# Algebraic $\mathcal{L}_{q}$-norms and complexity-like properties of Jacobi
polynomials-Degree and parameter asymptotics.

Nahual Sobrino  Donostia International Physics Center, Paseo Manuel de
Lardizabal 4, E-20018 San Sebastián, Spain  Nano-Bio Spectroscopy Group and
European Theoretical Spectroscopy Facility (ETSF), Departamento de Polímeros y
Materiales Avanzados: Física, Química y Tecnología, Universidad del País Vasco
UPV/EHU, Avenida de Tolosa 72, E-20018 San Sebastián, Spain  Jesús S. Dehesa
<EMAIL_ADDRESS>Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, Granada 18071, Spain  Departamento de Física Atómica,
Molecular y Nuclear, Universidad de Granada, Granada 18071, Spain

###### Abstract

The Jacobi polynomials $\hat{P}_{n}^{(\alpha,\beta)}(x)$ conform the canonical
family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter
weight function $(1-x)^{\alpha}(1+x)^{\beta},\alpha,\beta>-1,$ on the interval
$[-1,+1]$. The spreading of its associated probability density (i.e., the
Rakhmanov density) over the orthogonality support has been quantified, beyond
the dispersion measures (moments around the origin, variance), by the
algebraic $\mathfrak{L}_{q}$-norms (Shannon and Rényi entropies) and the
monotonic complexity-like measures of Cramér-Rao, Fisher-Shannon and LMC
(López-Ruiz, Mancini and Calbet) types. These quantities, however, have been
often determined in an analytically highbrow, non-handy way; specially when
the degree or the parameters $(\alpha,\beta)$ are large. In this work, we
determine in a simple, compact form the entropic and complexity-like
properties of the Jacobi polynomials in the two extremal situations:
($n\rightarrow\infty$; fixed $\alpha,\beta$) and ($\alpha\rightarrow\infty$;
fixed $n,\beta$). These two asymptotics are relevant per se and because they
control the physical entropy and complexity measures of the high energy
(Rydberg) and high dimensional (pseudoclassical) states of numerous
supersymmetric quantum-mechanical systems.

##  I Introduction

The formalization of the intuitive notion of simplicity/complexity of
probability distributions is a formidable task, not yet well established in
spite of a huge number of efforts in many scientific and technological areas
from quantum chemistry Sen2011 ; Antolin2009 ; Badii1997 ; Shiner1999  and
quantum technologies Nielsen2000 ; Cubitt2014 ; Bruss2019 ; Vitanyi2001 ;
Lloyd2001  to applied mathematics and approximation theory Dehesa2006 ;
Dehesa2015 ; Ghosh2019 ; Rudnicki2016 . To a large extent, this is because of
the great diversity of configurational shapes from perfect order to maximal
randomness (or perfect disorder) which have the bi- and multi-parametric
probability distributions associated to the quantum one- and many-body systems
and the special functions of applied mathematics and mathematical physics
Nikiforov1988 . This enormous amount of geometrical forms cannot be captured
by a single quantity, but it requires a number of measures of intrinsic and
extrinsic characters. The latter ones refer to algorithmic and computational
complexities Goldreich2008 ; Kaltchenko2021  which are closely related to the
time required for a computer to solve a given problem, so that they depend on
the chosen computer. The former ones refer to statistical measures of
complexity, extracted from the density functional theory of electronic systems
Parr1989 , which quantify the degree of structure or pattern of one- and many-
electron systems in terms of the single electron density. The main quantities
of this type are the Cramér-Rao Dehesa2006 ; Antolin2009 , Fisher-Shannon
Vignat2003 ; Angulo2008 ; Romera2004  and LMC (Lópezruiz, Mancini and Calbet)
Lopezruiz1995 ; Catalan2002  complexities and modifications of them Pipek1997
; Lopez2009 ; Lopez2005 ; Romera2008 ; Romera2009 ; Sanchez2014 ; Martin2003 ;
LopezRosa2009 ; Toranzo2017 ; Puertas2017c ; Puertas2017d . Here we will use
the basic density-dependent complexity measures (Cramér-Rao, Fisher-Shannon
and LMC) recently introduced in electronic structure (see e.g. Martin2006 ;
Dehesa2009 ; LopezRosa2009 ; Molina2012  and the reviews Angulo2011 ;
Dehesa2011 ), which are of intrinsic character in the sense that they do not
depend on the context but on the probability density of the system under
consideration. These quantities are given by the product of two spreading
measures of dispersion (variance) and entropic (Fisher information, Shannon
entropy, Rényi entropy) types, so that each measures two configurational
shapes of the system in a simultaneous manner.

The hypergeometric polynomials, $\\{p_{n}(x)\\}$, orthogonal with respect to
the weight function $h(x)$ on the support interval $\Lambda$, are known to
often control the quantum-mechanical wavefunctions of the bound states in
numerous quantum systems Nikiforov1988 ; Bagrov1990 ; Cooper2001 ;
Chaitanya2012 ; Yanez1994 ; Gangopadhyaya2017 . The Hermite, Laguerre and
Jacobi polynomials are the three canonical families of real hypergeometric
orthogonal polynomials (HOPs) Andrews1999 ; Ismail2005 ; Olver2010 ;
Koekoek2010 . Recently the entropy- and complexity-like properties of these
polynomials, which determine their spreading on the orthogonality interval,
have begun to be investigated by means of the entropy- and complexity-like
measures Dehesa2015 ; Dehesa2021 ; Sobrino2021  of the associated Rakhmanov
density $\rho_{n}(x)=p_{n}^{2}(x)\,h(x)$. This normalized-to-unity probability
density function governs the ($n\to+\infty$)-asymptotics of the ratio of two
polynomials with consecutive orders Rakhmanov1977 , and characterizes the
Born’s probability density of the bound stationary states of a great deal of
quantum-mechanical potentials which model numerous atomic and molecular
systems Nikiforov1988 ; Gangopadhyaya2017 ; Ghosh2019 ; Alhaidari2021 ;
Assi2021 ; Nath2021 . The numerical evaluation of the integral functionals
corresponding to the dispersion, entropic and complexity measures of the HOPs
by means of the standard quadratures is not convenient, because the highly
oscillatory nature of the integrand renders Gaussian quadrature ineffective as
the number of quadrature points grows linearly with $n$ and the evaluation of
high-degree polynomials are subject to round-off errors. Indeed, since all the
zeros of $p_{n}$ belong to the interval of orthogonality, the increasing
number of integrable singularities spoil any attempt to achieve reasonable
accuracy even for rather small $n$ Buyarov2004 ; Perrey2009 .

The entropic and complexity-like measures of the three canonical families of
the HOPs have been analytically calculated in terms of the degree and the
parameters which characterize their weight function. The resulting analytical
expressions for the Cramér-Rao complexity of Hermite and Laguerre polynomials
are simple and compact Dehesa2015 , but for the rest of complexity measures of
HOPs this is not at all true because the involved entropic components have a
somewhat highbrow, non-handy form, being mostly useful in an algorithmic sense
only (see the review Dehesa2021 ). This is especially true for the Fisher-
Shannon and LMC complexities of HOPs. However, recently, the Fisher-Shannon
and LMC measures of the Hermite $H_{n}(x)$, Laguerre $L_{n}^{(\alpha)}(x)$ and
Gegenbauer polynomials $C_{n}^{(\alpha)}(x)$ have been determined Sobrino2021
when $n\rightarrow\infty$ and when $\alpha\rightarrow\infty$, obtaining simple
and transparent expressions.

The goal of the present work is the analytical evaluation of the Cramér-Rao,
Fisher-Shannon and LMC complexities of the whole family of Jacobi polynomials
$\hat{P}_{n}^{(\alpha,\beta)}(x)$, with $\alpha,\beta>-1$, in the extreme
situations ($n\rightarrow\infty$; fixed $\alpha,\beta$) and
($\alpha\rightarrow\infty$; fixed $n,\beta$). These polynomials Jacobi1826 ;
Jacobi1859 ; Szego1939 ; Nikiforov1988 ; Olver2010 ; Yang2021  are known to be
orthonormal with respect to the weight function
$h_{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta}$ as

$\int_{-1}^{+1}\hat{P}_{n}^{(\alpha,\beta)}(x)\hat{P}_{m}^{(\alpha,\beta)}(x)\,h_{\alpha,\beta}(x)\,dx=\delta_{mn}.$
(1)

Then, the associated Rakhmanov probability density $\rho_{n}(x)$ is given by

$\rho_{n}(x)=\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{2}\,h_{\alpha,\beta}(x).$
(2)

Moreover we will denote the orthogonal Jacobi polynomials
$P_{n}^{(\alpha,\beta)}(x)=\hat{P}_{n}^{(\alpha,\beta)}(x)\,(\kappa_{n})^{1/2}$,
with the normalization constant given (see e.g. Olver2010 ) by

$\kappa_{n}=\int_{-1}^{+1}|P_{n}^{(\alpha,\beta)}(x)|^{2}\,h_{\alpha,\beta}(x)dx=\frac{2^{\alpha+\beta+1}\Gamma(\alpha+n+1)\Gamma(\beta+n+1)}{n!(\alpha+\beta+2n+1)\Gamma(\alpha+\beta+n+1)}.$
(3)

The special case $\alpha=\beta=\lambda-\frac{1}{2}$ corresponds to the
ultraspherical or Gegenbauer polynomials
$C_{n}^{(\lambda)}(x),\lambda>-\frac{1}{2},$ with slightly different
normalization, and the case $\alpha=\beta=0$ corresponds to the Legendre
polynomials (see e.g. Olver2010 ).

The structure of this paper is as follows. In Sections II, III and IV we
obtain the asymptotic behavior for the Cramér-Rao, Fisher-Shannon and LMC
complexities of the Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x)$ when
($n\rightarrow\infty;\mbox{fixed}\,\alpha,\beta$) and when
($\alpha\rightarrow\infty;\mbox{fixed}\,n,\beta$) in a simple, compact and
transparent form, respectively. Then, some concluding remarks and a few open
related issues are pointed out.

##  II Cramér-Rao complexity of Jacobi polynomials

The Cramér-Rao complexity of the Jacobi polynomials is given by the
corresponding quantity of its associated Rakhmanov density (2), which
quantifies the combined balance of the pointwise probability concentration
over its support interval jointly with the spreading of the probability around
the centroid. It is defined Dembo1991 ; Dehesa2006 ; Antolin2009  by

$\mathcal{C}_{CR}[\hat{P}_{n}^{(\alpha,\beta)}]=F[\hat{P}_{n}^{(\alpha,\beta)}]\times
V[\hat{P}_{n}^{(\alpha,\beta)}],$ (4)

where $F[\hat{P}_{n}^{(\alpha,\beta)}]$ and $V[\hat{P}_{n}^{(\alpha,\beta)}]$
are the Fisher information Fisher1925 ; Frieden2004  and the variance of the
Rakhmanov density (2), which are defined as

$F\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\int_{-1}^{+1}\frac{[\rho^{\prime}_{n}(x)]^{2}}{\rho_{n}(x)}dx,\quad\mbox{and}\quad
V[\hat{P}_{n}^{(\alpha,\beta)}]=\langle x^{2}\rangle-\langle x\rangle^{2},$

respectively, with the expectation value $\langle
x^{k}\rangle=\int_{-1}^{+1}x^{k}\rho_{n}(x)dx$ for $k=1,2$. In this section we
give the explicit expressions of these three spreading quantities and we find
in a simple compact form the values of the Cramér-Rao complexity of the Jacobi
polynomials in the two following asymptotical regimes: ($n\rightarrow\infty$;
fixed $\alpha,\beta$) and ($\alpha\rightarrow\infty$; fixed $n,\beta$).

