# Conservation and Integrability in TMG

M. R. Setare , S. N. Sajadi
Department of Science, Campus of Bijar, University of Kurdistan, Bijar, Iran
Electronic address<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
###### Abstract

In this work, following the paper by Romain Ruzziconi and Céline Zwikel [4] we
extend the questions of conservation, integrability and renormalization in
Bondi gauge and in GR to the theory of Topological Massive Gravity (TMG). We
construct the phase space and renormalize the divergences arising within the
symplectic structure through a holographic renormalization procedure. We show
that the charge expressions are generically finite, conserved and can be made
integrable by a field$-$dependent redefinition of the asymptotic symmetry
parameters.

##  1 Introduction

For the sake of simplicity the three dimensions space time is an arena for
studying various aspects of gravity. Maximally symmetric vacuum solutions of
Einstein field equations in three dimensions are characterized by their global
properties which encoded in the asymptotic charges. By specifying the boundary
conditions which form the asymptotic symmetries, the charges are computed. To
perform the boundary conditions and apply the further analyses, it is
customary to fix the gauge. Here, we used the Bondi gauge to study
asymptotically AdS spacetimes [1]. The analysis of asymptotically flat
spacetimes at null infinity led to the Bondi mass formula, which states that
due to the emission of gravitational waves the mass of the system decreases in
time [1]. The non$-$conservation of charges is an important part to explain
the dynamics of the system which is related to non$-$integrability of the
charges [2], [3]. Non$-$integrability is commonly considered as an unpleasant
property because it implies that the finite charge expressions rely on the
particular path that one chooses to integrate on the solution space, which is
a common feature of a dissipative system [4]. One way to solve the matter of
non$-$integrablity is to keep the full non$-$integrable expressions and try to
make sense of it. A vital technical result going into this direction is that
the Barnich$-$Troessaert bracket that enables one to derive mathematically
consistent charge algebras for non-integrable charges [5]. This bracket has
been employed in many different contexts and also the associated charge
algebras have been shown to be physically very relevant since they contain all
the information concerning the flux-balance laws of the theory [6]. Progress
in understanding the relation between non$-$conservation and
non$-$integrability of the charges has been made. In [7], it was conjectured
that no flux passing through the boundary is equivalent to the existence of a
specific section of the phase space for which the charges are integrable. A
natural however non$-$trivial question that we would like to explore in this
work is whether this conjecture is also applicable for asymptotic boundaries?
To get some insights into a attainable well$-$behaved quantum gravity,
countless number of works have been devoted to study the lower dimensional
models. Notably, studying cosmological Einstein’s theory in $3$-dimensions has
attracted large attentions. Although the $3D$ cosmological Einstein gravity
does not possess any local physical degree of freedom (dof), it has appealing
global properties. For example, it possesses the Banados-Teitelboim-Zanelli
(BTZ) black hole solutions [8] or $2D$ boundary CFT with the Brown$-$Henneaux
central charge ($c=3l/2G$) dual to the bulk $AdS_{3}$ spacetime [9]. Here, the
important purpose is to construct a $3D$ gravity theory that also possess a
local propagating dof as it has a viable boundary CFT. In this perspective,
the Topologically Massive Gravity (TMG) is particularly interesting because it
is a renormalizable theory and also describes a _local dynamical_ dof [10],
[11]. The so$-$called shortcoming in cosmological TMG, however, is that the
signs of energy of boundary graviton and mass of BTZ black hole are in
conflict with each others. For the cosmological TMG, the $AdS_{3}$ solution is
dual to a 2D CFT admitting two copies of Virasoro algebra.
When we were working on this paper, the paper [12] on the subject has been
published. In the mentioned paper the authors addressed the quetsion of
conservation and integrability of charges in Bondi gauge and in TMG. Unlike
our paper, they have perturbatively considered the field equations and
obtained an approximated solution.
The paper is organized as follows: in section 2, we apply the covariant phase
space methods on topological massive gravity in the Bondi gauge. We discuss
the solution space, renormalize the action principle and the symplectic
structure, obtain the corresponding finite surface charges associated with the
asymptotic symmetries. We conclude with some comments in section 3.

##  2 Phase space of TMG

Deser, Jackiw and Templeton constructed a $2+1$\- dimensional renormalizable
dynamical gravity theory with the help of the gravitational Chern$-$Simons
term [10]. The model is called Topologically Massive Gravity (TMG). The
Lagrangian density of TMG is presented as follows

$\displaystyle L_{TMG}=\dfrac{\sqrt{-g}}{16\pi
G}\left[\mathcal{R}-2\Lambda+\dfrac{1}{2\mu}\epsilon^{\lambda\mu\nu}\left(\Gamma^{\rho}_{\lambda\sigma}\partial_{\mu}\Gamma^{\sigma}_{\rho\nu}+\dfrac{2}{3}\Gamma^{\rho}_{\lambda\sigma}\Gamma^{\sigma}_{\mu\tau}\Gamma^{\tau}_{\nu\rho}\right)\right].$
(1)

Here $\epsilon^{\lambda\mu\nu}$ is a rank$-$3 tensor. This model describes a
topologically massive graviton possessing a single helicity mode and acquires
asymptotically AdS black hole solutions called Banados$-$Teitelboim-Zanelli
(BTZ) black holes [8]. Here we briefly review the phase space of TMG [25, 16].
By an infinitesimal variation of the above Lagrangian we obtain

$\delta L_{TMG}=\dfrac{\delta L_{TMG}}{\delta g_{\mu\nu}}\delta
g_{\mu\nu}+\partial_{\mu}\Theta^{\mu}_{TMG}$ (2)

Using the following Euler$-$Lagrange derivatives

$\displaystyle\dfrac{\delta L_{TMG}}{\delta
g_{\mu\nu}}=-\dfrac{\sqrt{-g}}{16\pi G}\left(G^{\mu\nu}+\Lambda
g^{\mu\nu}+\dfrac{1}{\mu}C^{\mu\nu}\right)$ (3)

one can find the TMG equations and the canonical TMG presymplectic potential
takes following form

$\displaystyle\Theta^{\mu}_{TMG}=\dfrac{\sqrt{-g}}{16\pi
G}\left(g^{\sigma\nu}\delta\Gamma^{\mu}_{\nu\sigma}-g^{\sigma\mu}\delta\Gamma^{\alpha}_{\alpha\sigma}+\dfrac{1}{2\mu}\left(\epsilon^{\lambda\mu\nu}\Gamma^{\rho}_{\lambda\sigma}\delta\Gamma^{\sigma}_{\nu\rho}+\epsilon^{\lambda\sigma\nu}R_{\nu\sigma}{}^{\rho\mu}\delta
g_{\lambda\rho}\right)\right)$ (4)

The TMG theory is invariant under diffeomorphisms which act on the metric with
a standard Lie derivative $\delta_{\xi}g_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$.
The weakly$-$vanishing Noether current $S^{\mu}_{\xi}$, is obtained as

$\dfrac{\delta L_{TMG}}{\delta
g_{\mu\nu}}\delta_{\xi}g_{\mu\nu}=\dfrac{\sqrt{-g}}{8\pi
G}\left(\nabla_{\mu}G^{\mu\nu}\xi_{\nu}+\dfrac{1}{\mathcal{\mu}}\nabla_{\mu}C^{\mu\nu}\xi_{\nu}\right)+\partial_{\mu}S^{\mu}_{\xi}$
(5)

The first and second terms in the right$-$hand side vanish due to the Bianchi
identities which correspond to the Noether identities of the theory. The total
derivative term gives $S^{\mu}_{\xi}$,

$S^{\mu}_{\xi}=2\dfrac{\delta L_{TMG}}{\delta
g_{\mu\nu}}=-\dfrac{\sqrt{-g}}{8\pi G}\left(G^{\mu\nu}+\Lambda
g^{\mu\nu}+\dfrac{1}{\mu}C^{\mu\nu}\right)\xi_{\nu}.$ (6)

The Barnich$-$Brandt codimension 2 form is given by

$k^{\mu\nu}_{BB,\xi}=\dfrac{1}{2}\delta\phi^{i}\dfrac{\delta}{\delta\phi^{i}_{\nu}}S^{\mu}_{\xi}+\left(\dfrac{2}{3}\partial_{\sigma}\delta\phi^{i}-\dfrac{1}{3}\delta\phi^{i}\partial_{\sigma}\right)\dfrac{\delta}{\delta\phi^{i}_{\nu\sigma}}S^{\mu}_{\xi}-(\mu\leftrightarrow\nu)$
(7)

this 2$-$form in our interesting case takes following form

$\displaystyle k_{BB,\xi}^{\mu\nu}=$ $\displaystyle
k^{\mu\nu}_{E}(\xi)+\epsilon^{\mu\nu\beta}{\cal
G}^{(l)}_{\alpha\beta}\bar{\xi}^{\alpha}+\epsilon^{\alpha\nu\beta}{\cal
G}^{(l)\mu}{}_{\beta}\bar{\xi}_{\alpha}+\epsilon^{\mu\alpha\beta}{\cal
G}^{\nu}{}_{\beta}\bar{\xi}_{\alpha}+k_{E}^{\mu\nu}(\epsilon\bar{\nabla}\bar{\xi})$
(8)

where

$\displaystyle k^{\mu\nu}_{E}(\xi)=$
$\displaystyle\bar{\xi}_{\alpha}\bar{\nabla}^{\mu}h^{\nu\alpha}-\bar{\xi}_{\alpha}\bar{\nabla}^{\nu}h^{\mu\alpha}+\bar{\xi}^{\mu}\bar{\nabla}^{\nu}h-\bar{\xi}^{\nu}\bar{\nabla}^{\mu}h+h^{\mu\alpha}\bar{\nabla}^{\nu}\bar{\xi}_{\nu}-h^{\nu\alpha}\bar{\nabla}^{\mu}\bar{\xi}_{\alpha}+\bar{\xi}^{\nu}\bar{\nabla}_{\alpha}h^{\mu\alpha}$
$\displaystyle-\bar{\xi}^{\mu}\bar{\nabla}_{\alpha}h^{\nu\alpha}+h\bar{\nabla}^{\mu}\bar{\xi}^{\nu},$
(9)

where

${\cal
G}^{(l)}_{\mu\nu}=R^{(l)}_{\mu\nu}-\dfrac{1}{2}\bar{g}_{\mu\nu}R^{(l)}-2\Lambda
h_{\mu\nu}\,,$ (10)

with

$\displaystyle R^{(l)}_{\mu\nu}$
$\displaystyle=\dfrac{1}{2}\left[-\bar{\nabla}^{2}h_{\mu\nu}-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}h+\bar{\nabla}_{\mu}\bar{\nabla}_{\sigma}h^{\sigma}_{\nu}+\bar{\nabla}_{\nu}\bar{\nabla}_{\sigma}h^{\sigma}_{\mu}\right]$
(11) $\displaystyle R^{(l)}$
$\displaystyle=-\bar{\nabla}^{2}h+\bar{\nabla}_{\rho}\bar{\nabla}_{\sigma}h^{\rho\sigma}-2\Lambda
h$ (12)

and $\delta g_{\mu\nu}=h_{\mu\nu}$ is a metric perturbation and
$h=g^{\mu\nu}h_{\mu\nu}$.

###  2.1 Bondi gauge in three dimensions

In the following, we will work in the Iyer$-$Wald approach that allows us to
renormalize the symplectic structure.

