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# Towards precision particle background estimation for future X-ray missions:
correlated variability between
Chandra ACIS and AMS

Catherine E. Grant  MIT Kavli Institute for Astrophysics & Space Research,
Cambridge, Massachusetts, USA  Eric D. Miller  MIT Kavli Institute for
Astrophysics & Space Research, Cambridge, Massachusetts, USA  Marshall W.
Bautz  MIT Kavli Institute for Astrophysics & Space Research, Cambridge,
Massachusetts, USA  Richard Foster  MIT Kavli Institute for Astrophysics &
Space Research, Cambridge, Massachusetts, USA  Ralph P. Kraft  Center for
Astrophysics $|$ Harvard & Smithsonian, Cambridge, Massachusetts, USA  Steven
Allen  Department of Physics, Stanford University, Stanford, California, USA
David N. Burrows  The Pennsylvania State University, University Park, PA, USA

###### Abstract

A science goal of many future X-ray observatories is mapping the cosmic web
through deep exposures of faint diffuse sources. Such observations require low
background and the best possible knowledge of the remaining unrejected
background. The dominant contribution to the background above 1-2 keV is from
Galactic Cosmic Ray protons. Their flux and spectrum are modulated by the
solar cycle but also by solar activity on shorter timescales. Understanding
this variability may prove crucial to reducing background uncertainty for
ESA’s Athena X-ray Observatory and other missions with large collecting area.
We examine of the variability of the particle background as measured by ACIS
on the Chandra X-ray Observatory and compare that variability to that measured
by the Alpha Magnetic Spectrometer (AMS), a precision particle detector on the
ISS. We show that cosmic ray proton variability measured by AMS is well
matched to the ACIS background and can be used to estimate proton energies
responsible for the background. We discuss how this can inform future
missions.

###### keywords:

X-rays, particle background, Chandra, ACIS, AMS

##  1 INTRODUCTION

Mapping faint diffuse structures such as clusters and groups of galaxies, and
the intergalactic medium is a science goal of many future X-ray observatories
and of the “Cosmic Ecosystems” theme of the Astro2020 Decadal Report “Pathways
to Discovery in Astronomy and Astrophysics for the 2020s”[1]. Such
observations require low background and the best possible knowledge of the
remaining unrejected background. For most detectors and orbits, the dominant
contribution to the unrejected background above 1-2 keV is from Galactic
Cosmic Ray (GCR) protons. The flux and spectrum of these protons are modulated
by the 11-year solar cycle but also with solar activity on much shorter
timescales of months, weeks, or even days. Understanding this variability may
prove crucial to reducing background uncertainty to the level required by
ESA’s Athena X-ray Observatory[2] and other proposed future large collecting
area missions like NASA’s Lynx[3] and the probe concept Advanced X-ray Imaging
Satellite (AXIS)[4].

These future missions are designed with ambitious goals in mind—finding the
first groups and clusters of galaxies, and mapping cosmic structures in
time—to take deep exposures and measure the properties of faint diffuse
objects. Any source of background, whether due to particles, stray light,
unresolved sources, or optical light leak, adds statistical and systematic
uncertainty to the final measurements of temperature, density, etc.
Observations of very low surface brightness sources are often currently
limited by systematic uncertainty in the background level. Measuring the
background in an “empty” portion of the observation field of view is not
always possible for extended emission sources, so the background must be
estimated by scaling from other observations separated in time. That scaling
factor adds additional systematic error to the final results. Meeting the
ambitious goals of future missions will require both reducing the background
as much as feasible and improving our knowledge of the remaining background.

There are many components of the background seen in X-ray observations. One
way to subdivide them is into two groups: focused X-rays, i.e. unresolved
point sources and the diffuse X-ray background, and everything else. For most
detectors, observatories, and orbits, the focused X-rays dominate the
background below 1 to 2 keV (Figure 1). The particle background itself has a
number of sources: Galactic cosmic rays (GCRs), solar energetic protons, and
the Earth’s trapped radiation. This part of the background is temporally
variable and its normalization depends on the presence or absence of
geomagnetic shielding. In addition, there’s an unfocused cosmic hard X-ray
background which produces a constant pedestal in the measured non-focused
background. Outside of solar storms and the Earth’s radiation belts, the
primary contribution to the non-X-ray background is GCR protons, either by
direct interaction in the detector, or by secondaries produced in interactions
within and around the detector.