The particularly elegant algebraic properties of the Jacobi polynomials (see
e.g. Nikiforov1988 ; Olver2010 ; Yang2021 ) have allowed to encounter the
following expression

$\displaystyle
F\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\left\\{\begin{array}[]{ll}2n(n+1)(2n+1),&\alpha,\beta=0,\\\\[8.53581pt]
\frac{2n+\beta+1}{4}\left[\frac{n^{2}}{\beta+1}+n+(4n+1)(n+\beta+1)+\frac{(n+1)^{2}}{\beta-1}\right],&\alpha=0,\beta>1,\\\\[8.53581pt]
\frac{2n+\alpha+\beta+1}{4(n+\alpha+\beta-1)}\left[n(n+\alpha+\beta-1)\left(\frac{n+\alpha}{\beta+1}+2+\frac{n+\beta}{\alpha+1}\right)\right.&\\\
+\left.(n+1)(n+\alpha+\beta)\left(\frac{n+\alpha}{\beta-1}+2+\frac{n+\beta}{\alpha-1}\right)\right],&\alpha,\beta>1,\end{array}\right.$
(9)

(and $\infty$ otherwise) for the Fisher information SanchezRuiz2005 ;
Guerrero2010 , and

$\displaystyle
V[\hat{P}_{n}^{(\alpha,\beta)}]=\frac{4(n+1)(n+\alpha+1)(n+\beta+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)^{2}(2n+\alpha+\beta+3)}$
$\displaystyle+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+\beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}(2n+\alpha+\beta+1)},$
(10)

for the variance Dehesa2006  of Jacobi polynomials. Then, from Eqs.(4)-(10)
one has Dehesa2015  the following values for the Cramér-Rao complexity of the
Jacobi polynomials

$\displaystyle
C_{CR}\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\left\\{\begin{array}[]{ll}2n(n+1)\left[\frac{(n+1)^{2}}{2n+3}+\frac{n^{2}}{2n-1}\right],&\alpha=\beta=0,\\\
&\\\
\left[\frac{(n+1)^{2}(n+\beta+1)^{2}}{(2n+\beta+2)^{2}(2n+\beta+3)}+\frac{n^{2}(n+\beta)^{2}}{(2n+\beta-1)(2n+\beta)^{2}}\right]&\\\
\times\left[\frac{n^{2}}{\beta+1}+n+(4n+1)(n+\beta+1)+\frac{(n+1)^{2}}{\beta-1}\right],&\alpha=0,\beta>1,\\\
&\\\
\left[\frac{(n+1)(n+\alpha+1)(n+\beta+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+2)^{2}(2n+\alpha+\beta+3)}+\frac{n(n+\alpha)(n+\beta)(n+\alpha+\beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}}\right]&\\\
\times\frac{1}{n+\alpha+\beta-1}\left[n(n+\alpha+\beta-1)\left(\frac{n+\alpha}{\beta+1}+2+\frac{n+\beta}{\alpha+1}\right)\right.&\\\
\left.+(n+1)(n+\alpha+\beta)\left(\frac{n+\alpha}{\beta-1}+2+\frac{n+\beta}{\alpha-1}\right)\right],&\alpha>1,\beta>1,\end{array}\right.$

From this expression we easily obtain the high-degree asymptotics
($n\rightarrow\infty$; fixed $\alpha,\beta$)

$\displaystyle
C_{CR}\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\left\\{\begin{array}[]{ll}2n^{3}+\mathcal{O}(n^{2}),&\alpha=\beta=0,\\\
&\\\
\frac{1}{2}\left(2+\frac{\beta}{\beta^{2}-1}\right)n^{3}+\mathcal{O}(n^{2}),&\alpha=0,\beta>1,\\\
&\\\
\frac{1}{2}\left(\frac{\alpha}{\alpha^{2}-1}+\frac{\beta}{\beta^{2}-1}\right)n^{3}+\mathcal{O}(n^{2}),&\alpha>1,\beta>1,\end{array}\right.$
(17)

and the high-parameter asymptotics ($\alpha\rightarrow\infty$; fixed
$n,\beta$)

$\displaystyle
C_{CR}\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\frac{(1+\beta+2n\beta)(1+\beta+2n(1+n+\beta))}{\beta^{2}-1},\qquad\beta>1,\qquad\alpha\to\infty,$
(18)

for the Cramér-Rao complexity of the Jacobi polynomials. Summarizing, we first
observe that in the limit $n\rightarrow\infty$ the Cramér-Rao complexity of
the Jacobi polynomials follow a qualitative $n^{3}$-law, similarly to the
corresponding quantity of Laguerre polynomials Dehesa2015 , despite the fact
that the respective weight functions are so different. This is because the two
factors (variance and Fisher information) of the Cramér-Rao complexity of
Laguerre polynomials have a linear (Fisher infomation) and quadratic
(variance) dependence on $n$, while for the Jacobi polynomials the Fisher
infomation has a cubic dependence on the degree and the variance is constant.
Moreover, in the limit ($\alpha\rightarrow\infty$; fixed $n,\beta$) the
Cramér-Rao complexity of Laguerre and Jacobi polynomials have also a mutual
similar behavior, having a constant leading term that depends on $n,\beta$ for
the Jacobi polynomals and $n$ for the Laguerre polynomials. In this limit the
Fisher information and variance have a direct and inverse quadratic dependence
on $\alpha$ for the Jacobi polynomials, while for the Laguerre polynomials the
Fisher information and the variance have an inverse and direct linear
dependence on $\alpha$, respectively.

##  III Fisher-Shannon complexity of Jacobi polynomials

This statistical quantity is given by the Fisher-Shannon complexity of the
Rakhmanov density (2) of the Jacobi polynomials, which is defined Angulo2008 ;
Romera2004  as

$\displaystyle\mathcal{C}_{FS}[\hat{P}_{n}^{(\alpha,\beta)}]=F[\hat{P}_{n}^{(\alpha,\beta)}]\times\frac{1}{2\pi
e}e^{2S[\hat{P}_{n}^{(\alpha,\beta)}]}=\frac{1}{2\pi
e}F[\hat{P}_{n}^{(\alpha,\beta)}]\times\left(\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}]\right)^{2},$
(19)

where the symbols $F[\hat{P}_{n}^{(\alpha,\beta)}]$ and
$\mathcal{L}_{S}[p_{n}]=e^{S[p_{n}]}$ denotes the Fisher information and the
Shannon entropic power or Shannon spreading length of the polynomial
$\hat{P}_{n}^{(\alpha,\beta)}(x)$, respectively. Note that
$\mathcal{C}_{FS}[\hat{P}_{n}^{(\alpha,\beta)}]$ measures the gradient content
of the Rakhmanov probability density $\rho_{n}(x)$ associated to the
polynomial $\hat{P}_{n}^{(\alpha,\beta)}(x)$ and its total extent along the
support interval $[-1,+1]$ simultaneously.

The explicit expression of the Fisher-Shannon complexity of the Jacobi
polynomials for generic values $(n,\,\alpha,\beta)$ is unknown up until now.
This is so because, although the Fisher information has been given by Eq. (9),
the Shannon entropy is not known in spite of many efforts. However, there are
two extreme situations in which the value of this quantity can be analytically
evaluated; namely, when ($\alpha\rightarrow\infty;\mbox{fixed}\,n,\beta$) and
when ($n\rightarrow\infty;\mbox{fixed}\,\alpha,\beta$). The goal of this
section is to obtain these two parameter and degree asymptotics in a compact
way for the Shannon spreading length, and then for the Fisher-Shannon
complexity (Eq. 19) of the orthonormal Jacobi polynomials
$\hat{P}_{n}^{(\alpha,\beta)}(x)$. The main results are briefly summarized in
Table 1.

Measure of $\hat{P}_{n}^{(\alpha,\beta)}(x)$ | n$\to\infty$ | $\alpha\to\infty$
---|---|---
$\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}]$ | $\left.\frac{\pi}{e}\right.$ | $\frac{1}{\alpha}$
$\mathcal{C}_{FS}[\hat{P}_{n}^{(\alpha,\beta)}]$ | $\begin{array}[]{ll}\frac{2\pi}{e^{3}}n^{3}&\alpha,\beta=0\\\ \frac{\pi}{4e^{3}}\left(4+\frac{1}{\beta-1}+\frac{1}{\beta+1}\right)n^{3}&\alpha=0,\beta>1\\\ \frac{\pi(\alpha+\beta)(\alpha\beta-1)}{2e^{3}(\alpha^{2}-1)(\beta^{2}-1)}n^{3}&\alpha,\beta>1\\\ \infty&{\rm otherwise}\end{array}$ | $\frac{(1+\beta+2n\beta)}{8\pi e(\beta^{2}-1)}$
Table 1: First order asymptotics for the Shannon spreading length
$\mathcal{L}_{S}$ and the Fisher-Shannon complexity $\mathcal{C}_{FS}$
measures of the orthonormal Jacobi polynomials
$\hat{P}_{n}^{(\alpha,\beta)}(x)$, when $n\to\infty$ and $\alpha\to\infty$.

First, we realize that the Shannon-like integral functional of the orthonormal
Jacobi polynomials $\hat{P}_{n}^{(\alpha,\beta)}(x)$ is given by

$\displaystyle S\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=$
$\displaystyle-\int_{-1}^{+1}\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{2}h_{\alpha,\beta}(x)\log\left\\{\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{2}h_{\alpha,\beta}(x)\right\\}dx=E\left[\hat{P}_{n}^{(\alpha,\beta)}\right]+I\left[\hat{P}_{n}^{(\alpha,\beta)}\right],$
(20)

with the functional Sanchez2000

$\displaystyle I\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=$
$\displaystyle-\int_{-1}^{+1}\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{2}h_{\alpha,\beta}(x)\log
h_{\alpha,\beta}(x)$ $\displaystyle=$
$\displaystyle(\alpha+\beta)\left(\frac{1}{2n+\alpha+\beta+1}+2\psi(2n+\alpha+\beta+1)-\psi(n+\alpha+\beta+1)-\log(2)\right)$
$\displaystyle-\left(\alpha\psi(n+\alpha+1)+\beta\psi(n+\beta+1)\right)$ (21)

and the Shannon entropy $E\left[\hat{P}_{n}^{(\alpha,\beta)}\right]$, which is
defined by

$\displaystyle E\left[\hat{P}_{n}^{(\alpha,\beta)}\right]$
$\displaystyle=-\int_{-1}^{+1}\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{2}h_{\alpha,\beta}(x)\log\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{2}dx.$
(22)

The analytical determination of this entropic measure is a formidable task.
Indeed, it has have been calculated for integer values of the polynomial’s
parameter in a somewhat highbrow manner only. However, we find below that they
can be expressed in a simple and compact way for the two extreme situations
mentioned above.

###  III.1 Asymptotics $n\rightarrow\infty$

The Shannon entropy of the Jacobi polynomials has been shown to have the
following degree asymptotics

$E({\hat{P}_{n}^{(\alpha,\beta)}})=\log(\pi)-1-(\alpha+\beta)\log(2)+\mathcal{O}(n^{-1}),\qquad
n\rightarrow\infty$ (23)

for fixed $(\alpha,\beta)$ Aptekarev1994 ; Dehesa2001 . Moreover, from Eq.
(III) and the known asymptotical behavior Olver2010  of the involved gamma
$\Gamma(x)$ and digamma $\psi(x)$ functions, we find that for fixed
$(\alpha,\beta)$ the integral functional
$I\left[\hat{P}_{n}^{(\alpha,\beta)}\right]$ fulfills the asymptotics

$\displaystyle
I\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=(\alpha+\beta)\log(2)+\mathcal{O}(n^{-1}),\qquad
n\rightarrow\infty$ (24)

Therefore, from Eqs. (20), (23) and (24) we find that the asymptotics for the
Shanon-like functional of the Jacobi polynomials is

$\displaystyle
S\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\log(\pi)-1+\mathcal{O}(n^{-1}),\qquad
n\rightarrow\infty$ (25)

so that the Shannon spreading length of the Jacobi polynopmials has the
behavior

$\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}]\sim\frac{\pi}{e},\qquad
n\rightarrow\infty$ (26)

On the other hand, from expression (9) we can obtain the following
($n\rightarrow\infty;\mbox{fixed}\,\alpha,\beta$) asymptotics for the Fisher
information of the Jacobi polynomials :

$\displaystyle
F\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\left\\{\begin{array}[]{ll}4n^{3}+\mathcal{O}(n^{2}),&\alpha,\beta=0,\\\
\frac{1}{2}\left(4+\frac{1}{\beta-1}+\frac{1}{\beta+1}\right)n^{3}+\mathcal{O}(n^{2}),&\alpha=0,\beta>1,\\\
\frac{(\alpha+\beta)(\alpha\beta-1)}{(\alpha^{2}-1)(\beta^{2}-1)}n^{3}+\mathcal{O}(n^{2}),&\alpha,\beta>1,\\\
\infty,&{\rm otherwise.}\end{array}\right.$ (31)

Finally, taking into account (19), (26) and (31), we have that the Fisher-
Shannon complexity of the Jacobi polynomials has the following asymptotics
($n\rightarrow\infty;\mbox{fixed}\,\alpha,\beta$) behavior

$\displaystyle\mathcal{C}_{FS}\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\left\\{\begin{array}[]{ll}\frac{2\pi}{e^{3}}n^{3}+\mathcal{O}(n^{2}),&\alpha,\beta=0,\\\
\frac{\pi}{4e^{3}}\left(4+\frac{1}{\beta-1}+\frac{1}{\beta+1}\right)n^{3}+\mathcal{O}(n^{2}),&\alpha=0,\beta>1,\\\
\frac{\pi(\alpha+\beta)(\alpha\beta-1)}{2e^{3}(\alpha^{2}-1)(\beta^{2}-1)}n^{3}+\mathcal{O}(n^{2}),&\alpha,\beta>1,\\\
\infty,&{\rm otherwise.}\end{array}\right.$ (36)

which extends and includes the corresponding asymptotical quantity recently
obtained for the subfamily of Gegenbauer polynomials Dehesa2021  to the whole
Jacobi family of orthogonal polynomials. Moreover, we observe that in the
limit $n\rightarrow\infty$ the Fisher-Shannon complexity of the Jacobi
polynomials behaves qualitatively similar to the corresponding quantity of
Laguerre polynomials Sobrino2021 , following a $n^{3}$-law, despite the fact
that the weight functions are very different in each case; this is because for
the Laguerre polynomials both the Fisher information and the Shannon spreading
length have a linear dependence on $n$, while for the Jacobi polynomials the
Fisher information has a cubic dependence on the degree and the Shannon
spreading length is constant.