####  2.1.1 Solution space

Asymptotically AdS spacetimes solving field equation are of the form

$ds^{2}=\frac{V}{r}e^{2\beta}du^{2}-2e^{2\beta}dudr+r^{2}e^{2\varphi}\left(d\phi-
Udu\right)^{2},$ (13)

with coordinates $(x^{\mu})=(u,r,\phi)$. In the metric, $V$, $\beta$ and $U$
are functions of $(u,r,\phi)$ and $\varphi$ is a function of $(u,\phi)$. The
Bondi gauge is obtained by requiring the following three gauge$-$fixing
conditions

$g_{rr}=0,\;\;\;\;\;g_{r\phi}=0,\;\;\;\;\;g_{\phi\phi}=r^{2}e^{2\varphi}.$
(14)

Applying the TMG’s field equations determine the metric for $\Lambda
l^{2}=-1,\mu l\neq 1$ as follows:

$\displaystyle\beta(u,r,\phi)=$ $\displaystyle\beta_{0}(u,\phi)$
$\displaystyle U(u,r,\phi)=$ $\displaystyle
U_{0}(u,\phi)+\dfrac{U_{1}(u,\phi)}{r}+\dfrac{U_{2}(u,\phi)}{r^{2}}$
$\displaystyle V(u,r,\phi)=$
$\displaystyle-\dfrac{U_{2}^{2}(u,\phi)}{r}e^{2\varphi(u,\phi)-2\beta_{0}(u,\phi)}-2U_{1}(u,\phi)U_{2}(u,\phi)e^{2\varphi(u,\phi)-2\beta_{0}(u,\phi)}+$
$\displaystyle V_{0}(u,\phi)r+V_{1}(u,\phi)r^{2}+V_{2}(u,\phi)r^{3}$ (15)

where

$\displaystyle U_{1}(u,\phi)$ $\displaystyle=$ $\displaystyle
2e^{2\beta_{0}(u,\phi)-2\varphi(u,\phi)}\partial_{\phi}\beta_{0}(u,\phi),$
(16) $\displaystyle U_{2}(u,\phi)$ $\displaystyle=$
$\displaystyle-e^{2\beta_{0}(u,\phi)-2\varphi(u,\phi)}N(u,\phi),$ (17)
$\displaystyle V_{1}(u,\phi)$ $\displaystyle=$
$\displaystyle\dfrac{1}{3U_{2}e^{4\varphi+2\beta_{0}}+2\mu
r^{2}e^{4\beta_{0}+3\varphi}}\left[(12U_{2}\partial_{u}\varphi-12U_{2}\partial_{u}\beta_{0}+6\partial_{\phi}(U_{0}U_{2})\right.$
(18)
$\displaystyle\left.+12U_{2}U_{0}(\partial_{\phi}\varphi-\partial_{\phi}\beta_{0})+6\partial_{u}U_{2})e^{4\varphi+2\beta_{0}}-4\mu
r^{2}(\partial_{u}\varphi+U_{0}\partial_{\phi}\varphi+\partial_{\phi}U_{0})e^{4\beta_{0}+3\varphi}\right.$
$\displaystyle\left.+(6V_{0}\partial_{\phi}\beta_{0}+3\partial_{\phi}V_{0})e^{4\beta_{0}+2\varphi}+48\partial_{\phi}\beta_{0}e^{6\beta_{0}}(2(\partial_{\phi}\beta_{0})^{2}-\partial_{\phi}\varphi\partial_{\phi}\beta_{0}+\partial^{2}_{\phi^{2}}\beta_{0})\right]$
$\displaystyle V_{2}(u,\phi)$ $\displaystyle=$ $\displaystyle\dfrac{1}{2\mu
l^{2}r^{3}e^{4\beta_{0}+3\varphi}}\left[-2l^{2}e^{4\varphi+2\beta_{0}}[\partial_{\phi}(U_{2}U_{0})+2U_{2}U_{0}\partial_{\phi}\varphi+2U_{2}\partial_{u}\varphi-\right.$
(19)
$\displaystyle\left.2U_{2}U_{0}\partial_{\phi}\beta_{0}-2U_{2}\partial_{u}\beta_{0}+\partial_{u}U_{2}-\dfrac{1}{2}U_{2}V_{1}]-l^{2}e^{4\beta_{0}+2\varphi}[2V_{0}\partial_{\phi}\beta_{0}+\partial_{\phi}V_{0}]\right.$
$\displaystyle\left.-2\mu
r^{3}e^{6\beta_{0}+3\varphi}+8l^{2}\partial_{\phi}\beta_{0}e^{6\beta_{0}}[-2(\partial_{\phi}\beta_{0})^{2}+\partial_{\phi}\varphi\partial_{\phi}\beta_{0}-\partial^{2}_{\phi^{2}}\beta_{0}]\right]$

We also have two constrained field equations which determine the time
evolution of $N$ and $V_{0}$ (see appendix A). The expansion of the metric
around the AdS spacetime is

$\displaystyle\dfrac{V}{r}$ $\displaystyle=$
$\displaystyle\left[-\dfrac{r^{2}e^{2\beta_{0}}}{l^{2}}-2r\left(U_{0}\partial_{\phi}\varphi+\partial_{u}\varphi+\partial_{\phi}U_{0}\right)+V_{0}+\dfrac{4N\partial_{\phi}\beta_{0}e^{2\beta_{0}-2\varphi}}{r}-\dfrac{N^{2}e^{2\beta_{0}-2\varphi}}{r^{2}}\right]$
(20) $\displaystyle+\dfrac{2e^{-\varphi}}{\mu
r}\left[e^{2\beta_{0}-2\varphi}\left(4\partial_{\phi}\beta_{0}\partial^{2}_{\phi}\beta_{0}+8(\partial_{\phi}\beta_{0})^{3}-4\partial_{\phi}\varphi(\partial_{\phi}\beta_{0})^{2}\right)+V_{0}\partial_{\phi}\beta_{0}-NU_{0}\partial_{\phi}\varphi-\right.$
$\displaystyle\left.U_{0}\partial_{\phi}N-N\partial_{u}\varphi-2N\partial_{\phi}U_{0}-\partial_{u}N+\dfrac{1}{2}\partial_{\phi}V_{0}\right]+\mathcal{O}(\mu^{-2}).$

The expression inside the first bracket is the solution of Einstein gravity
[23], and the second bracket comes from the TMG. If the $\partial_{u}N$ equals
to the equation
(23)111$\partial_{u}N+N\partial_{u}\varphi=e^{2\beta_{0}-2\varphi}\left(4\partial_{\phi}\beta_{0}\partial^{2}_{\phi}\beta_{0}+8(\partial_{\phi}\beta_{0})^{3}-4\partial_{\phi}\varphi(\partial_{\phi}\beta_{0})^{2}\right)+V_{0}\partial_{\phi}\beta_{0}-NU_{0}\partial_{\phi}\varphi-
U_{0}\partial_{\phi}N-2N\partial_{\phi}U_{0}+\dfrac{1}{2}\partial_{\phi}V_{0}$
of [23], then the second bracket equals zero. Also, the Cotton tensor for our
solution does not equal zero. The solution space is paramatrized by five
functions which three of them $(\beta_{0},U_{0},\varphi)$ characterize the
induced boundary metric through

$\bar{\gamma}_{ab}dx^{a}dx^{b}=\lim_{r\rightarrow\infty}\left(\dfrac{1}{r^{2}}ds^{2}\right)=\left(-\dfrac{e^{4\beta_{0}(u,\phi)}}{l^{2}}+U_{0}^{2}e^{2\varphi(u,\phi)}\right)du^{2}-2U_{0}e^{2\varphi}dud\phi+e^{2\varphi}d\phi^{2}\;,$
(21)

where $(x^{a})=(u,\phi)$ are the coordinates on boundary. The other two
functions $V_{0}$ and $N$ encode the bulk information on the mass and the
angular momentum.

###  2.2 Residual symmetries

The residual gauge diffeomorphisms
$\xi=\xi^{u}\partial_{u}+\xi^{\phi}\partial_{\phi}+\xi^{r}\partial_{r}$
preserving the Bondi gauge fixing (14) have to satisfy the conditions

$\mathcal{L}_{\xi}g_{rr}=\mathcal{L}_{\xi}g_{r\phi}=0,\;\;\;\;g^{\phi\phi}\mathcal{L}_{\xi}g_{\phi\phi}=2w(u,\phi),$
(22)

which the explicit solutions of them are given by

$\displaystyle\xi^{u}=$ $\displaystyle f$ $\displaystyle\xi^{\phi}=$
$\displaystyle Y-\dfrac{1}{r}\partial_{\phi}fe^{2\beta_{0}-2\varphi}$
$\displaystyle\xi^{r}=$ $\displaystyle
r(\partial_{\phi}fU_{0}+w-\partial_{\phi}Y-f\partial_{u}\varphi-Y\partial_{\phi}\varphi)+\partial_{\phi}fU_{1}+\dfrac{\partial_{\phi}fU_{2}}{r}+$
$\displaystyle
e^{2\beta_{0}-2\varphi}\left(\partial^{2}_{\phi}f+2\partial_{\phi}\beta_{0}\partial_{\phi}f-\partial_{\phi}f\partial_{\phi}\varphi\right).$
(23)

So, the residual diffeomorphisms are encoded in $f,Y,w$ which arbitrary
functions of the boundary coordinates. Using the modified Lie bracket

$[\xi_{1},\xi_{2}]_{\ast}=[\xi_{1},\xi_{2}]-\delta_{\xi_{1}}\xi_{2}+\delta_{\xi_{2}}\xi_{1},$
(24)

these vector fields satisfy the commutation relations
$[\xi(f_{1},Y_{1},w_{1}),\xi(f_{2},Y_{2},w_{2})]_{\ast}=\xi(f_{12},Y_{12},w_{12})$
where

$\displaystyle f_{12}=$ $\displaystyle
f_{1}\partial_{u}f_{2}+Y_{1}\partial_{\phi}f_{2}-\delta_{\xi_{1}}f_{2}-(1\leftrightarrow
2)$ $\displaystyle Y_{12}=$ $\displaystyle
f_{1}\partial_{u}Y_{2}+Y_{1}\partial_{\phi}Y_{2}-\delta_{\xi_{1}}Y_{2}-(1\leftrightarrow
2)$ $\displaystyle w_{12}=$
$\displaystyle-\delta_{\xi_{1}}w_{2}-(1\leftrightarrow 2).$ (25)

Under the infinitesimal residual gauge diffeomorphisms
($\mathcal{L}_{\xi}g_{ab}=\delta_{\xi}g_{ab}$), the boundary structure
transforms as

$\displaystyle\mathcal{L}_{\xi}g_{\phi\phi}=\delta_{\xi}g_{\phi\phi},\;\;\;\rightarrow\;\;\;\delta_{\xi}\varphi=$
$\displaystyle w$
$\displaystyle\mathcal{L}_{\xi}g_{ur}=\delta_{\xi}g_{ur},\;\;\;\rightarrow\;\;\;\delta_{\xi}\beta_{0}=$
$\displaystyle(f\partial_{u}+Y\partial_{\phi})\beta_{0}+\left(\dfrac{1}{2}\partial_{u}-\dfrac{1}{2}\partial_{u}\varphi+U_{0}\partial_{\phi}\right)f-\dfrac{1}{2}\left(\partial_{\phi}Y+Y\partial_{\phi}\varphi-w\right)$
$\displaystyle\mathcal{L}_{\xi}g_{\phi u}=\delta_{\xi}g_{\phi
u},\;\;\;\rightarrow\;\;\;\delta_{\xi}U_{0}=$ $\displaystyle
f\partial_{u}U_{0}+Y\partial_{\phi}U_{0}-U_{0}\partial_{\phi}Y-\partial_{u}Y+\dfrac{1}{l^{2}}e^{4\beta_{0}-2\varphi}\partial_{\phi}f+U_{0}(\partial_{u}f+U_{0}\partial_{\phi}f)$
(26)

and the variations of the $N$ and $V_{0}$ can be found in [17].

In Bondi gauge, the conditions

$\beta_{0}=0,\;\;\;\;\;U_{0}=0,\;\;\;\;\;\varphi=0,$ (27)

corresponds to the Brown$-$Henneaux boundary condition, i.e. the induced
boundary metric (21) is flat. These lead to the additional conditions as
follows

$\partial_{u}f=\partial_{\phi}Y,\;\;\;\;\partial_{u}Y=\dfrac{1}{l^{2}}\partial_{\phi}f\;\;\;\;\;\;w=0.$
(28)

###  2.3 Renormalization of the phase space

By evaluating the radial component of the TMG presymplectic potential (4) on
the solution space, we obtain some $\mathcal{O}(r^{2})$, $\mathcal{O}(r)$
terms that diverge when $r\rightarrow\infty$. Also, there is
$\mathcal{O}(\ln(r))$ divergence which comes from the Chern$-$Simons part of
on$-$shell action. Furthermore, the presymplectic potential admits some
$\mathcal{O}(l^{2})$ terms that make an prevention to take the flat limit
$l\rightarrow\infty$. Therefore, to remove these divergences, we should add
some counter-terms which we show below the exact form of them. In
asymptoticaly $AdS_{3}$ spacetimes, the minimal action principle in Bondi
gauge that satisfies the mentioned requirements is given by [22],[4],[25]

$\displaystyle S=$ $\displaystyle\dfrac{1}{16\pi
G}\int_{M}\sqrt{-g}\left[\mathcal{R}-2\Lambda+\dfrac{1}{2\mu}\epsilon^{\lambda\mu\nu}\left(\Gamma^{\rho}_{\lambda\sigma}\partial_{\mu}\Gamma^{\sigma}_{\rho\nu}+\dfrac{2}{3}\Gamma^{\rho}_{\lambda\sigma}\Gamma^{\sigma}_{\mu\tau}\Gamma^{\tau}_{\nu\rho}\right)\right]d^{3}x+\int_{\partial
M}a_{1}L_{GHY}d^{2}x$ $\displaystyle+\int_{\partial
M}a_{2}L_{ct}d^{2}x+\int_{\partial M}a_{3}L_{0}d^{2}x+\int_{\partial
M}a_{4}L_{b}d^{2}x+\int_{\partial
M}a_{5}L_{\mathcal{R}}d^{2}x+\dfrac{1}{\mu}\int_{\partial
M}a_{6}L_{ccs}d^{2}x$ $\displaystyle+\dfrac{1}{\mu}\int_{\partial
M}a_{7}L_{lccs}\log(r)d^{2}x$ (29)