Figure 1: Simulated spectra comparing the unresolved X-ray background with the
particle background in high-Earth orbit (HEO) and low-Earth orbit (LEO) for
one proposed observatory configuration. In this example, the particle
background is important above 1 to 2 keV. The normalization of the background
is different in LEO and HEO, due to the geomagnetic shielding present in LEO.
The dashed line is the current background requirement for the Athena WFI.

The choice of orbit for a proposed mission, while often primarily driven by
constraints other than the particle background, has major ramifications for
its overall normalization and the variability. In low-Earth orbit (LEO), the
GCR background is modulated by changing geomagnetic shielding along the orbit.
An example is Ref. 5 in which the cutoff rigidity along the orbit can be used
to model the non-X-ray background of the XIS on Suzaku with good accuracy. In
high-Earth orbit (HEO), most or all of the science observing time occurs while
the spacecraft is well outside radiation belts and there is no geomagnetic
shielding. Both NASA’s Chandra X-ray Observatory[6, 7] and ESA’s XMM-Newton[8,
9] have more than two decades of experience with the particle background in
HEO. The normalization of the background is higher than in LEO and the
variability is primarily driven by the solar cycle and solar activity. Neither
of these can be simply or accurately predicted in the same way that orbit and
Earth’s magnetic shielding can be used for prediction in LEO. Knowledge of the
background level and variability needs to come from other measurements.

(There is an additional time-varying component of the non-X-ray background as
seen by XMM-Newton and, to a lesser extent, Chandra that has thus far been
ignored: soft proton flares, which are brief enhancements of focused protons
with energies of order 100 keV[10]. This component is still not well
understood but is believed to be associated with the Earth’s magnetopause[11]
and should therefore be less important in L1/L2. As we are primarily
interested in GCR protons and are utilizing stowed ACIS data, we will not be
addressing soft protons.)

The launch of SRG/eROSITA in 2019 begins a new era as the first X-ray mission
to orbit the Sun-Earth Lagrangian L2.[12] The expectation was that the
particle background in L2 should look much the same as that seen in HEO by XMM
and Chandra, with the possible exception of soft proton flares. While there
are some questions about the normalization, which is higher than was expected
based on pre-launch Geant4 modeling, for the most part this appears to be
true. Ref. 13 reports the background has been declining as solar activity
increased, and found a $\sim 27$ day period in the particle rate, possibly
related to solar rotation.

Athena (Advanced Telescope for High ENergy Astrophysics), ESA’s next large
X-ray observatory, is scheduled to launch in the mid 2030s to Earth-Sun L1[2].
As some of the Athena science goals require study of faint diffuse emission,
such as that in clusters of galaxies, there has been a substantial effort to
better understand, model, and reduce the instrumental background produced by
particles and high energy photons. The background reduction studies include
detailed Geant4 simulations, experimental testing of shielding and coating
choices, as well as software mitigation strategies, both on-board and on the
ground. A summary of these ongoing efforts by the Athena WFI team and the WFI
Background Working Group is given in Ref. 14. The work described in this
proceedings paper was done as part of the US contribution to the Athena WFI
and as part of the WFI Background Working Group.

We examine the variability of the particle background as measured by the
Advanced CCD Imaging Spectrometer (ACIS) on the Chandra X-ray Observatory, and
then compare that variability to measurements by the Alpha Magnetic
Spectrometer (AMS), a precision particle physics detector on the International
Space Station. We show that the cosmic ray proton variability measured by AMS
is very well matched to the ACIS variability and can be used to estimate the
proton energies responsible for the ACIS unrejected background. We will
discuss how this knowledge can inform future missions, by inclusion of special
purpose particle monitors, as is under investigation for Athena, or by
comparison to other space-based particle monitors.

Using correlated variability to learn more about the sources of the NXB has
recently been done for both the XMM EPIC-pn[15] and EPIC-MOS[16] cameras, with
comparisons to the on-board XMM Radiation Monitor, the EPHIN particle
instrument on SOHO, and Chandra ACIS. After excluding solar energetic particle
events, they find tight correlations between $\sim$1 GeV protons and all the
X-ray background measures. The addition of the AMS to such comparisons with
its large collecting area, higher spectral resolution, and high statistical
precision, may allow for stronger conclusions on the proton energies of
importance, and on the effectiveness of instrument proxies on background
knowledge.