Interestingly, for the quantum systems with a solvable quantum-mechanical
potential with bound-states wavefunctions controlled by Jacobi polynomials
(e.g., some supersymmetric quantum systems) (see e.g. Ghosh2019 ;
Chaitanya2012 ; Cooper2001 ; Gangopadhyaya2017 ), the Fisher-Shannon measure
(36) allows one to find the quantum-classical limit of the physical Fisher-
Shannon complexity; they correspond to the high-energy or Rydberg states since
for such a limit $n\rightarrow\infty$ the wavelengths of particles are small
in comparison with the characteristic dimensions of the system and the
wavefunctions of the quasi-classical state.

###  III.2 Asymptotics $\alpha\rightarrow\infty$

Now we determine the Fisher-Shannon complexity
$\mathcal{C}_{FS}[\hat{P}_{n}^{(\alpha,\beta)}]$, given by (19), in the limit
$\alpha\rightarrow\infty$ with fixed degree $n,\beta$. We start by evaluating
the Shannon entropy (22) of the orthogonal Jacobi polynomials in this limit by
means of the relation

$E\left[P_{n}^{(\alpha,\beta)}\right]=2\frac{d}{dp}\left[\mathcal{N}_{p}\left[P_{n}^{(\alpha,\beta)}\right]\right]_{p=2},$
(37)

where the symbol $\mathcal{N}_{p}$ denotes the norm of the orthogonal Jacobi
polynomials defined as

$\displaystyle\mathcal{N}_{p}\left[P_{n}^{(\alpha,\beta)}\right]=\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}\left|P_{n}^{(\alpha,\beta)}(x)\right|^{p}dx.$
(38)

This quantity can be analytically estimated for $\alpha\to\infty$ by taking
into account the known relation Olver2010

$\lim_{\alpha\rightarrow\infty}\frac{P_{n}^{(\alpha,\beta)}(x)}{P_{n}^{(\alpha,\beta)}(1)}=\left(\frac{1+x}{2}\right)^{n},\qquad\text{with}\qquad
P_{n}^{(\alpha,\beta)}(1)=\frac{\Gamma(\alpha+n+1)}{n!\,\Gamma(\alpha+1)}.$
(39)

Then, from (38) and (39) we have the asymptotics

$\displaystyle\mathcal{N}_{p}\left[P_{n}^{(\alpha,\beta)}\right]$
$\displaystyle\sim
P_{n}^{(\alpha,\beta)}(1)\,2^{-np}\left(\frac{{}_{2}F_{1}(1,-\alpha,2+np+\beta;-1)}{1+np+\beta}+\frac{{}_{2}F_{1}(1,-np-\beta,2+\alpha;-1)}{1+\alpha}\right)$
$\displaystyle=P_{n}^{(\alpha,\beta)}(1)\,2^{-np}(1+np+\beta)^{-1}(1+\alpha)^{-1}\times$
$\displaystyle\left((1+\alpha)\,{}_{2}F_{1}(1,-\alpha,2+np+\beta;-1)+(1+np+\beta)\,{}_{2}F_{1}(1,-np-\beta,2+\alpha;-1)\right),$
(40)

where ${}_{2}F_{1}(a,b,c;x)$ denotes the Gaussian hypergeometric function
Olver2010 . Then, taking into account the known relation between the
hypergeometric functions

$\displaystyle(1-a){}_{2}F_{1}(1,a,2-b,-1)+(1-b){}_{2}F_{1}(1,b,2-a,-1)=\frac{2^{1-a-b}\Gamma(2-a)\Gamma(2-b)}{\Gamma(2-a-b)},$
(41)

with the parameters $a=-\alpha$ and $b=-np-\beta$, the asymptotical behavior
(III.2) simplifies as

$\displaystyle\mathcal{N}_{p}\left[P_{n}^{(\alpha,\beta)}\right]$
$\displaystyle\sim\frac{\Gamma(\alpha+n+1)}{n!}\frac{\Gamma(1+np+\beta)}{\Gamma(2+\alpha+np+\beta)}2^{1+\alpha+\beta}.$
(42)

Thus, according to Eqs. (37) and (42), one has that the Shannon entropy of the
orthogonal Jacobi polynomials $P_{n}^{(\alpha,\beta)}(x)$ in the current limit
is given as

$\displaystyle E\left[P_{n}^{(\alpha,\beta)}\right]$ $\displaystyle\sim
2^{2+\alpha+\beta}\frac{\Gamma(1+n+\alpha)\Gamma(1+2n+\beta)}{\Gamma(n)\Gamma(2+2n+\alpha+\beta)}(\psi(1+2n+\beta)-\psi(2+2n+\alpha+\beta))$
$\displaystyle=2^{2+\alpha+\beta}\alpha^{-n-\beta-1}\left(\frac{\Gamma(1+2n+\beta)}{\Gamma(n)}(\psi(1+2n+\beta)-\log(\alpha))+\mathcal{O}(\alpha^{-2})\right),\qquad\alpha\rightarrow\infty,$

so that we can express the Shannon entropy of the orthonormal Jacobi
polynomials as

$\displaystyle
E\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=\frac{1}{\kappa^{P}_{n}}E\left[P_{n}^{(\alpha,\beta)}\right]+\log(\kappa_{n})=$
$\displaystyle
2\alpha^{-n}\left(\frac{n\Gamma(1+2n+\beta)}{\Gamma(1+n+\beta)}(\psi(1+2n+\beta)-\log(\alpha))+\mathcal{O}(\alpha^{-2})\right)$
$\displaystyle+(1+\alpha+\beta)\log(2)+\log(\frac{\Gamma(1+n+\beta)}{n!})-(1+\beta)\log(\alpha).$
(43)

Moreover, from (III), we have the following asymptotics for the auxiliary
functional $I\left[P_{n}^{(\alpha,\beta)}\right]$

$\displaystyle
I\left[\hat{P}_{n}^{(\alpha,\beta)}\right]=-\alpha\log(2)+1+2n+\beta-\beta\log(2)+\beta\log(\alpha)-\beta\,\psi(1+n+\beta)+\mathcal{O}(\alpha^{-1}),\qquad\alpha\rightarrow\infty.$
(44)

A similar result follows for $\beta\to\infty$ by exchanging
$\alpha\leftrightarrow\beta$. Then, according to Eqs. (20), (43) and (44), we
find the following asymptotics for the Shannon-like integral functional of the
Jacobi polynomials

$\displaystyle
S\left[\hat{P}_{n}^{(\alpha,\beta)}\right]\sim-\log(\alpha)+\mathcal{O}(1),\qquad\alpha\rightarrow\infty,$
(45)

so that the Shannon entropy power or spreading length of Jacobi polynomials
behaves as

$\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}]\sim\frac{1}{\alpha},\qquad\alpha\rightarrow\infty.$
(46)

On the other hand, the asymptotics ($\alpha\rightarrow\infty$, fixed
$n,\beta$) for the Fisher information (9) of the Jacobi polynomials
F[$\hat{P}_{n}^{(\alpha,\beta)}$] turns out to be

$F[\hat{P}_{n}^{(\alpha,\beta)}]=\frac{(1+\beta+2n\beta)}{4(\beta^{2}-1)}\alpha^{2}+\mathcal{O}(\alpha),\qquad\alpha\rightarrow\infty.$
(47)

Finally, the substitution of the last two quantities into Eq. (19) gives rise
to the following ($\alpha\rightarrow\infty$, fixed $n,\beta$)-asymptotics for
the Fisher-Shannon complexity of the orthonormal Jacobi polynomials:

$\mathcal{C}_{FS}[\hat{P}_{n}^{(\alpha,\beta)}]\sim\frac{(1+\beta+2n\beta)}{8\pi
e(\beta^{2}-1)}+\mathcal{O}(\alpha^{-1}),\qquad\alpha\rightarrow\infty.$ (48)

The corresponding result for the polynomials with
($\alpha\rightarrow\infty;\beta\rightarrow\infty;\mbox{fixed}\,n$) remains to
be found; we have not been able to find it because the involved asymptotical
behavior of the second-order entropic moment
$W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]$ is a non-trivial task.

Finally, for multidimensional quantum systems with stationary states
controlled by Jacobi polynomials (e.g., some supersymmetric quantum systems)
(see e.g. Ghosh2019 ; Chaitanya2012 ; Cooper2001 ; Gangopadhyaya2017 ), the
Fisher-Shannon measure (48) and its asymptotical extension for
($\alpha\rightarrow\infty;\beta\rightarrow\infty;\mbox{fixed}\,n$) allow us to
find the pseudo-classical limit of the physical Fisher-Shannon complexity;
they correspond to the high-dimensional or pseudoclassical states. The latter
is because the wavefunctions of such extreme states involve polynomials
orthogonal with respect to a Jacobi weight function where both parameters
$\alpha$ and $\beta$ are directly proportional to the system’s dimensionality.

##  IV LMC complexity of Jacobi polynomials

In this section we investigate the LMC complexity of the orthonormal Jacobi
polynomials $\hat{P}_{n}^{(\alpha,\beta)}(x)$, which is defined as

$\mathcal{C}_{LMC}[\hat{P}_{n}^{(\alpha,\beta)}]=W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]\times\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}],$
(49)

where the second-order entropic moment $W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]$,
which measures the disequilibrium or deviation from uniformity, is given by

$W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]=\int_{-1}^{+1}\left(\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{2}h_{\alpha,\beta}(x)\right)^{2}\,dx=\int_{-1}^{+1}\,(1-x)^{2\alpha}(1+x)^{2\beta}\,\left[\hat{P}_{n}^{(\alpha,\beta)}(x)\right]^{4}\,dx.$
(50)

This statistical complexity quantifies the combined balance of the
disequilibrium and the total extent of the polynomials along its weight
function. The explicit expression of this measure in terms of the degree $n$
and the parameters $(\alpha,\beta)$ has not yet been determined in a handy
way, because neither $W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]$ nor the spreading
length $\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}]$ are analytically known.
In this section we obtain simple and compact analytical expressions for
$\mathcal{C}_{LMC}[\hat{P}_{n}^{(\alpha,\beta)}]$ in the two extreme
situations, ($n\rightarrow\infty;\mbox{fixed}\,\alpha,\beta$) and
($\alpha\rightarrow\infty;\mbox{fixed}\,n,\beta$). They are briefly summarized
in Table 2.

Measure of $\hat{P}_{n}^{(\alpha,\beta)}(x)$ | n$\to\infty$ | $\alpha\to\infty$
---|---|---
$W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]$ | $\begin{array}[]{ll}\frac{2^{\alpha+\beta-2}3}{\pi^{2}}\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}&\beta>0\\\ \log{(n)}&\beta=0\\\ n^{-2\beta}&-1<\beta<0\end{array}$ | $\frac{\Gamma(1+4n+2\beta)}{2^{2(1+2n+\beta)}n!^{2}\Gamma(1+n+\beta)}\alpha$
$\mathcal{C}_{LMC}[\hat{P}_{n}^{(\alpha,\beta)}]$ | $\begin{array}[]{ll}\frac{2^{\alpha+\beta-2}3}{\pi e}\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}&\beta>0\\\ \frac{\pi}{e}\log{(n)}&\beta=0\\\ \frac{\pi}{e}n^{-2\beta}&-1<\beta<0\end{array}$ | $\frac{\Gamma(1+4n+2\beta)}{2^{2(1+2n+\beta)}n!^{2}\Gamma(1+n+\beta)}$
Table 2: First order asymptotics for the disequilibrium $W_{2}$ and the LMC
complexity $\mathcal{C}_{LMC}$ measures of the orthonormal Jacobi polynomials
$\hat{P}_{n}^{(\alpha,\beta)}(x)$, when $n\to\infty$ and $\alpha\to\infty$.