The first term is the TMG bulk action and the second term is the
Gibbons$-$Hawking boundary term as

$L_{GHY}=\dfrac{1}{8\pi G}\sqrt{-\gamma}\gamma^{\mu\nu}n_{\nu;\mu}$ (30)

where
$n_{\mu}=\dfrac{1}{\sqrt{g^{rr}}}\delta^{r}_{\mu}=\sqrt{\dfrac{re^{2\beta_{0}(u,\phi)}}{-V(u,r,\phi)}}\delta^{r}_{\mu}$
and deteminent of induced metric
$\gamma=\left.rVe^{2(\varphi+\beta)}\right|_{r=R}$. The induced metric is
$\gamma_{ab}$ and obtained by taking $r=R=constant$ as

$ds_{B}^{2}=\dfrac{V(u,\phi)}{R}e^{2\beta_{0}}du^{2}+R^{2}e^{2\varphi}\left(d\phi-U(u,\phi)du\right)^{2}.$
(31)

The third term is the counter$-$term as

$L_{ct}=-\dfrac{1}{8\pi Gl}\sqrt{-\gamma}=-\dfrac{1}{8\pi
Gl}e^{\varphi+\beta}\sqrt{-RV}.$ (32)

The fourth term is a corner Lagrangian

$L_{0}=-\dfrac{1}{8\pi
G}\sqrt{-\gamma}D_{a}v^{a},\;\;\;\;\;v^{a}=\dfrac{Re^{\varphi}}{\sqrt{-\gamma}}(\delta^{a}_{u}+U\delta^{a}_{\phi}).$
(33)

The vector field $v^{a}$ appeared in (33) is tangent to the leaves of the
foliation and satisfies the two properties[24], [25]

$v^{a}\gamma_{ab}v^{b}=-1,\;\;\;\;\;\lim_{r\rightarrow\infty}\left(\dfrac{1}{r}\gamma_{ab}v^{a}\right)=-\dfrac{e^{2\beta_{0}}}{l}\delta^{u}_{a}.$
(34)

The next term in (2.3) is a kinetic term for the vector ($v^{a}$),

$L_{b}=\dfrac{l}{16\pi G}\sqrt{-\gamma}\left(D_{a}v^{a}\right)^{2}.$ (35)

The next term in (2.3) is the Gauss$-$Bonnet term

$L_{\cal{R}}=-\dfrac{l}{16\pi G}\sqrt{-\gamma}\mathcal{R}[\gamma].$ (36)

Finally, the last terms in (2.3) is a term to remove the divergence correspond
to $\mathcal{O}(R)$ and $\log(R)$ that are coming from the Chern-Simons term
of TMG

$L_{ccs}=\sqrt{-\bar{\gamma}}\bar{\epsilon}^{\lambda\sigma}N_{\rho}\bar{\Gamma}^{\rho}_{\lambda\sigma},\;\;\;\;\;N_{\rho}=-\dfrac{lr}{2}e^{-\Phi(u,\phi)}\delta^{u}_{\rho}$
(37)
$L_{lccs}=\partial_{a}\bar{m}^{a},\;\;\;\;\;\bar{m}^{a}=\partial_{\phi}\beta_{0}(\delta^{a}_{u}+U_{0}\delta^{a}_{\phi})$
(38)

The explicit terms of the Lagrangians are given in appendix B. Now we provide
some details on how to obtain the values of the coefficients (2.3) of the
different terms in the action.

  * •

By evaluating the action (2.3) on$-$shell, divergences in $\mathcal{O}(R^{2})$
arise. The $R^{2}$\- divergences by imposing $2a_{1}-a_{2}=1$ are removed,
while the $R$-divergences are removed by $2a_{1}-a_{2}-a_{3}=0$ and $a_{6}=1$.

  * •

The $\ln(R)$-divergences are removed by imposing: $a_{7}=1$.

  * •

The action (2.3) also exhibits some terms in $\mathcal{O}(l^{2})$ which can be
eliminated by imposing $a_{5}=-a_{1}$ and $2a_{1}-a_{2}-a_{4}=0$.

  * •

By using the Dirichlet boundary conditions (27) and $a_{1}=1$, the action is
stationary on solutions space.

Using all the constraints, we obtain $a_{i}=1$. Sending the cut$-$off to
infinity, $R\rightarrow\infty$, the expression of the renormalized action
(2.3) is given by

$\displaystyle S_{ren}$ $\displaystyle=$ $\displaystyle\dfrac{1}{16\pi G}\int
d^{2}x\left(-V_{0}e^{\varphi}+4e^{2\beta_{0}-\varphi}(-\partial^{2}_{\phi^{2}}\beta_{0}+\partial_{\phi}\beta_{0}\partial_{\phi}\varphi-2(\partial_{\phi}\beta_{0})^{2})\right)-\Gamma_{bulk}(R_{0})\;,$
(39)

where $\Gamma_{bulk}$ is the finite contribution of the bulk action evaluated
on its lower bound.

###  2.4 Renormalization of the symplectic structure

By inserting the Bondi metric (13) into (4), the radial component of the
presymplectic potential gives some divergences. The counter$-$terms to remove
the $\mathcal{O}(r^{2})$ and $\mathcal{O}(l^{2})$ divergences are similar to
those used in (2.3) to renormalize the action. We define the renormalized
presymplectic potential as [4],[25]

$\displaystyle\Theta^{r}_{ren}=$ $\displaystyle\Theta^{r}_{TMG}+\delta
L_{GHY}+\delta L_{ct}+\delta L_{0}+\delta L_{b}+\delta
L_{\mathcal{R}}+\dfrac{1}{\mu}\delta L_{ccs}$
$\displaystyle-\partial_{a}\Theta^{a}_{0}-\dfrac{1}{2}r\partial_{a}\bar{\Theta}^{a}_{0}$
(40)

where $g$ is a solution, $\delta g$ a perturbation around it. The first line
is the presymplectic potential prescribed by the renormalized action (2.3). To
understand the second term in (2.4), we define the following quantities
induced on the boundary

$\bar{v}^{a}=\lim_{r\rightarrow\infty}(rv^{a})=le^{-2\beta_{0}}(\delta^{a}_{u}+U_{0}\delta^{a}_{\phi}),\;\;\;\;\;\bar{\gamma}_{ab}=\lim_{r\rightarrow\infty}\left(\dfrac{1}{r^{2}}\gamma_{ab}\right)$
(41)

and $\bar{\gamma}=det(\bar{\gamma}_{ab})$ and $\bar{D}_{a}$ the determinant
and the covariant derivative associated with the induced boundary metric
$\bar{\gamma}_{ab}$, respectively. The term $\bar{\Theta}^{a}_{0}$ is the
presymplectic potential of the boundary Lagrangian

$\bar{L}_{0}=-\dfrac{1}{8\pi
G}\sqrt{-\bar{\gamma}}\bar{D}_{a}\bar{v}^{a},\;\;\;\;\;\delta\bar{L}_{0}=\partial_{a}\bar{\Theta}^{a}_{0}=-\dfrac{1}{8\pi
G}\sqrt{-\bar{\gamma}}\bar{D}_{a}\delta\bar{v}^{a}$ (42)

where the variation is taken with respect to $\bar{v}^{a}$ by keeping the
boundary metric $\bar{\gamma}_{ab}$ fixed (i.e. $\bar{\gamma}_{ab}$ is seen as
a background). By using above setup one can obtain the presymplectic potential
and the associated presymplectic current which explicitly provided as equation
(C) and (C) in Appendix (C). When we impose the conditions (27), at leading
order this expression vanishes. Hence, the associated charges are conserved
and the variational principle (2.3) is stationary on solutions.

###  2.5 Integrability and charge algebra

In this section, we talk about the renormalized charges and present a
particular slicing of the phase space in which they are integrable. The
renormalized co$-$dimension 2 form can be derived using

$\partial_{a}k^{ra}{}_{ren,\xi}[g;\delta
g]=w^{r}{}_{ren}[g;\delta_{\xi}g,\delta g]$ (43)

where $w^{r}{}_{ren}[g;\delta_{1}g,\delta_{2}g]$ given in (C). We obtain the
infinitesimal charges by integration on $S^{1}_{\infty}$,

$\delta/Q_{\xi}=\int_{0}^{2\pi}d\phi\;k^{ur}{}_{ren,\xi},$ (44)

where $\delta/$ indicates that the charge is not integrable. The explicite
expression has been provided as equation (C) in Appendix (C).

By using the renormalization procedure (2.4), the charges are finite. The
charges (C) seem to be non$-$integrable. This obstruction for integrability
can be cured by performing field$-$dependent redefinitions of the symmetry
parameters. In our case, we do the redefinition

$\tilde{f}=fe^{2\beta_{0}-\varphi},\;\;\;\;\tilde{Y}=Y-U_{0}f,\;\;\;\;\tilde{w}=w$
(45)

where $\tilde{f}$, $\tilde{Y}$ and $\tilde{w}$ are taken to be
field$-$independent, i.e. $\delta\tilde{f}=\delta\tilde{Y}=\delta\tilde{w}=0$.
In terms of these parameters, the commutation relations (2.2) become
$[\xi(\bar{f}_{1},\bar{Y}_{1},\bar{w}_{1}),\xi(\bar{f}_{2},\bar{Y}_{2},\bar{w}_{2})]_{\ast}=\xi(\bar{f}_{12},\bar{Y}_{12},\bar{w}_{12})$
with

$\displaystyle\bar{w}_{12}$ $\displaystyle=0$ $\displaystyle\bar{f}_{12}$
$\displaystyle=\bar{Y}_{1}\partial_{\phi}\bar{f}_{2}+\bar{f}_{1}\partial_{\phi}\bar{Y}_{2}-(1\leftrightarrow
2)$ $\displaystyle\bar{Y}_{12}$
$\displaystyle=\bar{Y}_{1}\partial_{\phi}\bar{Y}_{2}+\dfrac{1}{l^{2}}\bar{f}_{1}\partial_{\phi}\bar{f}_{2}-(1\leftrightarrow
2).$ (46)

The redefinition (45) gives the charges (C) integrable. We have explicitly
$\delta/Q_{\xi}\equiv\delta Q_{\xi}$ with

$\displaystyle\delta Q_{\xi}$ $\displaystyle=\dfrac{1}{8\pi
G}\int_{0}^{2\pi}d\phi\left[\tilde{Y}\delta(Ne^{\varphi})+\tilde{f}\delta\left(\dfrac{V_{0}}{2}e^{2\varphi-2\beta_{0}}+4(\partial_{\phi}\beta_{0})^{2}-2\partial_{\phi}\beta_{0}\partial_{\phi}\varphi+\dfrac{1}{2}(\partial_{\phi}\varphi)^{2}+\partial^{2}_{\phi}(2\beta_{0}-\varphi)\right)\right]$
$\displaystyle+\dfrac{1}{4\pi
G\mu}[\tilde{Y}\delta(V_{0}e^{-2\beta_{0}+2\varphi}+7(\partial_{\phi}\beta_{0})^{2}-3\partial_{\phi}\varphi\partial_{\phi}\beta_{0}+(\varphi-2\beta_{0})\partial_{\phi}\beta_{0}\partial_{\phi}\varphi-2(\varphi-2\beta_{0})(\partial_{\phi}\beta_{0})^{2}+$
$\displaystyle(\varphi-\beta_{0})\partial_{\phi}^{2}(2\beta_{0}-\varphi)+2\partial^{2}_{\phi}\beta_{0})+\tilde{w}\delta(\partial_{\phi}\beta_{0}+\partial_{\phi}\beta_{0}(\varphi-2\beta_{0}))+\tilde{f}\delta(-\partial_{\phi}U_{0}\partial^{2}_{\phi}e^{-2\beta_{0}+\varphi}+\partial^{2}_{\phi}U_{0}(\varphi-\beta_{0})$
$\displaystyle-\varphi\partial_{\phi}U_{0}\partial_{\phi}\beta_{0})]$ (47)

Integrating the expression (2.5) and using conditions (28) gives the finite
charge as