We begin in Section 2 with a description of the ACIS instrumental proxy for
the particle background, and of the AMS data. A comparison between the two is
made in Section 3 and we conclude with a discussion of future work in Section
4.

##  2 DATA

###  2.1 Chandra ACIS

For this project we use data from the Advanced CCD Imaging Spectrometer
(ACIS)[7] on the Chandra X-ray Observatory[6], which was launched into a
highly elliptical 2.6-day orbit in 1999. Chandra’s orbit samples a variety of
particle environments, but spends most of its time above the Earth’s radiation
belts. It is expected that the unfocused particle background experienced by
any future mission in HEO or L1/L2 should be very similar to that seen by
Chandra. GCR protons are the primary source of quiescent background, modulated
by the solar cycle and solar activity. The twenty-three year baseline provided
by Chandra data covers more than half of solar cycle 23, all of solar cycle
24, and the beginning of solar cycle 25, allowing for study of conditions at
solar maximum and minimum and for variances between solar cycles.

ACIS utilizes framestore charge-coupled devices (CCDs), ten CCDs in total,
each with 1024 by 1024 24-micron pixels. We focus on the back-illuminated (BI)
CCD, ACIS-S3, at the ACIS-S aimpoint, as the closest analog to future
detectors like the Athena WFI. It is 45-microns thick, fully depleted, and
runs in photon-counting mode, with frametimes of $\sim$3 second. Telemetry
limitations require discarding the contents of most pixels on-board. On-board
processing identifies potential X-ray event candidates and telemeters the
subset most likely to be X-rays and not particles. On-board filtering removes
events with pulseheights above an upper threshold and event morphologies that
including many contiguous pixels. While the filtered events themselves are not
telemetered, the instrument does keep an accounting of the numbers of events
removed and why. We use one of these counters, the high energy reject rate, as
a proxy for the particle rate seen by the spacecraft.

Figure 2: Time dependence of the particle background as measured by the rate
of high energy events on ACIS-S3. Each data point is a single ECS observation.
(top) The full mission history of high energy rates on S3 includes two solar
maxima and minima. The particle background is anti-correlated with the solar
cycle and varies by a factor of $\sim$2 over 11 years. The two cycles are
distinct; the first could not be used to accurately predict the rates in the
next cycle. (bottom left) A segment of the ACIS-S3 background during solar
maximum. The sawtooth pattern is typical of solar activity and individual
solar storms. The background variation is $\sim$10% over weeks to months.
(bottom right) A segment of the ACIS-S3 background during solar minimum, when
there was no significant solar activity. Background variations are smaller, of
order 5% over a matter of days.

The data used here were taken while ACIS was stowed, viewing its External
Calibration Source (ECS). No focused X-rays reach the detector—only particles
and fluorescent X-ray lines from the ECS, which are all well below the high
energy reject limit, set to $\sim$15 keV during these observations. While the
instrument is stowed, the background is dominated by Galactic cosmic rays,
primarily protons $>$ 10 MeV, plus any secondary particles produced through
interaction with the spacecraft and instrument. Focused low energy protons
($\sim$100 keV) which can cause short term increases in background count rates
are not seen while the instrument is stowed. These data are regularly taken
just before entry and just after exit from the radiation belts, and are
occasionally replaced by specialized instrument tests or are dropped entirely
to improve science observing efficiency. The observations are therefore not
evenly spaced in time and the coverage is sometimes irregular.

The full history of stowed ACIS-S3 high energy reject rate is shown in the top
panel of Figure 2. Each data point is the average rate for a single 3-10 ksec
observation: the mean (and median) exposure time is 8 ksec. The statistical
error on each point is smaller than the symbol size. The high-energy
background measured by ACIS is clearly anti-correlated with the solar cycle
and varies by a factor of $\sim$2 over 11 years. The solar maxima in 2001 and
2014 correspond to minima in background rates and vice versa. The solar cycles
seen by ACIS are distinct; the first could not be used to accurately predict
the rates in the next cycle.