###  IV.1 Asymptotics $n\rightarrow\infty$

To determine the ($n\rightarrow\infty;\mbox{fixed}\,\alpha,\beta$)-asymptotics
of the LMC complexity $\mathcal{C}_{LMC}[\hat{P}_{n}^{(\alpha,\beta)}]$ we
start by calculating the disequilibrium $W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]$
in the limit $n\rightarrow\infty$ by means of Theorem 3 of Aptekarev et al
Aptekarev2021 , obtaining

$\displaystyle
W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]\sim\left\\{\begin{array}[]{ll}\frac{2^{\alpha+\beta-2}3}{\pi^{2}}\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)},&\beta>0,\\\
\log{(n)},&\beta=0,\\\ n^{-2\beta},&-1<\beta<0,\end{array}\right.$ (54)

and then we keep in mind the corresponding asymptotics (26) for the Shannon
spreading length $\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}]$. The
substitution of the asymptotical values of these two entropic quantities into
Eq. (49) gives rise to the following asymptotical behavior
($n\rightarrow\infty$) of the LMC complexity of the orthonormal Jacobi
polynomials $\hat{P}_{n}^{(\alpha,\beta)}(x)$:

$\displaystyle\mathcal{C}_{LMC}[\hat{P}_{n}^{(\alpha,\beta)}]\sim\left\\{\begin{array}[]{ll}\frac{2^{\alpha+\beta-2}3}{\pi
e}\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)},&\beta>0,\\\
\frac{\pi}{e}\log{(n)},&\beta=0,\\\
\frac{\pi}{e}n^{-2\beta},&-1<\beta<0,\end{array}\right.$ (58)

Figure 1: Comparison of the LMC complexity $\mathcal{C}_{LMC}$ measures of the
orthonormal Gegenbauer polynomials $\hat{C}_{n}^{(\lambda)}(x)$ and the
orthonormal Jacobi polynomials $\hat{P}_{n}^{(\alpha,\beta)}(x)$, with
$(\alpha=\lambda-2$ and $\beta=\lambda-2,2,4,8)$, for various values of
$\lambda$ and $n\rightarrow\infty$.

which extends and includes the corresponding asymptotical quantity recently
obtained for the subfamily of Gegenbauer polynomials Dehesa2021  to the whole
Jacobi family of orthogonal polynomials. See also Figure 1, where the LMC
complexity $\mathcal{C}_{LMC}$ measures of the orthonormal Jacobi polynomials
$\hat{P}_{n}^{(\alpha,\beta)}(x)$, with $(\alpha=\lambda-2$ and
$\beta=\lambda-2,2,4,8)$, and the orthonormal Gegenbauer polynomials
$\hat{C}_{n}^{(\lambda)}(x)$ are compared for various values of $\lambda$ and
$n\rightarrow\infty$. Note that both Jacobi and Gegenbauer complexities match
when $\beta=\lambda-2$ as one would expect, what is a partial checking of our
results. On the other hand, we observe that in the limit $n\rightarrow\infty$
the LMC complexity of the Jacobi polynomials behaves very different to the
corresponding quantity of Laguerre polynomials Dehesa2021 , as one expects
because of their weight functions are so distinct, except in the special case
$\beta=0$. Indeed, for $\beta=0$ there happens the following phenomenon: the
disequilibrium has a logarithmic dependence on $n$ while the Shannon spreading
length is constant so that the total balance for the LMC complexity of the
Jacobi polynomials obeys the $\log n$-law as also occurs for the LMC
complexity of Laguerre polynomials Dehesa2021 .

Here again, for the quantum systems with a solvable quantum-mechanical
potential with bound-states wavefunctions controlled by Jacobi polynomials
(e.g., some supersymmetric quantum systems) (see e.g. Ghosh2019 ;
Chaitanya2012 ; Cooper2001 ; Gangopadhyaya2017 ), the LMC measure (36) allows
one to find the quantum-classical limit of the physical LMC complexity; they
correspond to the high-energy or Rydberg states.

###  IV.2 Asymptotics $\alpha\rightarrow\infty$

Here we determine the LMC complexity
$\mathcal{C}_{LMC}[\hat{P}_{n}^{(\alpha,\beta)}]$, given by Eq. (49), in the
limit ($\alpha\rightarrow\infty;\mbox{fixed}\,n,\beta$). First we realize that
the Shannon spreading length $\mathcal{L}_{S}[\hat{P}_{n}^{(\alpha,\beta)}]$,
has been already obtained in Eq. (46). Then, we calculate the second-order
entropic moment $W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]$, given by Eq. (50). We
use the limiting relation (39) into (50), obtaining the value

$\displaystyle W_{2}[P_{n}^{(\alpha,\beta)}]\sim$ $\displaystyle
P_{n}^{(\alpha,\beta)}(1)^{4}2^{-4n}\Gamma(1+2\alpha)\Gamma(1+4n+2\beta)\times$
$\displaystyle\left(\frac{1}{\Gamma(1+2\alpha)}{}_{2}\tilde{F}_{1}(1,-2\alpha,2+4n+2\beta,-1)+\frac{1}{\Gamma(1+4n+2\beta)}{}_{2}\tilde{F}_{1}(1,-4n-2\beta,2+2\alpha,-1)\right)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(\alpha+n+1)^{4}}{(n!)^{4}\Gamma(\alpha+1)^{4}}\frac{\Gamma(1+4n+2\beta)\Gamma(1+2\alpha)}{\Gamma(2+2\alpha+4n+2\beta)}2^{1+2\alpha+2\beta}$
(59)

for the orthogonal Jacobi polynomials. Now, the corresponding asymptotics for
the second-order entropic power of the orthonormal Jacobi polynomials is given
by

$W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]=\frac{1}{(\kappa_{n})^{2}}\,W_{2}[P_{n}^{(\alpha,\beta)}]\sim\frac{\Gamma(1+4n+2\beta)}{2^{2(1+2n+\beta)}\,(n!)^{2}\,\Gamma(1+n+\beta)}\alpha,\qquad\alpha\rightarrow\infty.$
(60)

Finally, the combination of Eqs. (49), (46) and (60) lead to the asymptotical
behavior

$\mathcal{C}_{LMC}[\hat{P}_{n}^{(\alpha,\beta)}]=\frac{\Gamma(1+4n+2\beta)}{2^{2(1+2n+\beta)}(n!)^{2}\,\Gamma(1+n+\beta)},\qquad\alpha\rightarrow\infty$
(61)

for the LMC complexity of the (orthonormal) Jacobi polynomials with
($\alpha\rightarrow\infty;\mbox{fixed}\,n,\beta$). It remains to find the
corresponding result for the polynomials with
($\alpha\rightarrow\infty;\beta\rightarrow\infty;\mbox{fixed}\,n$), which we
have not been able to find because the involved asymptotical behavior of the
the second-order entropic moment $W_{2}[\hat{P}_{n}^{(\alpha,\beta)}]$ is a
non-trivial task.

Finally, for multidimensional quantum systems with bound states controlled by
Jacobi polynomials (e.g., some supersymmetric quantum systems) (see e.g.
Ghosh2019 ; Chaitanya2012 ; Cooper2001 ; Gangopadhyaya2017 ), the LMC measure
(61) and its asymptotical extension when
($\alpha\rightarrow\infty;\beta\rightarrow\infty;\mbox{fixed}\,n$) allow us to
find the pseudo-classical limit of the physical LMC complexity; they
correspond to the high-dimensional or pseudoclassical states. This is because
the wavefunctions of such extreme states involve polynomials orthogonal with
respect to a Jacobi weight function where both parameters $\alpha$ and $\beta$
are directly proportional to the space dimensionality of the system.

##  V Conclusions and open problems

In this work we have determined the Cramér-Rao, Fisher-Shannon and LMC
complexity-like measures of the Jacobi polynomials
$\hat{P}_{n}^{(\alpha,\beta)}(x)$, with $\alpha,\beta>-1$, in the extreme
situations ($n\rightarrow\infty$; fixed $\alpha,\beta$) and
($\alpha\rightarrow\infty$; fixed $n,\beta$). They are given by the leading
term of the degree and parameter asymptotics of the corresponding statistical
properties of the associated probability density (Rakhmanov’s density),
respectively. Each of these complexity quantifiers capture in a simultaneous
way two polynomial’s configurational facets of dispersion (variance) and
entropic (Fisher, Shannon) types. Briefly, in the limit ($n\rightarrow\infty$;
fixed $\alpha,\beta$) we have found that both Cramér-Rao and Fisher-Shannon
complexities follow a qualitatively similar $n^{3}$-law behavior for all
$\beta$ (see Table 1), but the LMC complexity has a different asymptotical
$n$-behavior depending on $\beta$ (see Table 2). Moreover, in the limit
($\alpha\rightarrow\infty$; fixed $n,\beta$) we have found that the Fisher
information and the variance follow a direct and inverse quadratic dependence
on $\alpha$, respectively, while the second-order entropic moment and the
Shannon entropy power follow a direct and inverse linear dependence on
$\alpha$. The combination of the two dispersion/entropic factors involved for
the Cramér-Rao, Fisher-Shannon and LMC complexities lead to a constant leading
term for all the complexity measures.

Finally, a number of open related problems can be highlighted. First, the
extension of these results to the varying Jacobi polynomials Kuijlaars2004 ;
Kuijlaars2005 ; Buyarov1999  (i.e., when the parameters depend on the
polynomial degree) as well as to the exceptional Jacobi polynomials
Chaitanya2012 ; Sasaki2010 ; Quesne2008 , which are very useful to standard
and supersymmetric quantum mechanics Gangopadhyaya2017 . Second, the
determination of the general statistical complexity measures Sobrino2017  of
Fisher-Rényi Nagy2012 ; Puertas2017c ; Toranzo2017 ; Romera2008 ; Romera2009
and LMC-Rényi Pipek1997 ; Lopez2009 ; Romera2008 ; Nath2021 ; Zozor2017  types
for the standard and varying Jacobi polynomials; this includes the calculation
of the Rényi entropy of such polynomials. These open issues are not only
interesting per se but also because of their chemical and physical
applications, especially for the extreme quantum states of highly excited
Rydberg and high dimensional types of numerous atomic and molecular systems
whose bound states are described by wavefunctions controlled by these
polynomials.

## Funding information

The work of N. Sobrino has been partially supported by the Basque Government
and UPV/EHU under grant IT1249-19. The work of J. S. Dehesa has been partially
supported by the Agencia Estatal de Investigación (Spain) and the European
Regional Development Fund (FEDER) under the grant PID2020-113390GB-I00, and by
Agencia Andaluza del Conocimiento of the Junta de Andalucía (Spain) under
grant PY20-00082.

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#  $\Delta$-revolving sequences and
self-similar sets in the plane
\- Dedicated to the late Professor Shunji Ito -

Kiko Kawamura and Tobey Mathis
University of North Texas 111Address: Department of Mathematics, University of
North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017, USA; E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Initiated by Mizutani and Ito’s work in 1987, Kawamura and Allen recently
showed that certain self-similar sets generalized by two similar contractions
have a natural complex power series representation, which is parametrized by
past-dependent revolving sequences.

In this paper, we generalize the work of Kawamura and Allen to include a wider
collection of self-similar sets. We show that certain self-similar sets
consisting of more than two similar contractions also have a natural complex
power series representation, which is parametrized by $\Delta$-revolving
sequences. This result applies to several other famous self-similar sets such
as the Heighway dragon, Twindragon, and Fudgeflake.

Keywords.Revolving sequences, Tiling dragons, Self-similar sets

MSC (2022). Primary: $28A80$; Secondary: $37B10$.

##  1 Introduction

The history of systematic mathematical research on self-similar fractals dates
back to 1981, when Hutchinson proved the following celebrated theorem [5]: For
any finite family of similar contractions
$\\{\psi_{0},\psi_{1},\dots,\psi_{m-1}\\}$ on $\mathbb{C}$, which is called an
iterated function system or IFS, there exists a unique non-empty compact
solution $E$ of the set equation:

$E=\bigcup_{i=0}^{m-1}\psi_{i}(E).$

We call $E$ an attractor or a self-similar set for the IFS.

It is well known that any point of $E$ can be represented by at least one
coding sequence $(x_{i})_{i=1}^{\infty}$ such that

$E=\left\\{\lim_{n\to\infty}\psi_{x_{1}}\circ\psi_{x_{2}}\circ\cdots\circ\psi_{x_{n}}(0):(x_{i})_{i=1}^{\infty}\in\\{0,1,2,\cdots,m-1\\}^{\mathrm{I\\!N}}\right\\}.$
(1.1)

For more details, see Falconer’s book [3].

Eleven years prior to Hutchinson’s paper [5], C. Davis and D. E. Knuth [1] in
1970 introduced the idea that some tiling fractals are associated with the
complex number system. This idea was further discussed by Gilbert [4]. In [1],
Davis and Knuth introduced the concept of revolving sequences to represent
Gaussian integers: for any $z=x+iy$ with $x,y\in\mathbb{Z}$, there exists a
finite revolving sequence with length $N$:
$(\delta_{0},\delta_{1},\dots\delta_{N})$ such that

$z=\sum_{n=0}^{N}\delta_{N-n}(1+i)^{n},$

where $\delta_{j}\in\\{0,1,-1,i,-i\\}$ with the restriction that the non-zero
values must follow the cyclic pattern from left to right:

$\cdots\to 1\to(-i)\to(-1)\to i\to 1\to\cdots.$

Notice that a revolving sequence is created by a cycle of 90 degree clockwise
rotation on the unit circle.

