generalized Charlier
polynomials, and proved the following result.

###### Theorem 13

The generalized Charlier polynomials satisfy

$\frac{P_{n}\left(x;z\right)}{\varphi_{n}\left(x\right)}\sim{\displaystyle\sum\limits_{k\geq
0}}\omega_{k}\left(x;z\right)n^{-k},\quad n\rightarrow\infty,$ (43)

with

$\displaystyle\omega_{0}$
$\displaystyle=1,\quad\omega_{1}=z,\quad\omega_{2}=\left(x+1-b\right)z+\frac{z^{2}}{2},$
$\displaystyle\omega_{3}$
$\displaystyle=\left[\left(x+1\right)\left(x+1-b\right)+b^{2}\right]z+\left[2\left(x+1-b\right)+1\right]\frac{z^{2}}{2}+\frac{z^{3}}{6}.$

We have now all the elements to state our main result.

###### Theorem 14

Suppose that

$\frac{S_{n}\left(x;\lambda,z\right)}{\varphi_{n}\left(x\right)}\sim{\displaystyle\sum\limits_{k\geq
0}}\sigma_{k}\left(x;z\right)n^{-k},\quad n\rightarrow\infty.$ (44)

Then,

$\displaystyle\sigma_{0}$
$\displaystyle=1,\quad\sigma_{1}=z,\quad\sigma_{2}=\omega_{2}+z,$
$\displaystyle\sigma_{3}$
$\displaystyle=\omega_{3}+\left[x+z+\alpha_{2}\right]z,$
$\displaystyle\sigma_{4}$
$\displaystyle=\omega_{4}+\left[x^{2}+\left(z+\alpha_{2}\right)x+z\left(2+\alpha_{2}\right)+\omega_{2}+\alpha_{3}\right]z.$

Proof. Using the binomial theorem in (44), we can see that

$\frac{S_{n-1}}{\varphi_{n-1}\left(x\right)}\sim{\displaystyle\sum\limits_{k\geq
0}}\left[\sum_{j=0}^{k-1}\binom{k-1}{j}\sigma_{j+1}\right]n^{-k},\quad
n\rightarrow\infty.$ (45)

Using the recurrence

$\varphi_{n}\left(x\right)=\left(x-n\right)\varphi_{n-1}\left(x\right)$

in (32), we get

$\frac{\left(x-n\right)\left(P_{n}-S_{n}\right)}{\varphi_{n}\left(x\right)}=a_{n}\frac{S_{n-1}}{\varphi_{n-1}\left(x\right)}.$
(46)

Considering (43), (44) we have

$\frac{\left(x-n\right)\left(P_{n}-S_{n}\right)}{\varphi_{n}\left(x\right)}\sim(\sigma_{0}-\omega_{0})n+\sum_{k\geq
0}\left[x(\omega_{k}-\sigma_{k})-(\omega_{k+1}-\sigma_{k+1})\right]n^{-k},$

and from (40) and (45)

$a_{n}\frac{S_{n-1}}{\varphi_{n-1}\left(x\right)}\sim z\sum_{k\geq
1}\left[\alpha_{k}\sigma_{0}+\sum_{j=1}^{k-1}\alpha_{k-j}\sum_{i=0}^{j-1}\binom{j-1}{i}\sigma_{i+1}\right]n^{-k},$

thus, from (46) we deduce

$\sigma_{0}=\omega_{0}=1,$
$x(\omega_{0}-\sigma_{0})-(\omega_{1}-\sigma_{1})=0\Rightarrow\sigma_{1}=\omega_{1}=z,$
$x(\omega_{k}-\sigma_{k})-(\omega_{k+1}-\sigma_{k+1})=z\left[\alpha_{k}+\sum_{j=1}^{k-1}\alpha_{k-j}\sum_{i=0}^{j-1}\binom{j-1}{i}\sigma_{i+1}\right],\quad
k\geq 1.$

To obtain $\sigma_{k}$ with $k=2,3,4,$ it is enough to particularize the above
expression.

##  5 Acknowledgements

The work of the first author was supported by the strategic program
”Innovatives Ö– 2010 plus” from the Upper Austrian Government, and by the
grant SFB F50 (F5009-N15) from the Austrian Science Foundation (FWF). We thank
Prof. Carsten Schneider for his generous sponsorship.

The first author would also like to thank the Isaac Newton Institute for
Mathematical Sciences, Cambridge, for support and hospitality during the
programme ”Applicable resurgent asymptotics: towards a universal theory”
(ARA2), where work on this paper was completed. This work was supported by
EPSRC grant no EP/R014604/1.

The work of the second author was partially supported by the Ministry of
Science and Innovation of Spain and the European Regional Development Fund
(ERDF) (grant number XXX); by Consejería de Economía, Conocimiento, Empresas y
Universidad de la Junta de Andalucía (grant UAL18-FQM-B025-A); by Research
Group FQM-0229 (belonging to Campus of International Excellence CEIMAR); and
by the research centre CDTIME.

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