$Q_{\xi}[g]=\dfrac{1}{16\pi
G}\int_{0}^{2\pi}d\phi\left[\tilde{Y}\tilde{N}+\tilde{f}\tilde{V_{0}}\right]$
(48)

where

$\displaystyle\tilde{N}=$ $\displaystyle
Ne^{\varphi}+\frac{2}{\mu}(V_{0}e^{-2\beta_{0}+2\varphi}+7(\partial_{\phi}\beta_{0})^{2}-3\partial_{\phi}\varphi\partial_{\phi}\beta_{0}+(\varphi-2\beta_{0})\partial_{\phi}\beta_{0}\partial_{\phi}\varphi-2(\varphi-2\beta_{0})(\partial_{\phi}\beta_{0})^{2}+$
$\displaystyle(\varphi-\beta_{0})\partial_{\phi}^{2}(2\beta_{0}-\varphi)+2\partial^{2}_{\phi}\beta_{0}),\;\;\;\;\tilde{V}_{0}=\dfrac{V_{0}}{2}e^{2\varphi-2\beta_{0}}+4(\partial_{\phi}\beta_{0})^{2}-2\partial_{\phi}\beta_{0}\partial_{\phi}\varphi+\dfrac{1}{2}(\partial_{\phi}\varphi)^{2}+$
$\displaystyle\partial^{2}_{\phi}(2\beta_{0}-\varphi)+\frac{2}{\mu}(-\partial_{\phi}U_{0}\partial^{2}_{\phi}e^{-2\beta_{0}+\varphi}+\partial^{2}_{\phi}U_{0}(\varphi-\beta_{0})-\varphi\partial_{\phi}U_{0}\partial_{\phi}\beta_{0}).$
(49)

As can be seen there are only two independent charges.
According to the theorem of the covariant phase space method, the algebra of
charges is the same as the algebra of symmetry generators up to central terms
[18]. By using the explicit form of the charges and also using Dirichlet
boundary conditions (27) and (28), one can compute the charge algebra as
follows [19], [20], [21]

$\delta_{\xi_{2}}Q_{\xi_{1}}\equiv\\{Q_{\xi_{1}},Q_{\xi_{2}}\\}=Q_{\left[\xi_{1},\xi_{2}\right]_{\ast}}+C_{\xi_{1},\xi_{2}},$
(50)

where $\left[\xi_{1},\xi_{2}\right]_{\ast}$ is given by (2.5) and
$C_{\xi_{1},\xi_{2}}$ is the central extension

$C_{\xi_{1},\xi_{2}}=\dfrac{1}{8\pi
G}\int_{0}^{2\pi}d\phi\left(\partial^{2}_{\phi}\tilde{f}_{1}\partial_{\phi}\tilde{Y}_{2}-\partial^{2}_{\phi}\tilde{f}_{2}\partial_{\phi}\tilde{Y}_{1}\right)+\dfrac{1}{16\pi
G\mu
l^{2}}\int_{0}^{2\pi}d\phi\left(\partial_{\phi}\tilde{f}_{2}\partial^{2}_{\phi}\tilde{f}_{1}-\partial_{\phi}\tilde{f}_{1}\partial^{2}_{\phi}\tilde{f}_{2}\right).$
(51)

We should mention that this is the first time that the central extension with
the gravitational anomaly of TMG appears for such weak boundary conditions we
have considered in this paper.
As can be seen from (51), there are two kinds of central extension, the first
term arises from the Einstien term and the other one from the CS term and
appears in the Virasoro part. Defining

$\tilde{f}=\dfrac{l}{2}\left(Y^{+}+Y^{-}\right),\hskip
28.45274pt\tilde{Y}=\dfrac{1}{2}\left(Y^{+}-Y^{-}\right)$ (52)

and using the decomposition in modes $Y^{\pm}=\Sigma Y_{m}^{\pm}l^{\pm}_{m}$,
$l^{\pm}_{m}=e^{\pm imx^{\pm}}$ and writing $L_{\pm}=Q_{\xi(l^{\pm}_{m})}$,
the charge algebra becomes

$i\\{L^{\pm}_{m},L^{\pm}_{n}\\}=(m-n)L^{\pm}_{m+n}-\dfrac{c^{\pm}}{12}m^{3}\delta_{m+n,0},\hskip
14.22636pt\\{L_{m}^{\pm},L_{n}^{\mp}\\}=0,$ (53)

where $c^{\pm}=\dfrac{3l}{2G}\pm\dfrac{3}{2\mu G}$ which is the
Brown$-$Henneaux central charges.

##  3 Conclusion

In this work, we construct the maximal asymptotic symmetry algebra that one
will get in topological massive 3D gravity by imposing partial gauge fixing on
the components of the metric and very mild falloffs. After a renormalization
procedure of the action and the symplectic structure involving covariant
counter$-$terms, we obtain finite charge expressions that we present
integrable through a field$-$dependent redefinition of the symmetry
parameters. Then, we find the appropriate field$-$dependent redefinition of
the parameters to exhibit the charges integrable. Finally, we have obtained
the charge algebra which involves the Brown$-$Henneaux central extension in
asymptotically AdS3 spacetimes. This is the first time that the central
extension with the gravitational anomaly of TMG appears for such weak boundary
conditions we have considered in this paper.

##  4 Acknowledgments

We would like to thank Romain Ruzziconi and Kostas Skenderis for discussions
and comments.

##  Appendix A Constrained field equations

The constrained components of field equations are presented as follows:

$\displaystyle E_{u\phi}=$ $\displaystyle\dfrac{1}{8\mu
r^{3}}[6r^{5}\partial_{r}U(u,r,\phi)e^{-6\beta_{0}+3\varphi}(2\partial_{r}U(u,r,\phi)+r\partial^{2}_{r}U(u,r,\phi))-4\mu
r^{4}e^{-4\beta_{0}+2\varphi}(3\partial_{r}U(u,r,\phi)$
$\displaystyle+r\partial^{2}_{r}U)(6r\partial_{r}V-3r^{2}\partial^{2}_{r}V-6V+r^{3}\partial^{3}_{r}V)e^{-4\beta_{0}+\varphi}-8\mu
r^{2}\partial_{\phi}\beta_{0}e^{-2\beta_{0}}]=0,$ (54)

and

$\displaystyle E_{r\phi}$ $\displaystyle=\dfrac{1}{8\mu
r^{3}}[-6r^{6}\partial_{r}Ue^{-6\beta_{0}+3\varphi}(U\partial^{2}_{r\phi}U+\partial^{2}_{ur}U+\partial_{r}U(-U\partial_{\phi}\beta_{0}+\partial_{u}\varphi+U\partial_{\phi}\varphi-2\partial_{u}\beta_{0}+2\partial_{\phi}U))$
$\displaystyle 4\mu
r^{5}e^{-4\beta_{0}+2\varphi}(U\partial_{r\phi}^{2}U+\partial^{2}_{ru}U+\partial_{r}U(-2U\partial_{\phi}\beta_{0}+3\partial_{u}\varphi-2\partial_{u}\beta_{0}+3U\partial_{\phi}\varphi))+e^{-2\beta_{0}-\varphi}(2\partial^{3}_{r\phi^{2}}V$
$\displaystyle-2r\partial^{2}_{r\phi}V(\partial_{\phi}\varphi-\partial_{\phi}\beta_{0})-6\partial^{2}_{\phi}V+6\partial_{\phi}V(\partial_{\phi}\varphi-\partial_{\phi}\beta_{0}))-4\mu
re^{-2\beta_{0}}(-r\partial^{2}_{r\phi}V-2r^{2}U\partial^{2}_{\phi}\beta_{0}-$
$\displaystyle
2r^{2}\partial^{2}_{u\phi}\beta_{0}+2\partial_{\phi}V+2\partial_{\phi}\beta_{0}(Ur^{2}\partial_{\phi}\varphi+r^{2}\partial_{u}\varphi+V))+re^{-4\beta_{0}+\varphi}(r(12r^{2}\partial_{\phi}U(\partial_{\phi}\varphi-\partial_{\phi}\beta_{0})+$
$\displaystyle
9\partial_{\phi}V+r^{2}U(\partial_{\phi}\beta_{0})^{2}+\partial_{\phi}\beta_{0}(-2Ur^{2}\partial_{\phi}\varphi+r^{2}\partial_{u}\beta_{0}-r^{2}\partial_{u}\varphi-4V)+r^{2}U(\partial_{\phi}\varphi)^{2}+\partial_{\phi}\varphi(4V-$
$\displaystyle
8r^{2}\partial_{u}\beta_{0}+8r^{2}\partial_{u}\varphi))\partial_{r}U-12\partial_{u}V-r\partial^{2}_{r\phi}V(\partial_{r}U-4U)+12r\partial^{2}_{r\phi}U(-Ur^{2}\partial_{\phi}\beta_{0}+Ur^{2}\partial_{\phi}\varphi-$
$\displaystyle-2r^{2}\partial_{u}\beta_{0}+2r^{2}\partial_{u}\varphi+2r^{2}\partial_{\phi}U+V)+4r^{3}\partial^{2}_{ru}U(\partial_{\phi}\varphi-\partial_{\phi}\beta_{0})-r^{2}U\partial^{3}_{\phi
r^{2}}V+r^{3}\partial_{r}U\partial^{2}_{u\phi}\varphi$
$\displaystyle+4r^{3}U\partial^{3}_{r\phi^{2}}U-r^{3}\partial^{2}_{u\phi}\beta_{0}\partial_{r}U+4r^{3}\partial^{2}_{\phi}U\partial_{r}U+2r^{2}V\partial^{2}_{\phi
r^{2}}U-2r^{2}\partial^{2}_{r}V(-U\partial_{\phi}\beta_{0}+U\partial_{\phi}\varphi+$
$\displaystyle\partial_{u}\varphi-\partial_{u}\beta_{0}+\partial_{\phi}U)+r^{2}\partial^{2}_{r}U(3\partial_{\phi}V+V(\partial_{\phi}\varphi-\partial_{\phi}\beta_{0}))+12UV\partial_{\phi}\beta_{0})].$
(55)

##  Appendix B Explicit terms of Lagrangian

Here we provided the explicit terms of Lagrangian (2.3).

$\displaystyle L_{bulk}=$
$\displaystyle-\dfrac{e^{2\beta_{0}+\varphi}R^{2}}{8\pi
Gl^{2}}+\dfrac{(2e^{2\beta_{0}}\partial_{\phi}\beta_{0}+(U_{0}\partial_{\phi}U_{0}+\partial_{u}U_{0})l^{2}e^{-2\beta_{0}+2\varphi})R}{32\pi
G\mu l^{2}}+\dfrac{1}{16\pi G\mu}(U_{0}\partial^{2}_{\phi^{2}}\beta_{0}$
$\displaystyle+\partial_{\phi}U_{0}\partial_{\phi}\beta_{0}+\partial^{2}_{u\phi}\beta_{0})\ln(R)+\mathcal{O}\left(\dfrac{1}{R}\right),$
(56)

$\displaystyle L_{GHY}=$
$\displaystyle-\dfrac{e^{2\beta_{0}+\varphi}R^{2}}{4\pi
Gl^{2}}-\dfrac{e^{\varphi}}{4\pi
G}\left(\partial_{\phi}U_{0}+\partial_{u}\varphi+U_{0}\partial_{\phi}\varphi\right)R+\dfrac{1}{8\pi
G}[e^{2\beta_{0}-\varphi}(4(\partial_{\phi}\beta_{0})^{2}+2\partial^{2}_{\phi}\beta_{0}-2\partial_{\phi}\beta_{0}\partial_{\phi}\varphi)$
$\displaystyle+l^{2}e^{2\beta_{0}-\varphi}(-U_{0}^{2}\partial^{2}_{\phi}\varphi-
U_{0}\partial^{2}_{\phi}U_{0}-2\partial^{2}_{u\phi}\varphi
U_{0}-\partial^{2}_{u\phi}U_{0}-\partial^{2}_{u}\varphi+2\partial_{\phi}\beta_{0}(U_{0}\partial_{u}\varphi+U_{0}\partial_{\phi}U_{0}+U_{0}^{2}\partial_{\phi}\varphi)$
$\displaystyle+\partial_{\phi}\varphi(2U_{0}\partial_{u}\beta_{0}-\partial_{u}U_{0}-U_{0}\partial_{\phi}U_{0})+2\partial_{u}\beta_{0}(\partial_{u}\varphi+\partial_{\phi}U_{0}))+V_{0}e^{\varphi}]+\mathcal{O}\left(\dfrac{1}{R}\right),$
(57)

$\displaystyle L_{ct}=$
$\displaystyle-\dfrac{e^{2\beta_{0}+\varphi}R^{2}}{8\pi
Gl^{2}}-\dfrac{e^{\varphi}}{8\pi
G}(\partial_{u}\varphi+U_{0}\partial_{\phi}\varphi+\partial_{\phi}U_{0})R+\dfrac{e^{\varphi-2\beta_{0}}}{16\pi
G}(l^{2}U_{0}^{2}(\partial_{\phi}\varphi)^{2}+2l^{2}U_{0}\partial_{\phi}\varphi(\partial_{u}\varphi+\partial_{\phi}U_{0})$
$\displaystyle+l^{2}(\partial_{\phi}\varphi)^{2}+2l^{2}\partial_{u}\varphi\partial_{\phi}U_{0}+l^{2}(\partial_{\phi}U_{0})^{2}+V_{0}e^{2\beta_{0}})+\mathcal{O}\left(\dfrac{1}{R}\right),$
(58)