The lower panels in Figure 2 show shorter time sequences during solar maximum
and solar minimum. During solar maximum, a sawtooth pattern is caused by solar
activity and individual solar storms. The background variation is $\sim$10%
over timescales of weeks to months. During solar minimum the background
variations are smaller, but small scale “bubbling” continues at the 5% level
over a matter of days.

All of this variation, large and small, long and short timescales, is relevant
for modeling and removing the background contribution to scientific studies.
Reducing the uncertainty in our knowledge of this variation would benefit any
measurement of low surface brightness objects. Most detectors do a have a
proxy of the background rate, much like the ACIS high energy reject rate,
which does give an in situ simultaneous measure of the radiation environment,
but generally the uncertainty on these counters is still large enough to
influence the overall uncertainty of the final results.

###  2.2 AMS

X-ray detectors are optimized, in both hardware and software, to efficiently
and accurately measure X-ray signal. They are not designed to capture all the
signal from a particle interaction and cannot measure particle characteristics
very well, if at all. Specialized particle instruments can measure temporal
and spectral characteristics of the particles and determine their species.
This is why many X-ray observatories include a dedicated particle detector as
part of their instrument complement, such as the Radiation Monitor on XMM-
Newton.

Figure 3: AMS proton spectra at solar maximum (February 2014) and in the
transition to solar minimum (February 2017). The overall flux is higher at
solar minimum due to reduced magnetic shielding from the heliosphere. At solar
maximum, the flux is suppressed, particularly at lower energies where the
shielding is more effective. The statistical plus systematic errors on each
data bin are too small to be seen on this plot.

In this paper, we look at the Alpha Magnetic Spectrometer (AMS) on the
International Space Station (ISS). AMS is precision general purpose particle
instrument, sensitive to a wide range of particle energies and types, well
documented in a series of on-going papers and data releases.111See
https://ams02.space for a full list of references and results In particular,
our interest is AMS measurements of Galactic cosmic ray protons, which are
described in Ref. 17, 18, 19. The AMS team has extensively calibrated the
instrument, developed a highly detailed simulator, and carefully verified all
steps of the data analysis process.

At this point the question could be asked, how can AMS tell us about GCR
protons when it orbits in LEO? As already stated, our knowledge of the ISS
orbit and the geomagnetic shielding experienced at any point in the orbit
allows for filtering of the proton data, to include only those events with
energies well above the cut-off rigidity at that orbital position. As the ISS
orbit traverses a wide range of cut-off rigidities, the GCR proton spectrum,
as it would be seen outside geomagnetic influence, can be reconstructed. The
AMS team has done such a reconstruction, and includes estimates of the
systematic error introduced by this and all the other calibration and
filtering steps. AMS is well suited as an instrument for exploration of the
particle background experienced by any X-ray instrument.

Each publication by the AMS team is accompanied by a public release of the
relevant calibrated data. For GCR protons, there have been three releases:
first, the precise measurement of the proton spectrum;[17] next, the temporal
behavior of proton fluxes as a function of Bartels rotation (27 days, or
approximately the synodic solar rotation period);[18] and finally, the daily
behavior of proton fluxes.[19] We have initially focused on the data binned by
Bartels rotation[18], and are beginning to explore the daily binned data.[19]
The comparison to the daily binned data will be reported in a subsequent
publication.

Figure 3 shows an example of time-resolved spectra for two of the Bartels
rotation bins. The first is in February 2014, during solar maximum, and the
second is three years later, in February 2017, as rates are increasing towards
solar minimum. The overall flux is higher at solar minimum due to reduced
magnetic shielding from the heliosphere. At solar maximum, the flux is
suppressed, particularly at lower energies where the shielding is more
effective. The statistical plus systematic errors on each data bin are too
small to be seen on the plot. AMS proton data extend to much higher proton
energies, of order 1 TeV, but those data are not shown here due to low fluxes
and lack of any variability from heliospheric shielding.

##  3 Comparison of ACIS to AMS

Figure 4: A comparison of the ACIS-S3 high energy reject rate (orange) and the
AMS proton flux (blue). The ACIS data have been rebinned to match the Bartels
(27 day) binning of the AMS proton flux data. The 1-sigma error bars are
plotted, but are too small to see. This plot does not represent a fit, just a
scaling that allows both sets of data to fit in the same box. The
correspondence between the two data sets is quite striking, both the overall
shape and the shorter timescale features.