Let $W$ be the set of all revolving sequences. M. Mizutani and S. Ito in 1987
[10] considered the following set of points in the complex plane:

$X:=\left\\{\sum_{n=1}^{\infty}\delta_{n}(1+i)^{-n}:(\delta_{1},\delta_{2},\dots)\in
W\right\\}.$ (1.2)

Using techniques from symbolic dynamics, they proved that $X$ (which they
called Tetradragon) is tiled by four rotated paper-folding dragons. Paper-
folding dragon is generated by the IFS:

$\begin{cases}\psi_{0}(z)=(\frac{1-i}{2})z,&\\\
\psi_{1}(z)=(\frac{-1-i}{2})z+\frac{1-i}{2}.&\end{cases}$ (1.3)

Observe that revolving sequences $(\delta_{n})_{n=1}^{\infty}$ are past-
dependent sequences.

In the same paper, they mentioned an interesting conjecture: suppose
$\delta_{n}$ moves on the unit circle counterclockwise instead of clockwise,
then

$X^{R}:=\left\\{\sum_{n=1}^{\infty}\overline{\delta_{n}}(1+i)^{-n}:(\delta_{1},\delta_{2},\delta_{3},\dots)\in
W\right\\}$

is a union of four Lévy’s dragon curves. Lévy’s dragon is a well-known
continuous curve with positive area. Mizutani and Ito’s computer simulation of
$X^{R}$ empirically confirmed the conjecture; however, they could not give a
mathematical proof. Kawamura in 2002 [6] proved that their conjecture is
correct using a different approach from the viewpoint of functional equations.

These two concrete examples indicate that not only tiling fractals but more
generally some self-similar attractors can be parametrized by past-dependent
sequences.

Recently, Kawamura and Allen [7] defined Generalized Revolving Sequences
(GRS), where the 90 degree angle of rotation is replaced with a more general
angle $|\theta|=\frac{2\pi q}{p}\leq\pi$. In other words,
$\delta_{k}\in\\{0,1,e^{i\theta},e^{2i\theta},\cdots,e^{(p-1)i\theta}\\}$ with
the restriction that the non-zero values must follow the following cyclic
pattern from left to right:

$\cdots\to 1\to e^{i\theta}\to e^{2i\theta}\to e^{3i\theta}\to\cdots.$

Define $W_{\theta}$ to be the set of all generalized infinite revolving
sequences with parameter $\theta$. Kawamura and Allen considered the following
set.

$X_{\alpha,\theta}:=\left\\{\sum_{n=1}^{\infty}\delta_{n}\alpha^{n}:(\delta_{1},\delta_{2},\dots)\in
W_{\theta}\right\\}.$ (1.4)

where $\alpha\in\mathbb{C}$ such that $|\alpha|<1$ and $|\theta|=\frac{2\pi
q}{p}\leq\pi$. Note that generalized revolving sequences are past-dependent.

Kawamura and Allen proved that $X_{\alpha,\theta}$ is a union of $p$ rotated
copies of the self-similar attractor $M_{\alpha,\theta}$ of an IFS consisting
of two similarities with the same scale factor, one of which involves a
rotation through an angle $\theta$:

$\begin{cases}\psi_{0}(z)=\alpha z,&\\\ \psi_{1}(z)=(\alpha
e^{i\theta})z+\alpha.\end{cases}$ (1.5)

Figure 1: $X_{\alpha,\theta}$ for $\alpha=\frac{1-i}{2},\theta=-\frac{\pi}{2}$
(left), $\alpha=\frac{1-i}{2},\theta=\frac{\pi}{2}$ (right)

Figure 1 shows two examples of $X_{\alpha,\theta}$. Observe that this is a
generalization of Mizutani-Ito’s and Kawamura’s results: if $\alpha=(1-i)/2$
and $\theta=-\pi/2$ then $X_{\alpha,\theta}$ is a union of four paper-folding
dragons [10]. If $\alpha=(1-i)/2$ and $\theta=\pi/2$ then $X_{\alpha,\theta}$
is a union of four Lévy dragon curves.

In this paper, we continue building on the work initiated by Mizutani and Ito
to include a wider collection of self-similar sets. In particular, the
following questions are discussed.

  1. 1.

Does replacing the constant term $\alpha$ in (1.5) by an arbitrary constant
give any major influence to the property of generalized revolving sequences in
(1.4)?

  2. 2.

Consider self-similar sets generated by the IFS consisting of more than two
similar contractions. Can we find a complex power series representation, which
is parametrized by past-dependent sequences? How does the number of
contractions influence the properties of generalized revolving sequences?

In section 2, we introduce $\Delta$-revolving sequences allowing revolutions
on the unit circle by more than one angle, while keeping the current-
dependency. We illustrate the difference between generalized revolving
sequences and $\Delta$-revolving sequences with examples.

In section 3, we show that certain self-similar attractors of IFSs consisting
of $m\geq 2$ similar contractions have a natural complex power series
representation, which is parametrized by $\Delta$-revolving sequences. This
main theorem applies to several other famous self-similar sets such as the
Heighway dragon, Twindragon, and Fudgeflake.

Lastly in section 4, we introduce a slight modification of $\Delta$-revolving
sequences called $\Delta_{0}$-revolving sequences. $\Delta_{0}$-revolving
sequences are a more natural extension of the notion of generalized revolving
sequences than $\Delta$-revolving sequences. As a corollary of the main
theorem, we give a condition when certain self-similar attractors can also be
parametrized by $\Delta_{0}$-revolving sequences.

##  2 $\Delta$-Revolving Sequences

First, we review some notations and a result from [7].

######  Definition 2.1 (Revolving Angle).

We say $\theta\in(-\pi,\pi]$ is a revolving angle if $\theta$ is a nonzero
rational multiple of $2\pi$, that is, $|\theta|=(2\pi q)/p$ where
$p\in\mathbb{N},\>q\in\mathbb{N}_{0}$.

######  Definition 2.2 (Generalized Revolving Sequence).

Define

$\Delta_{\theta}:=\\{0,1,e^{i\theta},e^{2i\theta},\cdots e^{(p-1)i\theta}\\},$

where $\theta$ is a revolving angle. We say that a sequence
$(\delta_{n})\in\Delta_{\theta}^{\mathrm{I\\!N}}$ satisfies the Generalized
Revolving Condition (GRC), if

  1. 1.

$\delta_{1}$ is free to choose.

  2. 2.

If $\delta_{1}=\delta_{2}=\cdots=\delta_{n}=0$, then $\delta_{n+1}$ is free to
choose.

  3. 3.

Otherwise, $\delta_{n+1}=0$ or $\delta_{n+1}=\delta_{j_{0}(n)}e^{i\theta}$,
where $j_{0}(n)=\max\\{j\leq n:\delta_{j}\neq 0\\}$.

######  Example 2.3.

If $\theta=\pi/3$,
$\Delta_{\theta}:=\\{0,1,e^{i\pi/3},e^{2i\pi/3},e^{i\pi},e^{4i\pi/3},e^{5i\pi/3}\\}.$

$(\delta_{n})=(0,1,e^{i(\pi/3)},0,e^{i(2\pi/3)},0,0,e^{i\pi},e^{i(4\pi/3)},0,\dots)$
(2.1)

is a generalized revolving sequence.

Define $W_{\theta}$ as the set of all generalized revolving sequences with
parameter $\theta$:

$W_{\theta}:=\\{(\delta_{1},\delta_{2},\cdots)\in\Delta_{\theta}^{\mathrm{I\\!N}}:(\delta_{1},\delta_{2},\cdots)\mbox{
satisfies the GRC}\\}.$

Observe that $\delta_{n}$ moves on the unit circle counterclockwise if
$\theta>0$, and clockwise if $\theta<0$. Roughly speaking, $j_{0}(n)$ is the
last time before $n$ that $\delta_{j}$ is on the unit circle. Therefore,
generalized revolving sequences are past-dependent.

######  Theorem 2.4 (Kawamura-Allen, 2020).

For a given $\alpha\in\mathbb{C}$ such that $|\alpha|<1$ and $\theta$ is a
revolving angle, define

$X_{\alpha,\theta}:=\left\\{\sum_{n=1}^{\infty}\delta_{n}\alpha^{n}:(\delta_{n})\in
W_{\theta}\right\\}.$

Then $X_{\alpha,\theta}$ is a union of $p$ rotated copies of
$M_{\alpha,\theta}$:

$X_{\alpha,\theta}=\bigcup_{l=0}^{p-1}(e^{i\theta})^{l}M_{\alpha,\theta},$

where
$M_{\alpha,\theta}=\psi_{0}(M_{\alpha,\theta})\cup\psi_{1}(M_{\alpha,\theta})$
is the self-similar set generated by the IFS

$\begin{cases}\psi_{0}(z)=\alpha z,&\\\ \psi_{1}(z)=(\alpha
e^{i\theta})z+\alpha.\end{cases}$

Next, we loosen the restriction of a generalized revolving sequence, allowing
revolutions on the unit circle by more than one angle.

######  Definition 2.5 (Generator Set).

Let $S:=\\{\theta_{0},\theta_{1},\dots,\theta_{m-1}\\}$ where
$\theta_{0},\theta_{1},\dots,\theta_{m-1}$ are distinct angles and
$\theta_{0}=0$. Its elements can be written as $\theta_{j}=\frac{2\pi
q_{j}}{p_{j}}$. We refer to the set S as a generator set.

######  Definition 2.6 (Revolving Group Generated by $S$).

Let $S=\\{\theta_{0},\theta_{1},\dots,\theta_{m-1}\\}$ be a generator set.
Define

$\Delta:=\big{\\{}e^{i\sum_{n=1}^{m-1}k_{n}\theta_{n}}\>:\>k_{n}\in\\{0,1,\dots,p_{n}-1\\}\big{\\}}.$

######  Example 2.7.

If $S=\\{0,\pi,\frac{2\pi}{3}\\}$, then

$\Delta=\\{1,e^{i(\pi)},e^{i(\frac{2\pi}{3})},e^{i(\frac{4\pi}{3})},e^{i(\pi+\frac{2\pi}{3})},e^{i(\pi+\frac{4\pi}{3})}\\}.$

From this example, it is clear that $\Delta$ is a group, which we call the
revolving group generated by S. Notice that $\Delta$ does not contain $0$, and
$|\Delta|=\mathrm{lcm}(|p_{1}|,\cdots,|p_{m-1}|)$.

######  Definition 2.8 ($\Delta$-Revolving Sequence).

We say that a sequence $(\gamma_{n})\in\Delta^{\mathbb{N}}$ satisfies the
$\Delta$-Revolving Condition (DRC), if

  1. 1.

$\gamma_{1}$ is free to choose in $\Delta$,

  2. 2.

For $k\geq 1$, $\gamma_{k+1}=\gamma_{k}e^{i\theta_{j}}$ for some
$\theta_{j}\in S$.

If $(\gamma_{n})$ satisfies the DRC, we call $(\gamma_{n})$ a
$\Delta$-Revolving Sequence.

######  Example 2.9.

If $S=\\{0,\pi,\frac{2\pi}{3}\\}$,
$\Delta=\\{1,e^{i(\pi)},e^{i(\frac{2\pi}{3})},e^{i(\frac{4\pi}{3})},e^{i(\pi+\frac{2\pi}{3})},e^{i(\pi+\frac{4\pi}{3})}\\}$.
And, for examples,

$(\gamma_{n})=(1,e^{i\pi},e^{i\pi},e^{i(\pi+\frac{2\pi}{3})},e^{i(\pi+\frac{2\pi}{3})},e^{i(\pi+\frac{2\pi}{3})},e^{i(\frac{2\pi}{3})},\dots)$
(2.2)

is a $\Delta$-revolving sequence.

Define $W^{\Delta}$ as the set of all $\Delta$-revolving sequences with
generator set $S$ and $W_{1}^{\Delta}$ be the subset of $W^{\Delta}$ with the
restriction: $\gamma_{1}=1$.

Compare examples: (2.1) and (2.2). It is clear that the generalized revolving
condition is associated with a single angle of rotation on the unit circle,
while the $\Delta$-revolving condition is associated with multiple angles of
rotation on the unit circle. Observe also that generalized revolving sequences
allow that $\delta_{n}$ goes back to the origin and stays there as long as it
wants, while $\Delta$-revolving sequences do not allow that $\gamma_{n}$ goes
back to the origin; however, they allow ”staying in place”. Informally, each
digit $\gamma_{n}$ can choose freely either to stay ($\theta_{0}=0$) or move
($\theta_{j}\in S\setminus\\{\theta_{0}\\}$). However, the next digit
$\gamma_{k+1}$ is determined based on the current position $\gamma_{k}$.
Therefore, $\Delta$-revolving sequences are current-dependent, while the
generalized revolving sequences are past-dependent.