$\displaystyle L_{0}=$
$\displaystyle\dfrac{e^{\varphi}(\partial_{u}\varphi+U_{0}\partial_{\phi}\varphi+\partial_{\phi}U_{0})R}{8\pi
G}+\dfrac{e^{2\beta_{0}-\varphi}(\partial^{2}_{\phi^{2}}\beta_{0}-\partial_{\phi}\varphi\partial_{\phi}\beta_{0}+2(\partial_{\phi}\beta_{0})^{2})}{4\pi
G}+\mathcal{O}\left(\dfrac{1}{R}\right),$ (59)

$\displaystyle L_{b}=$
$\displaystyle-\dfrac{l^{2}e^{\varphi-2\beta_{0}}\left(\partial_{u}\varphi+U_{0}\partial_{\phi}\varphi+\partial_{\phi}U_{0}\right)^{2}}{16\pi
G}+\mathcal{O}\left(\sqrt{\dfrac{1}{R}}\right),$ (60)

$\displaystyle L_{\mathcal{R}}=$ $\displaystyle\dfrac{1}{8\pi
G}[e^{2\beta_{0}-\varphi}(-4(\partial_{\phi}\beta_{0})^{2}+2\partial_{\phi}\beta_{0}\partial_{\phi}\varphi-2\partial^{2}_{\phi}\beta_{0})+\dfrac{e^{-2\beta_{0}+\varphi}l^{2}}{8\pi
G}(U_{0}^{2}\partial^{2}_{\phi}\varphi+U_{0}\partial^{2}_{\phi}U_{0}+2U_{0}\partial^{2}_{u\phi}\varphi$
$\displaystyle+\partial^{2}_{u\phi}U_{0}+\partial^{2}_{u}\varphi+U_{0}^{2}(\partial_{\phi}\varphi)^{2}+\partial_{\phi}\varphi(-2U_{0}\partial_{u}\beta_{0}+\partial_{u}U_{0}+3U_{0}\partial_{\phi}U_{0}+2U_{0}\partial_{u}\varphi-2\partial_{\phi}\beta_{0}U_{0}^{2})$
$\displaystyle-2\partial_{\phi}\beta_{0}U_{0}(\partial_{u}\varphi+\partial_{\phi}U_{0})+(\partial_{\phi}U_{0})^{2}+2\partial_{\phi}U_{0}(\partial_{u}\varphi-\partial_{u}\beta_{0})+(\partial_{u}\varphi)^{2}-2\partial_{u}\beta_{0}\partial_{u}\varphi)]+\mathcal{O}\left(\dfrac{1}{R^{2}}\right),$
(61)

$L_{ccs}=2e^{2\beta_{0}}\partial_{\phi}\beta_{0}+l^{2}e^{2\varphi-2\beta_{0}}\left(U_{0}\partial_{\phi}U_{0}+\partial_{u}U_{0}\right)$
(62)
$L_{lccs}=\partial^{2}_{u\phi}\beta_{0}+\partial_{\phi}\beta_{0}\partial_{\phi}U_{0}+U_{0}\partial^{2}_{\phi^{2}}\beta_{0}$
(63)

The boundary Lagrangian (42) is given as

$\displaystyle\bar{L}_{0}=$ $\displaystyle\dfrac{e^{-4\beta_{0}-\varphi}}{8\pi
G}(l^{2}e^{2\varphi}((U_{0}+U^{3}_{0})\partial_{\phi}\varphi+(1+3U^{2}_{0})\partial_{\phi}U_{0}-4(U_{0}+U^{3}_{0})\partial_{\phi}\beta_{0}-4(1+U^{2}_{0})\partial_{u}\beta_{0}$
$\displaystyle+\partial_{u}U^{2}_{0}+\partial_{u}\varphi+U^{2}_{0}\partial_{u}\varphi)+e^{4\beta_{0}}(U_{0}\partial_{\phi}\varphi-\partial_{\phi}U_{0})).$
(64)

##  Appendix C Explicit terms of potentials

The renormalized presymplectic potential as

$\displaystyle\Theta^{r}_{ren}(g,\delta g)$ $\displaystyle=\dfrac{1}{16\pi
G}[V_{0}e^{\varphi}\delta(\varphi-2\beta_{0})+2Ne^{\varphi}\delta
U_{0}+2e^{2\beta_{0}-\varphi}(6\partial_{\phi}\beta_{0}\partial_{\phi}\delta\beta_{0}-\partial_{\phi}\varphi\partial_{\phi}\delta\beta_{0}+\partial^{2}_{\phi}\delta\beta_{0})]$
$\displaystyle+\dfrac{1}{64\pi\mu G}[-2V_{0}e^{2\varphi-2\beta_{0}}\delta
U_{0}-2U_{0}\partial^{2}_{\phi}\delta\beta_{0}+2U_{0}\partial^{2}_{\phi}\delta\varphi-2\partial^{2}_{u\phi}\delta\beta_{0}+2\partial^{2}_{\phi}\delta
U_{0}+2\partial^{2}_{u\phi}\delta\varphi-$
$\displaystyle\partial_{\phi}\delta\beta_{0}(-4U_{0}\partial_{\phi}\varphi-4\partial_{u}\varphi-2\partial_{\phi}U_{0})+\partial^{2}_{\phi}\beta_{0}(8U_{0}\delta\varphi-8U_{0}\delta\beta_{0}-6\delta
U_{0})+\partial^{2}_{\phi}\varphi(-4U_{0}\delta\varphi$
$\displaystyle+4U_{0}\delta\beta_{0}+2\delta
U_{0})+\partial^{2}_{\phi}\delta\varphi(4U_{0}\partial_{\phi}\beta_{0}-2\partial_{u}\varphi+2\partial_{\phi}U_{0})+8\partial^{2}_{u\phi}\beta_{0}\delta(\varphi-\beta_{0})-4\partial^{2}_{\phi}U_{0}\delta(\varphi-\beta_{0})$
$\displaystyle-4\partial^{2}_{u\phi}\varphi\delta(\varphi-\beta_{0})+\partial_{u}\delta\varphi(2\partial_{\phi}\varphi+4\partial_{\phi}\beta_{0})+\partial_{\phi}\varphi\partial_{\phi}\delta
U_{0}+(\partial_{\phi}\varphi)^{2}(-2U_{0}\delta\varphi+2U_{0}\delta\beta_{0}+\delta
U_{0})$
$\displaystyle+\partial_{\phi}\varphi(\delta\beta_{0}(-4U_{0}\partial_{\phi}\beta_{0}+2\partial_{u}\varphi+6\partial_{\phi}U_{0})+\delta\varphi(4U_{0}\partial_{\phi}\beta_{0}-2\partial_{u}\varphi-6\partial_{\phi}U_{0})+2\partial_{\phi}\beta_{0}\delta
U_{0})-$ $\displaystyle
4\partial_{\phi}\beta_{0}\delta\beta_{0}(\partial_{u}\varphi+2\partial_{\phi}U_{0})+4\partial_{\phi}\beta_{0}\delta\varphi(\partial_{u}\varphi+2\partial_{\phi}U_{0})-16\delta
U_{0}(\partial_{\phi}\beta_{0})^{2}]+\mathcal{O}\left(r^{-1}\right),$ (65)

the associated presymplectic current reads as

$\displaystyle w^{r}_{ren}(g,\delta_{1}g,\delta_{2}g)$
$\displaystyle=\dfrac{1}{16\pi
G}[\delta_{2}(V_{0}e^{\varphi})\delta_{1}(\varphi-2\beta_{0})+2\delta_{2}(Ne^{\varphi})\delta_{1}U_{0}-2e^{2\beta_{0}-\varphi}\partial_{\phi}\delta_{2}\varphi\partial_{\phi}\delta_{1}\beta_{0}+$
$\displaystyle
2e^{2\beta_{0}-\varphi}\delta_{2}(2\beta_{0}-\varphi)(6\partial_{\phi}\beta_{0}\partial_{\phi}\delta_{1}\beta_{0}-\partial_{\phi}\varphi\partial_{\phi}\delta_{1}\beta_{0}+\partial^{2}_{\phi}\delta_{1}\beta_{0})]+$
$\displaystyle\dfrac{1}{64\pi
G\mu}[-2\delta_{2}(V_{0}e^{2\varphi-2\beta_{0}})\delta_{1}U_{0}-2\delta_{2}U_{0}\partial^{2}_{\phi}\delta_{1}\beta_{0}+2\delta_{2}U_{0}\partial^{2}_{\phi}\delta_{1}\varphi+$
$\displaystyle(-4\delta_{2}(U_{0}\partial_{\phi}\varphi)-4\partial_{u}\delta_{2}\varphi-2\partial_{\phi}\delta_{2}U_{0})\partial_{\phi}\delta_{1}\beta_{0}+8\delta_{2}(U_{0}\partial^{2}_{\phi}\beta_{0})\delta_{1}\varphi-8\delta_{2}(U_{0}\partial^{2}_{\phi}\beta_{0})\delta_{1}\beta_{0}$
$\displaystyle-6\partial^{2}_{\phi}\delta_{2}\beta_{0}\delta_{1}U_{0}+4\delta_{2}(U_{0}\partial^{2}_{\phi}\varphi)\delta_{1}\varphi+4\delta_{2}U_{0}\delta_{1}\beta_{0}+\partial^{2}_{\phi}\delta_{1}\varphi(4\delta_{2}(U_{0}\partial_{\phi}\beta_{0})-2\delta_{2}(\partial_{u}\varphi)$
$\displaystyle+2\delta_{2}(\partial_{\phi}U_{0}))+8\delta_{2}(\partial^{2}_{u\phi}\beta_{0})\delta_{1}(\varphi-\beta_{0})-4\delta_{2}(\partial^{2}_{\phi}U_{0})\delta_{1}(\varphi-\beta_{0})-4\delta_{2}(\partial^{2}_{u\phi}\varphi)\delta_{1}(\varphi-\beta_{0})$
$\displaystyle+\partial_{u}\delta_{1}\varphi(2\delta_{2}(\partial_{\phi}\varphi)+4\delta_{2}(\partial_{\phi}\beta_{0}))+\partial_{\phi}\delta_{2}\varphi\partial_{\phi}\delta_{1}U_{0}-2\delta_{2}(U_{0}(\partial_{\phi}\varphi)^{2})\delta_{1}\varphi+2\delta_{2}(U_{0}(\partial_{\phi}\varphi)^{2})\delta_{1}\beta_{0}$
$\displaystyle+\delta_{2}((\partial_{\phi}\varphi)^{2})\delta_{1}U_{0}+\partial_{\phi}\delta_{2}\varphi\delta_{1}\beta_{0}(-4U_{0}\partial_{\phi}\beta_{0}+2\partial_{u}\varphi+6\partial_{\phi}U_{0})+\partial_{\phi}\varphi\delta_{1}\beta_{0}(-4\delta_{2}U_{0}\partial_{\phi}\beta_{0})$
$\displaystyle+2\partial_{u}\delta_{2}\varphi+6\partial_{\phi}\delta_{2}U_{0})+\delta_{1}\varphi(4\delta_{2}(U_{0}\partial_{\phi}\beta_{0})-2\partial_{u}\delta_{2}\varphi-6\partial_{\phi}\delta_{2}U_{0})+2\partial_{\phi}\delta_{2}\beta_{0}\delta_{1}U_{0})-$
$\displaystyle
4\partial_{\phi}\beta_{0}\delta_{1}\beta_{0}(\partial_{u}\delta_{2}\varphi+2\partial_{\phi}\delta_{2}U_{0})+4\partial_{\phi}\delta_{2}\beta_{0}\delta_{1}\varphi(\partial_{u}\varphi+2\partial_{\phi}U_{0})+4\partial_{\phi}\beta_{0}\delta_{1}\varphi(\partial_{u}\delta_{2}\varphi+2\partial_{\phi}\delta_{2}U_{0})$
$\displaystyle-16\delta_{1}U_{0}\delta_{2}((\partial_{\phi}\beta_{0})^{2})]-(1\leftrightarrow
2)+\mathcal{O}(r^{-1}).$ (66)

The explicit expression for the infinitesimal charges (44) reads as

$\displaystyle\delta/Q_{\xi}[g]$ $\displaystyle=\int_{0}^{2\pi}d\phi
k^{ur}_{ren,\xi}[g;\delta g]=\dfrac{1}{8\pi
G}\int_{0}^{2\pi}d\phi[Y\delta(Ne^{\varphi})+\partial_{\phi}(e^{2\beta_{0}-\varphi}\partial_{\phi}f)\delta(\beta_{0}-\varphi)+f(\dfrac{1}{2}\delta
V_{0}e^{\varphi}$ $\displaystyle-
V_{0}e^{\varphi}\delta(\beta_{0}-\varphi)-U_{0}\delta(Ne^{\varphi})+2e^{2\beta_{0}-\varphi}(6\partial_{\phi}\beta_{0}\partial_{\phi}\delta\beta_{0}-\partial_{\phi}\varphi\partial_{\phi}\delta\beta_{0}+\partial^{2}_{\phi}\delta\beta_{0}))]+\dfrac{1}{4\pi
G\mu}[$
$\displaystyle(Y-fU_{0})\partial^{2}_{\phi}\delta\beta_{0}+((7Y-6fU_{0})\partial_{\phi}\beta_{0}+(fU_{0}-\dfrac{3}{2}Y)\partial_{\phi}\varphi+\dfrac{1}{2}(\partial_{u}f-\partial_{\phi}Y-w+2U_{0}\partial_{\phi}f$
$\displaystyle+2\partial_{u}\beta_{0}f-f\partial_{u}\varphi))\partial^{2}_{\phi}\delta\beta_{0}+(Y-2fU_{0})\partial^{2}_{\phi}\beta_{0}-\dfrac{1}{2}(Y-2fU_{0})\partial^{2}_{\phi}\varphi-f\partial^{2}_{\phi
u}\beta_{0}+f\partial^{2}_{\phi}U_{0}$
$\displaystyle+\dfrac{f}{2}\partial^{2}_{\phi
u}\varphi-\dfrac{1}{2}\partial^{2}_{\phi}Y-\dfrac{1}{2}\partial^{2}_{u\phi}f+(2Y(\partial_{\phi}\beta_{0})^{2}-\partial_{\phi}\beta_{0}(2\partial_{\phi}Y-2f\partial_{\phi}U_{0}+Y\partial_{\phi}\varphi-\partial_{u}f+f\partial_{u}\varphi$
$\displaystyle-2f\partial_{u}\beta_{0}-4U_{0}\partial_{\phi}f+w)-\partial_{\phi}\varphi(U_{0}\partial_{\phi}f-\dfrac{1}{2}\partial_{\phi}Y+f\partial_{\phi}U_{0})-\partial_{\phi}f\partial_{\phi}U_{0}-\dfrac{1}{2}\partial_{\phi}w-\dfrac{1}{2}\partial_{\phi}f\partial_{u}\varphi$
$\displaystyle+\partial_{\phi}f\partial_{u}\beta_{0})\delta\beta_{0}+\dfrac{\delta\varphi}{2}(-2Y(\partial_{\phi}\beta_{0})^{2}+\partial_{\phi}\beta_{0}(-2\partial_{u}\beta_{0}f-4f\partial_{\phi}U_{0}+Y\partial_{\phi}\varphi+3\partial_{\phi}Y+w+f\partial_{u}\varphi$
$\displaystyle-6U_{0}\partial_{\phi}f-\partial_{u}f)+\partial_{\phi}\varphi(2U_{0}\partial_{\phi}f-\partial_{\phi}Y+2f\partial_{\phi}U_{0})+2\partial_{\phi}f\partial_{\phi}U_{0}+\partial_{\phi}w+\partial_{\phi}f\partial_{u}\varphi-2\partial_{\phi}f\partial_{u}\beta_{0})$
$\displaystyle+(-V_{0}\delta\beta_{0}+\dfrac{1}{2}\delta
V_{0}+V_{0}\delta\varphi)(Y-fU_{0})e^{2\varphi-2\beta_{0}}].$ (67)

## References

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# Closed-form expressions for the pure time delay in terms of the input and
output Laguerre spectra

Alexander Medvedev<EMAIL_ADDRESS>Information Technology, Uppsala
University, Uppsala, SWEDEN

###### Abstract

The pure time delay operator is considered in continuous and discrete time
under the assumption of the input signal being integrable (summable) with
square. By making use of a discrete convolution operator with polynomial
Markov parameters, a common framework for handling the continuous and discrete
case is set. Closed-form expressions for the delay value are derived in terms
of the Laguerre spectra of the output and input signals. The expressions hold
for any feasible value of the Laguerre parameter and can be utilized for e.g.
building time-delay estimators that allow for non-persistent input. A
simulation example is provided to illustrate the principle of Laguerre-domain
time delay modeling and analysis.

###### keywords:

delay systems, infinite-dimensional systems, time-invariant systems

††thanks: This paper was not presented at any IFAC meeting.

##  1 Introduction

Describing properties of signals and systems as functions of time is a natural
option since time domain is the most common and easily interpretable
observation framework. Yet, it does not necessarily yield the best setup for
system analysis, estimation, and design when the involved signals are known to
possess a certain property. For instance, continuous periodic signals are
efficiently represented by Fourier series which description has led to the
development of frequency-domain methods [18]. Besides introducing parsimonious
signal representations, the use of frequency domain has simplified the
mathematical tools for analysis of dynamical system by substituting time-
domain convolution operators with multiplication of the corresponding Fourier
transforms.

Ubiquitous in systems theory signals integrable with square have suggested the
use of Laguerre functions [5] in their modeling. In [15], the role of Laguerre
functions as a versatile tool for studying properties of discrete linear time-
invariant systems was convincingly demonstrated. In fact, the Laguerre shift
operator applied to generate a Laguerre (functional) basis [11] is, in a well-
defined sense, equivalent to the forward shift ($q$) operator and can be
generally employed for describing signals and systems in both continuous and
discrete time. As argued in [13], the Laguerre shift is one of the bilinear
operators, along with $\delta$-operator, $\gamma$-operator, Tustun’s operator,
proposed in systems theory, primarily to gain beneficial numerical properties
of the manipulated objects and unify continuous and discrete frameworks.

A promising but underdeveloped application area of Laguerre functions is time-
delay systems. In contrast with the solid results available regarding
approximation of time-delay systems by means of orthogonal functional bases,
see e.g. [11], [10], publications on time-delay systems in Laguerre domain,
i.e. when the involved signals are represented by their Laguerre spectra, are
scarce. References to relevant papers along this avenue of research that
originates from [6] are provided in the next section.

Tme-of-flight estimation of signals with finite energy (pulses) that
constitutes the core of radar, sonar, ultrasound, and lidar technology [14] is
essentially the estimation of the delay between an emitted and reflected
pulse. Even though delay estimation methods based on Laguerre functions were
numerically benchmarked against conventional techniques in [3], lack of system
theoretical grounds still hinders their practical applications.

This paper focuses on the mathematical description of the time-delay operator
in Laguerre domain. The main contributions of this work are as stated below.

  * •

A common framework for describing pure continuous and discrete time-delay
operators in Laguerre domain in the form of a convolution operator with
polynomial Markov parameters is introduced. It enables analysis, design, and
estimation approaches uniformly applicable to both continuous and discrete
delay models.

  * •

Based on the properties of the proposed convolution description, closed-form
expressions for the time-delay value as a function of the input and output
Laguerre spectra delay block are derived.

Notice that closed-form expressions for the delay are not intended as
estimators since they do not take into account model uncertainty and
disturbance properties. Yet, as [1] demonstrates, discrete delay estimation in
the face of heavily correlated and non-stationary additive measurement
disturbances can be built in Laguerre domain by exploiting the analytical
results.

The paper is organized as follows. First necessary notions of Laguerre domain
are introduced. Then, the Laguerre domain representations of the continuous
and discrete pure delay operators are revisited and cast in a common
framework. Making use of the latter, the delay value is analytically related
to the input and output Laguerre spectra of the delay block. The presented
concepts are illustrated by simulation.

##  2 Laguerre domain

### Continuous time:

The Laplace transform of the $k$-th continuous Laguerre function is given by

$\ell_{k}(s)=\frac{\sqrt{2p_{c}}}{s+p_{c}}T_{c}^{k}(s),\quad
T_{c}(s)\triangleq\frac{s-p_{c}}{s+p_{c}},$

for $k\in\mathbb{N}$, where $p_{c}>0$ represents the Laguerre parameter, and
$T_{c}$ is the continuous Laguerre shift operator.

Let $\mathbb{H}_{c}^{2}$ be the Hardy space of functions analytic in the open
left half-plane. The set $\\{\ell_{k}\\}_{k\in\mathbb{N}}$ is an orthonormal
complete basis in $\mathbb{H}_{c}^{2}$ with respect to the inner product

$\displaystyle\langle W,V\rangle_{c}\triangleq\frac{1}{2\pi
i}\int_{-\infty}^{\infty}W(s)V(-s)~{}{\rm d}s.$ (1)

Any function $W\in\mathbb{H}^{2}_{c}$ can be represented as a series

$W(s)=\sum_{k=0}^{\infty}w_{k}\ell_{k}(s),\quad w_{j}=\langle
W,\ell_{j}\rangle_{c},$

and the set $\\{w_{j}\\}_{j\in\mathbb{N}}$ is then referred to as the
continuous Laguerre spectrum of $W$.

### Discrete time:

The discrete Laguerre functions are specified in $\mathscr{Z}$-domain by

$L_{j}(z)=\frac{\sqrt{1-p_{d}}}{z-\sqrt{p_{d}}}T_{d}^{j}(z),\quad
T_{d}(z)\triangleq\frac{1-\sqrt{p_{d}}z}{z-\sqrt{p_{d}}},$ (2)

for all $j\in\mathbb{N}$, where the constant $0<p_{d}<1$ is the discrete
Laguerre parameter, and $T_{d}(z)$ is the discrete shift operator.

Let $\mathbb{H}_{d}^{2}$ be the Hardy space of analytic functions on the
complement of the unit disc that are square-integrable on the unit circle and
equipped with the inner product

$\displaystyle\langle W,V\rangle_{d}=\frac{1}{2\pi i}\oint_{D}\
W(z)\overline{V(z)}~{}\frac{{\mathrm{d}}z}{z},$ (3)

where ${\overline{V(z)}=V(z^{-1})}$ and $D$ is the unit circle. Then,
$\\{L_{k}(z)\\}_{k\in\mathbb{N}}$ is an orthonormal complete basis in
$\mathbb{H}^{2}_{d}$.

Any function $W\in\mathbb{H}^{2}_{d}$ can be represented as a series

$W(z)=\sum_{k=0}^{\infty}w_{k}L_{k}(z),\quad w_{j}=\langle
W,L_{j}\rangle_{d},$

and the set $\\{w_{j}\\}_{j\in\mathbb{N}}$ is referred to as the discrete
Laguerre spectrum of $W$.

A system is said to be represented in Laguerre domain when the involved in the
system model signals are represented by their Laguerre spectra.

Further on, both the continuous and discrete case are presented in a uniform
manner and no notational difference is made when the framework is clearly
stated.

##  3 Time delay in Laguerre domain

### Continuous time:

The well-known associated Laguerre polynomials (see e.g. [17]) are explicitly
given by

$\mathrm{L}_{m}(\xi;\alpha)=\sum_{n=0}^{m}\frac{1}{n!}\binom{m+\alpha}{m-n}(-\xi)^{n},\
\forall m\in\mathbb{N},\quad\xi\in\mathbb{R}.$ (4)

In what follows, only the polynomials with a particular value of $\alpha$ are
utilized and the shorthand notation
$\mathrm{L}_{m}(\xi)\triangleq\mathrm{L}_{m}(-\xi;\alpha)|_{\alpha=-1}$ is
introduced.

Consider the signal $u(t)\in\mathbb{L}_{2}[0,\infty)$ given by its Laguerre
spectrum $\\{u_{j}\\}_{j\in\mathbb{N}_{0}}$. Being passed through to a delay
block

$y(t)=u(t-\tau),\quad\tau\geq 0,\quad t\in[0,\infty),$ (5)

the input $u(t)$ results in the output $y(t)\in\mathbb{L}_{2}[0,\infty)$ with
the spectrum $\\{y_{j}\\}_{j\in\mathbb{N}_{0}}$. Then, according to [7], the
following relationship holds between the spectra

$\displaystyle y_{j}=\sum_{k=0}^{j-1}h_{j-k}(\kappa)u_{k}+h_{0}(\kappa)u_{j},\
\forall j\in\mathbb{N}_{0},$ (6)

where $h_{k}(\kappa)=\mathrm{e}^{-\frac{\kappa}{2}}\mathrm{L}_{k}(\kappa)$,
and $\kappa=2p\tau$. Notice that $\mathrm{L}_{0}(\kappa)=1$ and, therefore,
$h_{0}(\kappa)=\mathrm{e}^{-\frac{\kappa}{2}}$.