Alternatively, due to the group structure of $\Delta$, a $\Delta$-Revolving
Sequence can be thought of as an infinite walk on the Cayley Digraph of the
group $\Delta$ generated by the set with elements corresponding to $S$.

Figure 2: Cayley Digraph for generator set $S=\\{0,\frac{2\pi}{3},\pi\\}$

##  3 Main results

######  Lemma 3.1.

Consider a standard IFS $\\{\psi_{0},\psi_{1},\dots,\psi_{m-1}\\}$ given by

$\psi_{k}(z)=\alpha_{k}z+c_{k},$ (3.1)

where $\alpha_{k}\in\mathbb{C}$ such that $|\alpha_{k}|<1$ and
$c_{k}\in\mathbb{C}$ for $k=0,1,\dots,m-1$.

Then the self-similar set $E=\cup_{i=0}^{m-1}\psi_{i}(E)$ generated by (3.1)
has the following natural series representation:

$E=\left\\{\sum_{n=1}^{\infty}c_{x_{n}}\prod_{k=0}^{m-1}\alpha_{k}^{I_{k}(x,n-1)}\>:\>x=(x_{n})\in\\{0,1,\dots,m-1\\}^{\mathbb{N}}\right\\},$
(3.2)

where $I_{k}(x,n):=\\#\\{j\leq n\>:\>x_{j}=k\\}$ for $k=0,1,\dots m-1$ and
$n\in\mathrm{I\\!N}$.

Notice that $I_{k}(x,n)$ is the number of $k$’s occurring in the first $n$
digits of the sequence $(x_{n})$.

###### Proof.

From (1.1), it follows that every point in $E$ can be represented by at least
one coding sequence $(x_{n})$ given in (3.2). Vice versa, the set on the right
of (3.2) satisfies the set equation $E=\cup_{i=0}^{m-1}\psi_{i}(E)$. Let
$M:=\\{0,1,\dots,m-1\\}^{\mathrm{I\\!N}}$. It is well known that $M$ is
compact in the product topology. Define the function $f:M\to\mathbb{C}$ by

$f(x)=\sum_{n=1}^{\infty}c_{x_{n}}\prod_{k=0}^{m-1}\alpha_{k}^{I_{k}(x,n-1)}.$

Since it is easily seen that $f$ is continuous and $f(M)=E$, it follows that
$E$ is compact so it is closed. ∎

Before presenting the main theorem, let us define one more notation.

######  Definition 3.2 (Constant Sequence of $(\gamma_{n})$).

Let $S=\\{\theta_{0},\theta_{1},\dots,\theta_{m-1}\\}$ be a generator set and
$\Delta$ be the revolving group generated by S.

For $\Sigma:=(\gamma_{n})\in W^{\Delta}$ and $c_{k}\in\mathbb{C}$ for
$k=0,1,\dots m-1$, define the function
$s_{\Sigma}:\mathbb{N}\longrightarrow\\{c_{0},c_{1},\dots,c_{m-1}\\}$ by

$s_{\Sigma}(n)=c_{k},\text{ if }\gamma_{n+1}=\gamma_{n}e^{i\theta_{k}}.$

We say that the sequence $(s_{n})$ given by $s_{n}=s_{\Sigma}(n)$ is the
constant sequence of $(\gamma_{n})$. Notice that we allow $c_{l}=c_{k}$ for
$l\neq k$.

######  Theorem 3.3.

Let $S=\\{\theta_{0},\theta_{1},\dots,\theta_{m-1}\\}$ be a generator set and
$\Delta$ be the revolving group generated by S. For $\alpha\in\mathbb{C}$ with
$|\alpha|<1$ and $c_{0},c_{1},\dots,c_{m-1}\in\mathbb{C}$, define
$X_{\alpha,S}$ as follows.

$X_{\alpha,S}=\left\\{\sum_{n=1}^{\infty}\alpha^{n-1}s_{n}\gamma_{n}\>:\>\Sigma=(\gamma_{n})\in
W^{\Delta},\>s_{n}=s_{\Sigma}(n)\right\\}.$ (3.3)

Then $X_{\alpha,S}$ is the union of $|\Delta|$ many rotated copies of the
self-similar attractor $T_{\alpha,S}$:

$X_{\alpha,S}=\bigcup_{\gamma\in\Delta}\gamma\cdot T_{\alpha,S},$

where $T_{\alpha,S}$ is generated by the IFS:

$\begin{cases}\psi_{0}(z)=\alpha z+c_{0},&\\\ \psi_{k}(z)=(\alpha
e^{i\theta_{k}})z+c_{k},\qquad k=1,2,\dots,m-1.\end{cases}$

###### Proof..

For a given generator set $S=\\{\theta_{0},\theta_{1},\dots,\theta_{m-1}\\}$,
set $\alpha_{k}=\alpha e^{i\theta_{k}}$ for $k=0,1,\dots,m-1$ in Lemma 3.1.
Note that $\alpha_{0}=\alpha$ since $\theta_{0}=0$. Then it follows from (3.2)
that the self-similar set $T_{\alpha,S}$ has the following representation.

$T_{\alpha,S}=\left\\{\sum_{n=1}^{\infty}c_{x_{n}}\prod_{k=0}^{m-1}(\alpha
e^{i\theta_{k}})^{I_{k}(x,n-1)}\>:\>x=(x_{n})\in\\{0,1,\dots,m-1\\}^{\mathrm{I\\!N}}\right\\}.$

Observe that $\sum_{k=0}^{m-1}{I_{k}(x,n-1)}=n-1$ and

$\prod_{k=0}^{m-1}e^{i\theta_{k}\cdot
I_{k}(x,n-1)}=e^{i\sum_{k=0}^{m-1}I_{k}(x,n-1)\cdot\theta_{k}}=e^{i\sum_{j=1}^{n-1}\theta_{x_{j}}},$

Thus,

$T_{\alpha,S}=\left\\{\sum_{n=1}^{\infty}\alpha^{n-1}c_{x_{n}}e^{i\sum_{j=1}^{n-1}\theta_{x_{j}}}\>:\>(x_{n})\in\\{0,1,\dots,m-1\\}^{\mathrm{I\\!N}}\right\\}.$
(3.4)

Define $X_{1,\alpha,S}$ given by:

$X_{1,\alpha,S}=\left\\{\sum_{n=1}^{\infty}\alpha^{n-1}s_{n}\gamma_{n}\>:\>\Sigma=(\gamma_{n})\in
W^{\Delta},\gamma_{1}=1,\>s_{n}=s_{\Sigma}(n)\right\\}.$

We prove that $X_{1,\alpha,S}=T_{\alpha,S}$. Let
$\gamma_{n}=e^{i\sum_{j=1}^{n-1}\theta_{x_{j}}}$. Then it is clear that there
exists a coding:

$\gamma_{1}=1,\qquad\gamma_{n+1}=\gamma_{n}\cdot e^{i\theta_{x_{n}}}$

for $n=1,2,\dots$. Therefore, $(\gamma_{n})$ satisfies the DRC with the
restriction that the first term is $1$.

Let $s_{n}=c_{x_{n}}$. It is clear that $(s_{n})$ is the constant sequence of
$(\gamma_{n})$.

Conversely, take a point $z\in X_{1,\alpha,S}$, which is generated by a
sequence $(\gamma_{n})_{n\in\mathrm{I\\!N}}$. Construct a sequence
$(x_{n})\in\\{0,1,\dots,m-1\\}^{\mathrm{I\\!N}}$ by

$x_{n}=k,\qquad\mbox{if }\gamma_{n+1}=\gamma_{n}e^{i\theta_{k}},$

for $k=0,1,2,\dots(m-1)$. Notice that $\theta_{k}$ is a unique revolving angle
for each $k$ so that $(x_{n})$ is determined uniquely by $(\gamma_{n})$.

Recall that for a given generator set $S$, the revolving group generated by
$S$ has $\mathrm{lcm}(|p_{1}|,\cdots,|p_{m-1}|)$ many elements where $p_{i}$
is the denominator of $\theta_{i}$ in reduced form. Thus,

$X_{\alpha,S}=\bigcup_{\gamma\in\Delta}\gamma\cdot
X_{1,\alpha,S}=\bigcup_{\gamma\in\Delta}\gamma\cdot T_{\alpha,S}.$

∎

Figure 3: Examples of $X_{\alpha,S}$:
$(m,c_{0},c_{1},c_{2},\alpha)=(3,0,\alpha,\bar{\alpha},\frac{1}{2}-\frac{\sqrt{3}i}{6})$.
$(\theta_{1},\theta_{2})=(\pi/3,-2\pi/3)$: a union of 3 fudgeflake (top),
$(\theta_{1},\theta_{2})=(\pi/2,-\pi/2)$: (bottom left),
$(\theta_{1},\theta_{2})=(\pi/3,-\pi/3)$: (bottom right).

######  Example 3.4.

Mandelbrot introduced a tiling fractal, known as the fudgeflake in his classic
book (see page 72 in [8]). The fudgeflake is a self-similar attractor
generated by three similar contractions:

$\begin{cases}\psi_{1}(z)=\alpha z,&\\\ \psi_{2}(z)=(\alpha
e^{i\theta_{1}})z+\alpha,&\\\ \psi_{3}(z)=(\alpha
e^{i\theta_{2}})z+\bar{\alpha},\end{cases}$ (3.5)

where $\alpha=\frac{1}{2}-\frac{\sqrt{3}i}{6}$, $\theta_{1}=\pi/3$ and
$\theta_{2}=-2\pi/3$. For more details. see page 22-23 in Edger’s book [2].

It is clear that Theorem 3.3 includes this particular case. Figure 3 shows
several examples of $X_{\alpha,S}$; in particular, the self-similar attractors
of (3.5).

Terdragon is another famous tiling self-similar attractor generated by (3.5)
if $\alpha=\frac{1}{2}-\frac{\sqrt{3}i}{6}$, $\theta_{1}=2\pi/3$ and
$\theta_{2}=0$.(See page 163 in [2]). However, notice that Theorem 3.3
excludes this case since $\theta_{2}=0$.

######  Example 3.5.

The Heighway dragon and the Twindragon are famous self-similar attractors, not
generated by (1.5) but by a slightly different pair of two similar
contractions:

$\begin{cases}\phi_{0}(z)=\alpha z,&\\\ \phi_{1}(z)=(\alpha
e^{i\theta})z+1.\end{cases}$ (3.6)

In particular, if $\alpha=(1+i)/2$ and $\theta=\pi/2$, then the IFS generates
the Heighway dragon. If $\alpha=(1+i)/2$ and $\theta=\pi$, then the Twindragon
is generated [2].

Let $S=\\{0,\theta\\},c_{0}=0$ and $c_{1}=1$ in Theorem 3.3. Since

$s_{n}=s_{\Sigma}(n)=\begin{cases}0,&\mbox{ if }\gamma_{n+1}=\gamma_{n},\\\
1,&\mbox{ if }\gamma_{n+1}=\gamma_{n}e^{i\theta},\end{cases}$

$X_{\alpha,S}$ can also be parametrized by generalized revolving sequences
$(\delta_{n})$.

$X_{\alpha,S}=\left\\{\sum_{n=0}^{\infty}\delta_{n}\alpha^{n}:(\delta_{n})\in
W_{\theta}\right\\}.$ (3.7)

Note that the only difference between (3.7) and (1.4) is whether the sum
begins from $n=0$ or $n=1$. It is not surprising. Let
$H_{\alpha,\theta}=\phi_{0}(H_{\alpha,\theta})\cup\phi_{1}(H_{\alpha,\theta})$
be the self-similar set generated by the IFS (3.6). Notice that
$\psi_{0}(z)=\alpha\phi_{0}(z/\alpha)$ and
$\psi_{1}(z)=\alpha\phi_{1}(z/\alpha)$ in (1.5) and (3.6). Thus, it is easy to
see that $H_{\alpha,\theta}=M_{\alpha,\theta}/\alpha$.

Figure 4: $X_{\alpha,S}$:
$(m,c_{0},c_{1},\alpha,\theta)=(2,0,1,\frac{1+i}{2},\frac{\pi}{2})$: a union
of 4 Heighway dragons

######  Remark 3.6.

Theorem 2.4 is a special case of Theorem 3.3. Let
$S=\\{0,\theta_{1}\\},c_{0}=0$ and $c_{1}=\alpha$ so that

$s_{n}=\begin{cases}0,&\mbox{ if }\gamma_{n+1}=\gamma_{n},\\\ \alpha,&\mbox{
if }\gamma_{n+1}=\gamma_{n}e^{i\theta_{1}}.\end{cases}$

Observe that $\delta_{n}=\alpha^{-1}\gamma_{n}s_{n}$ since (i) $\delta_{n}=0$
if $\gamma_{n+1}=\gamma_{n}$ and (ii) $\delta_{n}=\gamma_{n}$ if
$\gamma_{n+1}=\gamma_{n}e^{i\theta_{1}}$. Therefore,

$\left\\{\sum_{n=1}^{\infty}\alpha^{n-1}\gamma_{n}s_{n}\>:\>\Sigma=(\gamma_{n})\in
W^{\Delta},\>s_{n}=s_{\Sigma}(n)\right\\}=\left\\{\sum_{n=1}^{\infty}\delta_{n}\alpha^{n}:(\delta_{n})\in
W_{\theta}\right\\},$

where $W^{\Delta}$ is the set of all $\Delta$-revolving sequences, and
$W_{\theta}$ is the set of all generalized revolving sequences.