### Discrete time:

Introduce the polynomials

$L_{m}^{(\tau)}(\xi)=(-\xi)^{m-\tau}\sum_{n=0}^{\tau-1}\binom{m+n}{n}\binom{m-1}{\tau-n-1}(-\xi^{2})^{n},$

where it is agreed that $\binom{n}{k}=0$ for $k>n$ by definition. For the
discrete delay operator in time domain, it holds

$y(t)=u(t-\tau),\quad t,\tau\in\mathbb{N}_{0}.$ (7)

Assuming $u(t)\in\mathbb{I}^{2}[0,\infty)$ and with the same notation for the
input and output Laguerre spectra, it is shown in [13] that

$\displaystyle y_{j}=\sum_{k=0}^{j-1}h_{j-k}(\xi)u_{k}+h_{0}(\xi)u_{j},\
\forall j\in\mathbb{N}_{0},$ (8)

where $h_{0}(\xi)=\xi^{\tau}$, $h_{j}=(1-p)L_{j}^{(\tau)}(\xi),j\geq 1$, and
$\xi=\sqrt{p}$.

A readily observed difference between the continuous and discrete time cases
is that the argument of the polynomials $\mathrm{L}_{k}(\kappa)$ carries
information about $\tau$ whereas the argument of $L_{m}^{(\tau)}(\xi)$ is
solely defined by the Laguerre parameter $p$. In the continuous-time case of
(6), the delay value appears only in a product with the Laguerre parameter
thus highlighting the role of the latter as a time scale degree of freedom. On
the contrary, in the discrete case of (6), $\tau$ influences the order of
$L_{m}^{(\tau)}(\xi)$ as the time scale is fixed by the discrete time variable
$t$.

The common convolution form of the continuous and discrete delay operators,
i.e. (6) and (8), implies that these descriptions exhibit some kind of
“causality”. Indeed, the output coefficient $y_{k}$ depends only on the input
coefficients $u_{k},u_{k-1},\dots,u_{0}$. Naturally, this is not a temporal
casualty since each Laguerre coefficient is evaluated on the time interval
$t\in[0,\infty)$. Assuming that the input signal is formed so that
$u_{i}=0,i=0,\ldots,m-1$, for some $m\in\mathbb{N}_{+}$, the first $m$
coefficients of the output, i.e. $y_{i},i=0,\ldots,m-1$, are independent of
the input and constitute instead the first $m$ coefficients of the Laguerre
spectrum of an additive disturbance. Thus, the signal shape of the realization
can be reconstructed (see Section 5) and utilized, e.g. for noise reduction.

Despite the identical form of Laguerre-domain representations of the
continuous and discrete delay in (5) and (7), the continuous operator is an
infinite-dimensional system and the discrete operator admits a minimal state-
space realization of order $\tau$. The difference roots in the properties of
the polynomials $\mathrm{L}_{m}(\cdot)$ and $L_{m}^{(\tau)}(\cdot)$.

Recalling that the output of a discrete linear time-varying system under zero
initial conditions is given by the convolution of the input sequence with the
system’s Markov parameters, $h_{0},h_{1},\dots$ are further referred to as the
Laguerre-domain Markov parameters of delay operators (5), (7).

Consider now a Hankel matrix composed of the Markov parameters

$\mathcal{H}_{n}=\begin{bmatrix}h_{0}&h_{1}&\ldots&h_{n-1}\\\
h_{1}&h_{2}&\ldots&h_{n}\\\ \vdots&\vdots&\ddots&\vdots\\\
h_{n-1}&h_{n}&\ldots&h_{2n-2}\\\ \end{bmatrix}.$

######  Proposition 1.

For the Markov parameters in (6)

$\operatorname{rank}\mathcal{H}_{n}=n,\quad\forall n.$

For the Markov paraemters in (8)

$\operatorname{rank}\mathcal{H}_{n}=\begin{cases}n&\text{ if $n<\tau$},\\\
\tau&\text{if $\tau\leq n$.}\end{cases}$

The Ho-Kalman algorithm [8] relates the rank of $\mathcal{H}_{n}$ to the order
of the minimal (continuous or discrete) state-space realization (McMillan
degree) that yields the Markov parameters.

### Continuous time:

In view of the definition of the Markov parameters in (6), it suffices to show
that the polynomials $\mathrm{L}_{m}$ constitute an orthogonal functional
basis on the positive real axis.

In [16, p. 205], there is a proof for the orthogonality property of the
associated Laguerre polynomials $\mathrm{L}_{m}(\xi;\alpha)$ (as defined in
(4)) assuming $\alpha>-1$. However, the case of $\alpha=-1$ is not covered
there, while it is exactly the type of Laguerre polynomials that are
encountered in time-delay systems. Notably, for $\alpha<-1$, Laguerre
polynomials are no longer orthogonal on the positive real axis but rather
satisfy a non-Hermitian orthogonality on certain contours in the complex
plane, cf. [9]. Some confusion yet arises with respect to what notion of
orthogonality applies in case of $\alpha=-1$, cf. [9, p. 205 and p. 209].
Therefore, orthogonality of the polynomials $\mathrm{L}_{m}$ is proven in
Appendix B thus securing the fact that the matrix $\mathcal{H}_{n}$ is always
full rank for the continuous time delay case.

### Discrete time:

As shown in [12], a minimal realization of order $\tau$ satisfying (8) is
given by the Laguerre-domain state-space equations

$\displaystyle x_{j+1}$ $\displaystyle=Fx_{j}+Gu_{j},$ (9) $\displaystyle
y_{j}$ $\displaystyle=Hx_{j}+{Ju_{j}},$

where

$\displaystyle F$
$\displaystyle=\begin{bmatrix}-\sqrt{p}&1-p&\dots&\sqrt{p}^{\tau-2}(1-p)\\\
0&-\sqrt{p}&\dots&\sqrt{p}^{\tau-3}(1-p)\\\ \vdots&\vdots&\vdots&\vdots\\\
0&0&\dots&-\sqrt{p}\end{bmatrix},$ $\displaystyle G^{\intercal}$
$\displaystyle=(1-p)\begin{bmatrix}\sqrt{p}^{\tau-1}&\sqrt{p}^{\tau-2}&\dots&1\end{bmatrix},$
$\displaystyle H$
$\displaystyle=\begin{bmatrix}1&\sqrt{p}&\dots&\sqrt{p}^{\tau-1}\end{bmatrix},$

and $J=\sqrt{p}^{\tau}$ $\blacksquare$

The realization in (9) reveals that the delay $\tau$ appears in the discrete-
time case both as the system order and system parameter. This is in a sharp
contrast with the time-domain description, where $\tau$ is the order of the
system and not a parameter. System order estimation techniques [2] are less
elaborate compared to those of parameter estimation. Thus, the representation
of the pure time-delay operator in Laguerre enables applying the well-
developed technology of parameter estimation [1]. The next section
demonstrates that the delay value can be obtained both in continuous and
discrete time without an intermediate Markov parameter representation and
directly from the input and output spectra.

##  4 Main result

######  Proposition 2.

Consider a system given by the Markov parameters $\\{h_{0},h_{1},\dots\\}$,
$h_{i}\in\mathbb{R},i=0,1,\dots$, driven by the input sequence
$\\{u_{0},u_{1},\dots\\}$, $u_{i}\in\mathbb{R},i=0,1,\dots$, $u_{0}\neq 0$,
and producing the output sequence $\\{y_{0},y_{1},\dots\\}$,
$y_{i}\in\mathbb{R},i=0,1,\dots$ according to

$\displaystyle y_{j}=\sum_{k=0}^{j-1}h_{k}u_{j-k}+h_{0}u_{j}.$ (10)

Then, it applies that

$h_{k}=\sum_{j=0}^{k}g_{k-j}y_{j},$ (11)

and

$\displaystyle g_{0}$ $\displaystyle=\frac{1}{u_{0}},$ $\displaystyle g_{k}$
$\displaystyle=-\frac{1}{u_{0}}\sum_{j=0}^{k-1}u_{k-j}g_{j},\quad k\geq 1.$

Introduce the following notation

$Y_{N}\\!\\!=\\!\\!\begin{bmatrix}y_{0}\\\ y_{1}\\\ \vdots\\\
y_{N}\end{bmatrix};\ T(U_{N})\\!\\!=\\!\\!\begin{bmatrix}u_{0}&0&\dots&0\\\
u_{1}&u_{0}&\dots&0\\\ \vdots&\vdots&\ddots&0\\\
u_{N}&u_{N-1}&\dots&u_{0}\end{bmatrix};\
H_{N}\\!\\!=\\!\\!\begin{bmatrix}h_{0}\\\ h_{1}\\\ \vdots\\\
h_{N}\end{bmatrix}.$

Then, (10) implies that

$Y_{N}=T(U_{N})H_{N}.$ (12)

The assumption $u_{0}\neq 0$ secures non-singularity of $T(U_{N})$ and the
Markov coefficients can be uniquely recovered from the Laguerre spectra of the
input and output

$H_{N}=T^{-1}(U_{N})Y_{N}.$

The result now follows by a direct application of Lemma 4 in Appendix A
$\blacksquare$

The assumption of $u_{0}\neq 0$ is not restrictive. The leading zero
coefficients can be skipped thus implying that corresponding coefficients of
the output are also zero, according to the “causality” property in Laguerre
domain.

Now the main results of the paper can be formulated stating that the value of
$\tau$ in (7) or (5) can be analytically evaluated from the input and the
output spectra.

######  Proposition 3.

Consider three subsequent Laguerre-domain Markov parameters
$h_{m-1},h_{m},h_{m+1}$ evaluated for any $m\geq 1$ from the Laguerre spectra
of the input and output signals of either (5) or (7)
$\\{u_{0},u_{1},\dots\\}$, $\\{y_{0},y_{1},\dots\\}$ according to (11). The
following relationships hold then for any admissible value of the Laguerre
parameter $p$:

Continuous time:

$\kappa=-\frac{(m+1)h_{m+1}+(m-1)h_{m-1}-2mh_{m}}{h_{m}},$ (13)

where $\kappa=2p\tau$.

Discrete time:

$\tau=-\frac{(m+1)h_{m+1}+(m-1)h_{m-1}+m(\xi+\xi^{-1})h_{m}}{(\xi-\xi^{-1})h_{m}},$
(14)

where $\xi=\sqrt{p}$.

From (6) and (8), it is readily observed that the Markov parameters
$h_{0},h_{1},\dots$ are proportional to the corresponding polynomials
$\mathrm{L}_{m}$ and $L_{m}^{(\tau)}$. Then it suffices to show that
identities (13) and (14) apply to the polynomials.

### Continuous time

Recall that the Markov parameters of the continuous delay are defined as

$h_{m}(\kappa)=\mathrm{e}^{-\frac{\kappa}{2}}\mathrm{L}_{m}(\kappa)=\mathrm{e}^{-\frac{\kappa}{2}}\mathrm{L}_{m}(-\kappa;\alpha)_{\alpha=-1}.$

It is well known, see e.g. [16, Theorem 68], that the zeros of
$\mathrm{L}_{m}(x;\alpha),\alpha>-1$ are positive and distinct, which excludes
the case at hand. Yet, in view of Proposition 5, the polynomials
$\mathrm{L}_{m}(\kappa)$ possess orthogonality on $[0,\infty)$. Then, in
virtue of [16, Theorem 55], all the zeros of $\mathrm{L}_{m}(\kappa)$ are
distinct and negative. Therefore, it is guaranteed that $h_{m}(\kappa)\neq
0,m=0,1,\dots$ for all feasible values of $\kappa$ and the denominator of (13)
does not turn to zero.

The associated Laguerre polynomials obey the following three-term relationship
[16]

$\mathrm{L}_{m+1}(\kappa)=\frac{1}{m+1}(\kappa+2m)\mathrm{L}_{m}(\kappa)-\frac{m-1}{m+1}\mathrm{L}_{m-1}(\kappa).$
(15)

Solving (15) with respect to $\kappa$ and substituting the Markov parameters
in it gives (13). Notice that the term corresponding to
$\mathrm{L}_{m-1}(\kappa)$ becomes zero for $m=1$. Therefore, the value of
$\mathrm{L}_{2}(\kappa)$ is completely defined by that of
$\mathrm{L}_{1}(\kappa)$. Consequently, $\mathrm{L}_{0}(\kappa)$ and, thus,
$h_{0}$ are immaterial to the recursion.

### Discrete time

To secure that $h_{m}\neq 0$ in (14), it is sufficient to recall that the
discrete delay operator in Laguerre domain admits a finite-dimensional minimal
realization given by (9). Then, none of the Markov parameters of it turn to
zero due to the eigenvalue-revealing structure of the matrix $F$.

Similarly to the continuous case, for (7), the Markov parameters are
proportional to the polynomials $L_{m}^{(\tau)}$ that are subject to the
following recursion [13]

$L_{m+1}^{(\tau)}(\xi)=a_{m}^{(\tau)}(\xi)L_{m}^{(\tau)}(\xi)+b_{m}L_{m-1}^{(\tau)}(\xi)$
(16)

holds with

$a_{m}^{(\tau)}(\xi)\triangleq a_{m,1}^{(\tau)}\xi+a_{m,2}^{(\tau)}\xi^{-1}$
$a_{m,1}^{(\tau)}=-\frac{m+\tau}{m+1},\quad
a_{m,2}^{(\tau)}=-\frac{m-\tau}{m+1},\quad b_{m}=-\frac{m-1}{m+1}.$ (17)

Exactly as in the continuous case, since $b_{1}=0$, the polynomial
$L_{0}^{(\tau)}(\xi)$ is not included in the recursion and
$L_{2}^{(\tau)}(\xi)$ is completely defined by $L_{1}^{(\tau)}(\xi)$
$\blacksquare$

The fact that the discrete Markov parameters $h_{m}$ are always nonzero has
consequences for the zeros of the polynomials $L_{m}^{(\tau)}$. The
polynomials are definitely not orthogonal, as Proposition 1 implies. However,
one can prove that all the roots of the equation $L_{m}^{(\tau)}(\xi)=0$ are
real and distinct. Further, they are symmetric with respect to origin and
located outside of the interval $(-1,1)$ where the admissible values of
$\pm\sqrt{p}$ lie. These are the properties of zeros that one would otherwise
expect from an orthogonal polynomial family.

In view of the recursive relationships appearing in the proof above, it is
instructive to recall Favard’s theorem [4]. It secures the orthogonality of a
polynomial sequence $\phi_{n},n=0,1,\dots$ with respect to some positive
weight function if they satisfy

$\eta\phi_{n}(\eta)=a_{n}\phi_{n+1}(\eta)+c_{n}\phi_{n}(\eta)+d_{n}\phi_{n-1}(\eta)$
(18)

for some numbers $a_{n},b_{n},c_{n}$ where $a_{n}\neq 0$, $c_{n}\neq 0$. Then,
it readily follows that the polynomials $L_{m}^{(\tau)}(\xi)$ in (15) form an
orthogonal basis. Yet, Favard’s theorem does not say how to find the weight
function, which problem is resolved in Proposition 5.

On the contrary, the three-term relationship in (16) does not comply with the
conditions of Favard’s theorem since $a_{m}^{(\tau)}(\xi)$ is not a first-
order polynomial in $\xi$. In fact, the polynomials $L_{m}^{(\tau)}(\xi)$ do
not even build a polynomial family since order of the polynomials is defined
by both $m$ and $\tau$. The question of whether or not the polynomials in the
discrete case are orthogonal is negatively answered by the existence of
finite-dimensional realization (9) reproducing the sequence of the Markov
parameters, see Proposition 1.

Notice also that there exist much simpler relations between the delay value
and some of the Markov parameters than those specified by Proposition 3, see
[12]. For instance, in continuous time, it applies that

$\tau=-\frac{\ln h_{0}}{p},$

and the corresponding identity in discrete time is

$\tau=\frac{2\ln h_{0}}{\ln p}.$

##  5 Numerical example

To illustrate the analytical results presented above, consider continuous
delay (5) with $\tau=5$. The input signal is selected to be a linear
combination of the first four Laguerre functions

$U(s)=6\ell_{0}(s)-3\ell_{1}(s)+2\ell_{2}(s)-\ell_{3}(s).$

The time evolution of the input as well as the output $y(t)$ are shown in Fig.
1. The Laguerre spectra of $u(t)$ and $y(t)$ are presented in Fig. 2.

Consider now the identity in (15) with $m=2$, i.e.

$\kappa=-\frac{3h_{3}+h_{1}-4h_{2}}{h_{2}},$

with $h_{1}=-0.7318,h_{2}=-0.0732,h_{3}=0.1903$ (see Fig. 2) and $p=0.08$. It
holds for the numerical values of the example. The same is true for any other
feasible $m$. One should although notice here that the numerical procedures
for evaluating Laguerre spectra are not sufficiently developed and the
estimates of the coefficients of order over 30 are usually unreliable.

Figure 1: The input signal $u(t)$ of the continuous time-delay operator and
its output $y(t)$ (dashed line). An approximation of $y$ by the first 25
Laguerre functions with $p=0.08$ is also shown. Figure 2: First 25
coefficients of the Laguerre spectra ($p=0.08$) of $u(t)$ and $y(t)$. Only the
four first coefficients of the input spectrum are non-zero. The first 25
Markov coefficients $h_{k}\times 10$ calculated from the input and the output
spectra are as well shown. Figure 3: Ten realizations of the disturbance model
representing random linear combination of the first four Laguerre functions,
$p=0.18$. The coefficients of the disturbance model are uniformly distributed
random variables in the interval $[-15,15]$. Figure 4: Input signal
$u(t)=6\ell_{4}(t)-3\ell_{5}(t)+2\ell_{6}(t)-\ell_{7}(t)$, $p=0.18$. The delay
block output $y(t)$, $\tau=5$. The output disturbed with a disturbance
realization in Fig. 3. The $\mathbb{L}_{2}$ signal-to-noise ratio is
$\|y\|_{2}/\|d\|_{2}=0.3703$. Figure 5: Laguerre spectrum of the disturbed
output. The disturbance does not influence the Laguerre coefficients that
belong to the delayed input signal (the delay block output).

As discussed in Section 3, both the continuous and discrete descriptions of
the time-delay operator exhibit “casuality” in Laguerre domain. To illustrate
the utility of this property, consider a continuous time-delay block with a
measurement disturbance

$y(t)=u(t-\tau)+d(t),$

where $d(t)\in\mathbb{L}_{2}[0,\infty)$ is a random signal. A Laguerre-domain
disturbance model is introduced in [1] constituting a linear combination of
certain Laguerrre functions with random variables as weights. Fig. 3 depicts
time-domain realizations of the disturbance model

$d(t)=d_{1}\ell_{0}(t)+d_{2}\ell_{1}(t)+d_{3}\ell_{2}(t)+d_{4}\ell_{3}(t),$

where $d_{i},i=1,\dots,4$ are random numbers uniformly distributed in the
interval $[-15,15]$. Now shape the input of the delay block as

$u(t)=6\ell_{4}(t)-3\ell_{5}(t)+2\ell_{6}(t)-\ell_{7}(t),$

so that all the spectral components of $d(t)$ are of lower order compared to
those of $u(t)$. Then, in the Laguerre spectrum of the output signal (see Fig.
5), the components that belong to the disturbance do not influence the
components of the delayed input which implies that the delay value can be
recovered in the same manner than in the disturbance-free case treated above.
This is despite the fact that the disturbance completely dominates the output
in time domain, Fig. 4. Indeed, the disturbance energy is almost three times
higher than the energy of the input signal, i.e. $\|d\|_{2}=19.0958$ and
$\|u\|_{2}=7.0711$. The invariance property is due to the orthogonality of the
Laguerre function basis. A detailed explanation of how the spectral
decomposition can be exploited in noise reduction is provided in [1].

##  6 Conclusions

This paper gives closed-form expressions for the delay in terms of the
Laguerre spectra of the input and output. Both discrete and continuous time
cases are treated. Similarities and differences arising due the infinite-
dimensional nature of the continuous delay and a finite-dimensional
realization of it in discrete time are highlighted. The derived expressions
exhibit the inherent connections between the continuous and discrete delay
pure operators by providing a common Laguerre-domain modeling framework
exploiting convolution representations with polynomial Markov parameters.

This work was partially supported the Swedish Research Council under grant
2019-04451.

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##  Appendix A Inverse of lower triangular Toeplitz matrix

######  Lemma 4.

If $u_{0}\neq 0$, the inverse of $T(U_{N})$ is given by

$T^{-1}(U_{N})\triangleq G(U_{N})=\begin{bmatrix}g_{0}&0&\dots&0\\\
g_{1}&g_{0}&\dots&0\\\ \vdots&\vdots&\ddots&0\\\
g_{N}&g_{N-1}&\dots&g_{0}\end{bmatrix},$

where

$\displaystyle g_{0}$ $\displaystyle=\frac{1}{u_{0}}$ $\displaystyle g_{k}$
$\displaystyle=-\frac{1}{u_{0}}\sum_{j=0}^{k-1}u_{k-j}g_{j},\quad k\geq 1.$

Introduce the matrix $E_{N}\in\mathbb{R}^{N\times N}$

$E_{N}=\begin{bmatrix}0&0&\dots&0&0\\\ 1&0&\dots&0&0\\\
\vdots&\vdots&\ddots&\vdots&0\\\ 0&0&\dots&1&0\end{bmatrix}.$

Then the low-triangular Toeplitz matrix $T(U_{N})$ can be written as a matrix-
valued polynomial in $E_{N}$

$T(U_{N})=\sum_{k=0}^{N-1}u_{k}E_{N}^{k}.$

Since $T^{-1}(U_{N})$ is also a low-triangular Toeplitz matrix matrix, it
follows

$T^{-1}(U_{N})=\sum_{k=0}^{N-1}g_{k}E_{N}^{k}.$

Observing that $E_{N}$ is nilpotent

$\displaystyle T(U_{N})G(U_{N})=G(U_{N})T(U_{N})$
$\displaystyle=\sum_{k=0}^{N-1}u_{k}E_{N}^{k}\sum_{k=0}^{N-1}g_{k}E_{N}^{k}$
$\displaystyle=\sum_{k=0}^{N-1}\sum_{i+j=k}u_{i}g_{j}E_{N}^{k}=I.$ (19)

The inner sum is taken over all partitions of $k$ into two parts, i.e. the
indices $i$ and $j$. For the equality to hold, one has to ensure that
$u_{0}g_{0}=1$ and

$\sum_{i+j=k}u_{i}g_{j}=0,\quad k\geq 1.$

Singling out the term with $g_{k}$ results in

$u_{0}g_{k}=-(u_{1}g_{k-1}+\dots+u_{k}g_{0}).$

Solving the equation with respect to $g_{k}$ completes the proof.
$\blacksquare$

##  Appendix B Orthogonality of $\mathrm{L}_{m}$

######  Proposition 5.

Laguerre polynomials $\mathrm{L}_{n}(\cdot),n\in\mathbb{N}_{0}$ are orthogonal
on the positive real axis in the sense that

$\int_{0}^{\infty}\frac{\e^{-x}}{x}\mathrm{L}_{n}(x)\mathrm{L}_{m}(x)~{}\mathrm{d}x=\begin{cases}0,&n\neq
m\\\ \frac{1}{n!},&n=m.\end{cases}$ (20)

In [16], it is shown that

$\displaystyle(m-n)\int_{a}^{b}x^{\alpha}\e^{-x}\mathrm{L}_{n}(x;\alpha)\mathrm{L}_{m}(x;\alpha)~{}\mathrm{d}x=$
$\displaystyle
x^{k+1}\e^{-x}\left(\mathrm{L}_{m}(x;\alpha)\dot{\mathrm{L}}_{n}(x;\alpha)-\mathrm{L}_{n}(x;\alpha)\mathrm{\dot{L}}_{m}(x;\alpha)\right)\biggr{\rvert}_{a}^{b}\biggl{.}$

Specialized to $\alpha=-1$, the integration result reads

$\varphi_{m,n}(x)=\e^{-x}\left(\mathrm{L}_{m}(x)\dot{\mathrm{L}}_{n}(x)-\mathrm{L}_{n}(x)\dot{\mathrm{L}}_{m}(x)\right)$

Clearly, due to the exponential factor

$\lim_{x\to\infty}\varphi_{m,n}(x)\to 0.$ (21)

Furthermore, the least order term in $\mathrm{L}_{m}(x),\ m>0$ is always $x$.
Therefore, the least order term in $\dot{\mathrm{L}}_{m}(x),\ m>0$ is always
$1$. The least order term in the product
$\mathrm{L}_{m}(x)\dot{\mathrm{L}}_{n}(x),\ n\neq 0,m\neq 0,m\neq n$ is, once
again, $x$. Then $\varphi_{m,n}(0)=0$ which fact, together with (21) implies
orthogonality of $\mathrm{L}_{n}(\cdot)$ and $\mathrm{L}_{m}(\cdot)$ for
$m\neq n$, i.e. (20). It can also be shown that for $m=n$

$\displaystyle\int_{0}^{\infty}\frac{\e^{-x}}{x}\mathrm{L}_{n}(x)\mathrm{L}_{n}(x)~{}\mathrm{d}x=\frac{1}{n!}.$

$\blacksquare$