##  4 Remark

First, we define a slight modification of $\Delta$-revolving sequences as
follows.

######  Definition 4.1 ($\Delta_{0}$-Revolving Sequence).

Let $\Delta_{0}:=\Delta\cup\\{0\\}$. We say that a sequence
$(\delta_{n})\in\Delta_{0}^{\mathbb{N}}$ satisfies the $\Delta_{0}$-Revolving
Condition (DZRC) if

  1. 1.

$\delta_{1}$ is free to choose in $\Delta_{0}$

  2. 2.

If $\delta_{1}=\delta_{2}=\dots=\delta_{n}=0$, then $\delta_{n+1}$ is free to
choose in $\Delta_{0}$.

  3. 3.

Otherwise, $\delta_{n+1}=0$ or $\delta_{n+1}=\delta_{j_{0}(n)}e^{i\theta_{k}}$
for some $\theta_{k}\in S\backslash\\{0\\}$,

where $j_{0}(n)=max\\{j\leq n:\delta_{j}\neq 0\\}$.

######  Example 4.2.

If $S=\\{0,\pi,\frac{2\pi}{3}\\}$,
$\Delta_{0}=\\{0,1,e^{i(\pi)},e^{i(\frac{2\pi}{3})},e^{i(\frac{4\pi}{3})},e^{i(\pi+\frac{2\pi}{3})},e^{i(\pi+\frac{4\pi}{3})}\\}$,
and, for examples,

$(\delta_{n})=(1,e^{i\pi},0,e^{i(\pi+\frac{2\pi}{3})},0,0,e^{i(\frac{2\pi}{3})},\cdots)$
(4.1)

is a $\Delta_{0}$-revolving sequence.

We denote the set of all $\Delta_{0}$-revolving sequences with generator set
$S$ by $W^{\Delta_{0}}$.

Compare examples: (2.2) and (4.1). Observe that $\Delta_{0}$-revolving
sequences are a more natural extension of the notation of generalized
revolving sequences than $\Delta$-revolving sequences. In fact,
$\Delta_{0}$-revolving sequences are past-dependent.

Now, a question arises naturally: for a given $\alpha\in\mathbb{C}$ such that
$|\alpha|<1$ and a generator set $S$, define

$X^{*}_{\alpha,S}:=\left\\{\sum_{n=1}^{\infty}\delta_{n}\alpha^{n}:(\delta_{n})\in
W^{\Delta_{0}}\right\\}.$

It is not hard to imagine that $X^{*}_{\alpha,S}$ is a union of self-similar
sets; however, it is not immediately clear which IFS generates these self-
similar sets.

######  Corollary 4.3.

$X^{*}_{\alpha,S}$ is the union of $|\Delta|$ many rotated copies of the self-
similar attractor $T^{*}_{\alpha,S}$:

$X^{*}_{\alpha,S}=\bigcup_{\gamma\in\Delta}\gamma\cdot T^{*}_{\alpha,S},$

where $T^{*}_{\alpha,S}$ is generated by the IFS:

$\begin{cases}\psi_{0}(z)=\alpha z&\\\ \psi_{k}(z)=(\alpha
e^{i\theta_{k}})z+\alpha,\qquad k=1,2,\dots,m-1.\end{cases}$

######  Remark 4.4.

In general, self-similar attractors generated by

$\begin{cases}\psi_{0}(z)=\alpha z&\\\ \psi_{k}(z)=(\alpha
e^{i\theta_{k}})z+c_{k},\qquad k=1,2,\dots,m-1.\end{cases}$

cannot be parametrized by $\Delta_{0}$-revolving sequences. Consider (3.4) in
the proof of Theorem 3.3. Since $c_{0}=0$, let
$\delta_{n}=\mathrm{I}(x_{n})e^{i\sum_{j=1}^{n-1}\theta_{x_{j}}}$, where

$\mathrm{I}(x_{n})=\begin{cases}0,&\mbox{ if }x_{n}=0,\\\ 1,&\mbox{ if
}x_{n}\neq 0.\end{cases}$

Then it is clear that there exists a $\Delta_{0}$-revolving sequence
$(\delta_{n})$ given by

$\delta_{n}=\begin{cases}0,&\mbox{ if }x_{n}=0,\\\
\delta_{j_{0}(n-1)}e^{i\theta_{x_{j_{0}(n-1)}}},&\mbox{ if $j_{0}(n-1)$
exists},\\\ 1,&\mbox{ if $j_{0}(n-1)$ does not exist},\end{cases}$

where $j_{0}(n)=\max\\{j\leq n:x_{j}\neq 0\\}$. However, notice that there is
an issue to define $(x_{n})$ from $(\delta_{n})$, if $(\delta_{n})$ is a
sequence whose terms are eventually all $0$.

## Acknowledgment

The first author would like to express her gratitude to the late Prof. S. Ito,
who introduced one of his results in [10]. He inspired and encouraged many
female mathematicians to study problems related to fractal geometry and number
theory. Lastly, We greatly appreciate Prof. P. Allaart for his helpful
comments and suggestions in preparing this paper.

## References

  * [1] C. Davis and D. E. Knuth, Number representations and dragon curves I, II, J. Recreational Math., 3, pp. 66-81 (Part I), pp. 133-149 (Part II) (1970).
  * [2] G. Edgar, Measure, Topology, and Fractal Geometry, Springer-Verlag, (1990)
  * [3] K. J. Falconer, Fractal Geometry. Mathematical Foundations and Applications, 3rd Edition, Wiley (2014)
  * [4] W. J. Gilbert, Fractal geometry derived from complex bases, Math. Intelligencer, 4, pp. 78-86 (1982).
  * [5] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30, pp. 713-747 (1981).
  * [6] K. Kawamura, On the classification of self-similar sets determined by two contractions on the plane, J. Math. Kyoto Univ., 42, pp. 255-286 (2002).
  * [7] K. Kawamura and A. Allen, Revolving Fractals, J. Fractal Geom., 8, pp. 289-304 (2021)
  * [8] B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, (1982)
  * [9] T. Mathis and K. Kawamura, Revolving sequences and Terdragon, Preprint (2021) https://arxiv.org/abs/2106.10549
  * [10] M. Mizutani and S. Ito, Dynamical Systems on Dragon Domains, Japan J. Appl. Math., 4, pp. 23-46 (1987).


# Does diffusion mechanism favour the emergent scenario of the universe?

Subhayan<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata-700032, West Bengal, India.  Subenoy
<EMAIL_ADDRESS>Department of Mathematics, Jadavpur
University, Kolkata-700032, West Bengal, India.

###### Abstract

In the present work, the flat FLRW Universe has been modelled with cosmic
matter in the form of diffusive barotropic fluid. The diffusive fluid
undergoes dissipation due to diffusion mechanism in the form of cosmological
scalar field $\phi$. From the perspective of non-equilibrium thermodynamics,
the evolution equations of the universe have been formulated. By a suitable
choice of the cosmological scalar field, emergent scenario of the universe has
been obtained.

diffusive fluid, emergent scenario.

Diffusion can be considered as one of the basic macroscopic forces in nature.
Several physical and biological processes are caused due to diffusion. Some
well known examples of dynamical processes (in physics) are heat conduction,
Brownian motion and various transport phenomena [2, 3, 4, 1] in biological
systems where diffusion is the driving mechanism . The random collisions
between the particles of the system and those of the background is caused due
to diffusion mechanism at the microscopic level. On the other hand, random
effects are averaged at the macroscopic scale and diffusion is characterised
by heat equation or Fokker-Planck equation . Although there is a wide variety
of phenomena having diffusive behaviour, still there does not exist a
consistent diffusion theory in general relativity. However from cosmological
point of view, it is speculated that diffusion may have a basic role in the
evolution dynamics of the large scale structure formation of the universe.
Further, in standard cosmology, galaxies are assumed as point particles of a
fluid, undergoing velocity diffusion [2, 3, 4, 1, 5].

To consider diffusion in general relativity, one has to consider macroscopic
continuum description provided by the Fokker-planck equation. So, in diffusive
process, the energy - momentum tensor is not covariantly conserved (i.e.
$\nabla_{\mu}T^{\mu\nu}\neq 0$), rather it satisfies the Fokker - Planck
equation, namely [1, 5, 2, 3, 4]

$\nabla_{\mu}T^{\mu\nu}=3\sigma J^{\nu},$ (1)

where $\sigma(>0)$ is the diffusion constant and $J^{\nu}$ , the current
density of the matter satisfies

$\nabla_{\mu}J^{\mu}=0.$ (2)

Thus one can not have usual Einstein equations i.e.
$R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}=T_{\mu\nu}$ for diffusive process, due to
Bianchi identity. The simplest modification of the Einstein equation is to
introduce two interacting matter components of which one is the usual
diffusive fluid having conservation (non-conservation) equation given by
equation (1) while the simplest choice for the other component (in analogy
with cosmological constant) is a cosmological scalar field. so the modified
Einstein field equations take the form

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\phi g_{\mu\nu}=T_{\mu\nu},$ (3)

where the scalar field $\phi$ has the evolution equation [1, 2, 3, 4]
(dimension factor in $\phi$ has been chosen to be unity for convenience.)

$\nabla_{\mu}\phi=3\sigma J_{\mu},$ (4)

and $T_{\mu\nu}$ satisfies the above Fokker - Planck equation (given by
equation (1)).

Here $3\sigma$ measures the energy transferred from the scalar field to the
matter per unit time due to diffusion. Note that in vacuum or in the absence
of diffusion, the above modified Einstein field equations (3) become Einstein
equations with a cosmological constant while in general equation (3) may be
termed as Einstein equations with variable ‘cosmological constant’.

The above diffusion process is usually termed as kinetic model with
microscopic velocity of the fluid particles undergoing diffusion. Here the
diffusion mechanism takes place on the tangent bundle of the space time and as
a result Lorentz invariance of the space-time is preserved. Now choosing the
cosmic fluid as perfect fluid, one has the energy-momentum tensor

$\mathcal{T}_{\mu\nu}=\rho u_{\mu}u_{\nu}+p(g_{\mu\mu}+u_{\mu}u_{\nu})$ (5)

with current density $J^{\mu}=nu^{\mu}$. Here $n$ is the particle number
density of the fluid. $u^{\mu}$ is the four velocity of the fluid. $\rho$ and
$p$ are the energy density and thermodynamic pressure of the fluid
respectively. Now projecting equation (1) along the fluid $4$-velocity
$u^{\mu}$ and on the hyper surface orthogonal to $u^{\mu}$ , one gets [1, 2,
3, 4]

$\nabla_{\mu}(\rho u^{\mu})+p\nabla_{\mu}u^{\mu}=3\sigma.n$ (6)

and

$(p+\rho)u^{\mu}\nabla_{\mu}u^{\nu}+(u^{\mu}u^{\nu}+g^{\mu\nu})\nabla_{\mu}p=0,$
(7)

Here equation (7), the Euler equation does not change due to diffusion process
as diffusion force acts along the matter flow. It is to be noted that there
are several diffusion models in the literature namely for unification of dark
energy and dark matter from diffusive cosmology see ref. [6]. Ref. [7] deals
with transition between bouncing hyper-inflation to $\Lambda_{CDM}$ from
diffusive scalar fields while unified DE-DM with diffusive interactions and
interacting diffusive unified dark energy and dark matter from scalar fields
can be found in ref.[8] and [9] respectively. In particular, a Lagrangian
formulation of diffusion mechanism can be found in ref. [6].

In the background of homogeneous and isotropic flat FLRW model, the modified
Friedmann equations with diffusion dynamics take the form,

$3H^{2}=\rho+\phi$ (8)

and

$2\dot{H}=-(\rho+p)$ (9)

Now equation (3) for the present geometry simplifies to

$na^{3}(t)=\mbox{constant, i.e.~{}}n=n_{0}a^{-3}.$ (10)

Hence the modified matter conservation equation (1) for the matter field (5)
takes the form,

$\dot{\rho}+3H(p+\rho)=\sigma n_{0}a^{-3}=\sigma_{0}a^{-3}$ (11)

which on integration yields

$\rho=a^{-3(1+\omega)}\left[\rho_{0}+\int\limits_{t_{0}}^{t}\sigma_{0}a^{3\omega}dt\right].$
(12)

Here $\omega=\dfrac{p}{\rho}$, is the constant equation of state parameter of
the fluid. $\rho_{0}$ is the energy density at reference epoch of time
$t=t_{0}$ and $a(t_{0})=1$. $n(t_{0})=n_{0}$ is assumed. Now eliminating
$\rho$ between equations (8) and (9), one gets the cosmic evolution equation
as

$2\dot{H}+3(1+\omega)H^{2}=\phi(1+\omega)$ (13)

On the other hand, the above modified Friedmann equations (i.e. equations (8)
and (9)) for diffusive mechanism can be rewritten as,

$3H^{2}=\rho_{d}~{}\mbox{,}~{}2\dot{H}=-(\rho_{d}+p_{d}+\pi_{d})$ (14)

while the conservation equation(11) becomes

$\dot{\rho_{d}}+3H(\rho_{d}+p_{d}+\pi_{d})=0,$ (15)

with $\rho_{d}=\rho+\phi$, $p_{d}=p$ and $\pi_{d}=-\phi$ . Thus interacting
two fluid system in diffusion mechanism [10] is equivalent to a single
dissipative fluid in Einstein gravity. Here dissipation is chosen in the form
of bulk viscous pressure $\pi_{d}$. Further one may consider the above
dissipative pressure (i.e. bulk viscous pressure) due to non-equilibrium
thermodynamics with particle creation mechanism. In fact, for adiabatic
thermodynamic process, the dissipative pressure $\pi_{d}$ is related linearly
to the particle creation rate $\Gamma_{d}$ as [12, 11]

$\pi_{d}=-\frac{\Gamma_{d}}{3H}(\rho_{d}+p_{d}).$ (16)

Using the $1$st friedmann equation in (14) of equivalent Einstein gravity into
the above equation (16) with $p_{d}=p=\omega\rho$ and $\pi_{d}=-\phi$, the
cosmological scalar field is related to the particle creation rate as

$\Gamma_{d}=\frac{3H\phi}{3H^{2}(1+\omega)-\omega\phi}.$ (17)

Hence the present interacting diffusive mechanism [10] with cosmological
scalar field can be considered as non-equilibrium thermodynamic description of
Einstein gravity with particle creation formalism.

To overcome the classical singularity of Einstein gravity, cosmologists
propose two models namely the bouncing Universe or the emergent Universe. In
the present work, for non-singular solution we shall consider the model of
emergent scenario as it is very much relevant as pre-inflationary era. An
emergent Universe [13, 14, 15, 16, 12, 17, 18] is a modelled Universe with no
time like singularity having static Einstein era in the infinite past (i.e.
$t\rightarrow-\infty$). The present work examines whether emergent scenario is
possible or not in the present cosmological scalar field diffusion mechanism.
In order to have a cosmological solution one may choose phenomenologically the
form of $\phi$ as

$\phi=3\alpha H,$ (18)

with $\alpha$, a constant. Using this choice of $\phi$ in the field equations
(8),(9)and (11) one gets

$\sigma_{0}a^{-3}=-3\alpha\dot{H},$ (19)

which shows that $\alpha$ and $\sigma_{0}$ are of same sign (due to
$\dot{H}<0$).

For this choice of $\phi$, the solutions of the cosmic evolution equation (13)
yield the form of Hubble parameter and scale factor as,

(i) For $\alpha\geq H_{0}$ :

$H=\frac{\alpha}{1+\left(\frac{\alpha}{H_{0}}-1\right)e^{-\frac{3}{2}\alpha(1+\omega)(t-t_{0})}}~{}~{},$
(20)
$a=\left[\frac{\left(\frac{\alpha}{H_{0}}-1\right)+e^{\frac{3}{2}\alpha(1+\omega)(t-t_{0})}}{\left(\frac{\alpha}{H_{0}}-1\right)+1}\right]^{\frac{2}{3(1+\omega)}}$
(21)

(ii) For $0<\alpha<H_{0}$ :

$H=\frac{\alpha}{1-\left(1-\frac{\alpha}{H_{0}}\right)e^{-\frac{3}{2}\alpha(1+\omega)(t-t_{0})}}~{}~{},$
(22)

$a=\left[\frac{\left(1-\frac{\alpha}{H_{0}}\right)-e^{\frac{3}{2}\alpha(1+\omega)(t-t_{0})}}{\left(1-\frac{\alpha}{H_{0}}\right)-1}\right]^{\frac{2}{3(1+\omega)}}$
(23)

and

(iii) For $\alpha<0$ :

$\displaystyle
H=\frac{|\alpha|}{1-\left(\frac{|\alpha|}{H_{0}}+1\right)e^{-\frac{3}{2}\alpha(1+\omega)(t-t_{0})}}~{}~{}\mbox{,}~{}~{}~{}$
(24) $\displaystyle
a=\left[\frac{e^{\frac{3}{2}\alpha(1+\omega)(t-t_{0})}-\left(\frac{|\alpha|}{H_{0}}+1\right)}{1-\left(\frac{|\alpha|}{H_{0}}+1\right)}\right]^{\frac{2}{3(1+\omega)}}.$
(25)

Here $H_{0}$ is the value of $H$ at reference epoch of time $t_{0}$ .

For $\alpha<0$, the above cosmological solution (25) has a big-bang
singularity at the epoch,

$t_{s}=t_{0}-\frac{2}{3(1+\omega)|\alpha|}\ln\left(1+\frac{|\alpha|}{H_{0}}\right).$
(26)

Note that $\alpha<0$ (i.e. $\sigma_{0}<0$) is not physically realistic, so we
shall present the above solution for $\alpha<0$ only for mathematical
completeness.

Again for $0<\alpha<H_{0}$, big-rip singularity exists for the cosmological
solution (23) at the epoch,

$t_{s}=t_{0}+\frac{2}{3(1+\omega)\alpha}\ln\left[\left(1-\frac{\alpha}{H_{0}}\right)\right].$
(27)

In the case $H_{0}<\alpha$, the cosmological solution (21) has no singularity
at any real time.

Clearly, this solution [(20) , (21)] yields the Emergent scenario as it
follows the following criteria [12] :

$\displaystyle H$ $\displaystyle\rightarrow
0~{},~{}a\rightarrow\left[\frac{\alpha-
H_{0}}{\alpha}\right]^{\frac{2}{3(1+\omega)}}\mbox{when}~{}t\rightarrow-\infty$
(28a) $\displaystyle H$ $\displaystyle\rightarrow
0~{},~{}a\rightarrow\left[\frac{\alpha-
H_{0}}{\alpha}\right]^{\frac{2}{3(1+\omega)}}\mbox{when}~{}t<<t_{0}~{}\mbox{and}~{}$
(28b) $\displaystyle H$
$\displaystyle\sim\alpha~{},~{}a\simeq\left[\frac{H_{0}}{\alpha}\right]^{\frac{2}{3(1+\omega)}}\exp\left[{\alpha(t-t_{0})}\right]~{}~{}\mbox{when}~{}t>>t_{0}$
(28c)

So evidently the explicit solution for emergent scenario should be in the form
(also considering , $\alpha=H_{0}+\delta$ with $\delta\geq 0$) :

$\displaystyle H^{(E)}=\frac{(H_{0}+\delta)H_{0}}{H_{0}+\delta
e^{-\frac{3}{2}(H_{0}+\delta)(1+\omega)(t-t_{0})}}$ (29a)
$\displaystyle\mbox{and}~{}a^{(E)}=\left[\frac{\delta+H_{0}e^{\frac{3}{2}(H_{0}+\delta)(1+\omega)(t-t_{0})}}{H_{0}+\delta}\right]^{\frac{2}{3(1+\omega)}}$
(29b)

under the diffusive non-singular scalar field ,

$\phi=3(H_{0}+\delta)H.$ (30)

The nature of corresponding particle creation rate can be found from equation
(17) as,

$\Gamma_{d}=\frac{3H}{1-(1+\omega)(1-\frac{H}{\alpha})}.$ (31)

(a)

(b)

(c)

(d)

Figure 1:  Evolution of different physical parameters namely (a) Scale factor
$a$ (top left), (b) Hubble parameter $H$ (top right), (c) Cosmological scalar
field $\phi$ (bottom left), (d) Particle creation rate $\Gamma_{d}$ (bottom
right) for $\omega=-0.5$, $\alpha=0.9$, $t_{0}=1$ with three different values
of $H_{0}$ : $0.5$, $0.4$, $0.3$.

.

(a)

(b)

Figure 2:  Evolution of different thermodynamic parameters namely (a) Energy
density $\rho$ (left) and (b) Temperature $T$ (right) as a functions of time
$t$ and barotropic index of the fluid $\omega$ for $H_{0}=0.5,t_{0}=1$ with
$\delta=0.4$.

Further, one can write down the evolution equation (13) as the evolution of
Hubble parameter with the scale factor as

$\frac{dH}{da}+\frac{3}{2}(1+\omega)\frac{H}{a}=\frac{3}{2}\alpha(1+\omega)\frac{1}{a},$
(32)

which on integration gives

$H=\alpha-\delta(1+z)^{\frac{3}{2}(1+\omega)},$ (33)

where $z$ is the amount of cosmological red shift
$\left(z=\dfrac{1}{a}-1\right)$. Now introducing the dimensionless density
parameter,
$\Omega=\dfrac{\rho}{\rho_{c}}~{}\mbox{with}~{}\rho_{c}=\dfrac{3H^{2}}{8\pi
G},\mbox{the critical density}$, the above equation can be written as

$\frac{H^{2}}{H_{0}^{2}}=\Omega_{\Lambda_{0}}+\Omega_{M}(1+z)^{3(1+\omega)}+\Omega_{MP}(1+z)^{3(1+\omega_{MP})}$
(34)

where $\Omega_{\Lambda_{0}}=\left(1+{\dfrac{\delta}{H_{0}}}\right)^{2}$,
$\Omega_{M}=\left(\dfrac{\delta}{H_{0}}\right)^{2}$,
$\Omega_{MP}=2\dfrac{\delta}{H_{0}}\left(1+\dfrac{\delta}{H_{0}}\right)$ and
$\omega_{MP}=\left(\frac{\omega-1}{2}\right)$ with
$\Omega_{\Lambda_{0}}+\Omega_{M}+\Omega_{MP}=1$. From equation (34) one can
see that as $z\rightarrow-1$ i.e. $a\rightarrow\infty$, the present model
approaches $\Lambda_{CDM}$ model. The evolution of the instantaneous
equilibrium temperature of a system under non-equilibrium thermodynamic
prescription can be written as [11] ,

$\frac{\dot{T}}{T}+\omega\left(3H-\Gamma_{d}\right)=0.$ (35)

In the emergent scenario, one has (integrating equation (35))

$T=T_{0}(1+z)^{3\omega}e^{\beta\omega(t-t_{0})}$ (36)

where $T_{0}$ is the present measured value of temperature (at $t=t_{0}$) and
$\beta$ is a constant. So, equations (34) and (36) represent the Hubble
parameter and temperature respectively in terms of today’s measured value. The
time evolution of $a$, $H$, $\phi$ and $\Gamma_{d}$ has been exhibited
graphically in figure 1. Also the variation of thermodynamic parameters namely
energy density $\rho$ and temperature $T$ with time $(t)$ and with equation of
state parameter ( $\omega$) of the cosmic fluid have been shown in a $3d$ plot
in figure 2.

## Discussion

The present work is an attempt to examine whether emergent scenario of the
Universe is possible under diffusive process. Considering kinetic model of the
diffusion process, cosmological scalar field is chosen linearly to the Hubble
parameter to obtain emergent scenario of the cosmic evolution  and their
variations with respect to time and equation of state parameter have been
shown graphically ($3$d plot) in figure 2. Different thermodynamic parameters
like energy density and temperature also have been determined under emergent
scenario.

Further it has been established that such scalar field diffusion process
corresponds to the particle creation mechanism [12, 11] in the non-equilibrium
thermodynamic description. It is interesting to note that, for the non-
singular particle creation process, [see equation (31)] the barotropic index
of the fluid can be restricted to $\omega<0$ in the present scenario.  Also
for non-singular solution, the cosmological scalar field is chosen
phenomenologically as proportional to the Hubble parameter and the
proportionality constant is found to be positive.  Finally, this work
establishes that the dissipative processes like diffusion, particle creation
etc. may correspond to the evolution pattern of the universe as per the
present observation. For future works, it may be attempted to find the
Lagrangian formulation of such non-equilibrium thermodynamic phenomena to
study the microscopic behaviour of the universe.

## Acknowledgements

The author SM acknowledges UGC for awarding Research fellowship and SC thanks
Science and Engineering Research Board (SERB),India for awarding MATRICS
Research grant support (File no. MTR/2017/000407).

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