Comparison of the ACIS-S3 high energy reject rate to AMS proton fluxes may
help estimate the proton energies most responsible for the ACIS background and
the strength of any correlation may indicate the effectiveness of the
instrumental proxy on background knowledge. Before this comparison, however,
the ACIS data requires filtering and rebinning. As seen in Figure 2, there is
a locus of data points following the quiescent background level, but also
numerous outliers above the quiescent rate. The outliers represent particular
solar storms that elevated the particle rate. As we are interested in the
quiescent variations, these positive outliers were removed. Negative outliers
were not removed, since these represent longer term suppression of the
particle background after particularly vigorous solar storms. Thirteen ACIS
observations were removed, leaving a total of 1,044 observations over the time
period covered by the AMS 27-day binned data.

Figure 4 is a first comparison between the ACIS high energy reject rates and
the AMS proton fluxes from 0.4 to 60 GeV, both binned to the 27-day Bartels
rotation period. This plot does not represent a fit, just a scaling that
allows both sets of data to fit in the same box. One-sigma error bars are
included, but are too small to see (statistical plus systematic error for AMS,
statistical only for ACIS). There is clearly a strong correlation, both in the
long-term solar cycle changes and in the shorter timescale features. This plot
also highlights how the shorter term behavior is different between the period
with declining rates, with many sharp features, and the period with ascending
rates, which is smoother.

Figure 5: ACIS-S3 high energy count rates as a function of AMS proton flux.
The data are clearly well correlated. A linear fit is also shown. The scatter
about the fit may be due to uneven sampling of the ACIS data within the time
bin, or real differences between the two samples.

Figure 5 directly plots the ACIS high energy count rates against the AMS
proton flux. This further demonstrates correlation between the ACIS background
proxy and the actual particle measurements. A linear fit is also shown, which
has a high degree of significance and no obvious curvature or other large
scale deviation. There is some scatter from the fit, possibly due to the
uneven sampling of the ACIS data within each time bin, or possibly real
differences between the two samples. The linear fit, if extrapolated to zero
proton flux, leaves a residual ACIS count rate of $\sim 25$ counts/frame which
is at least partially due to the unvarying pedestal provided by the hard X-ray
background.

To better understand how much of the scatter in the correlation is due to
variation within each 27-day time bin, an additional comparison is shown in
Figure 6. This figure is quite similar to Figure 4, except that the original
unbinned ACIS data are shown against the Bartels-rotation-binned AMS data.
While the correlation is still clear, there is substantial variation in the
ACIS data on timescales shorter than 27 days. This is certainly an argument
for examing the more recent AMS data with daily binning.

Figure 6: A comparison of the ACIS-S3 high energy reject rate (orange) and the
AMS proton flux (blue). The only difference between this figure and Figure 4
is that the ACIS data are unbinned and show the size of the variation in the
ACIS data within a 27 day Bartel rotation time bin.

A further check on the scatter in the correlation is shown in Figure 7. Here
the difference between the binned ACIS data and the linear fit is shown as a
function of the AMS proton flux. What’s new is that instead of statistical
error bars, this shows the RMS of the individual ACIS observations within each
bin. The RMS is always larger than the difference between the ACIS data and
the linear fit, indicating that the shorter timescale variations may explain
some of the scatter in the correlation.

Figure 7: The difference between the ACIS high energy count rate and the
linear model, plotted as a function of the AMS proton flux. The data and the
model are the same as that in Figure 5, but instead of statistical error bars,
the measured RMS of the individual observations in each bin is shown. The RMS
within each bin is much larger than the scatter about the fit.

A final question that can be addressed is whether these data can be used to
estimate the proton energies most responsible for the ACIS particle
background. To address this, we perform linear fits to the binned ACIS high
energy proxy and the AMS proton fluxes, but separately for each proton energy
bin. If some proton energies are better correlated with ACIS, that can be an
indication that those energies are more important in producing the ACIS
particle background. In Figure 8 we show the RMS deviation from each of these
linear fits, as a function of the proton energy in that bin. A minimum can be
seen around 1 GeV, but there’s a broader range of proton energies that can
reasonably fit the ACIS data, from the lowest bin at 0.5 GeV to a few GeV. In
reality, the ACIS focal plane experiences a range of proton energies, but
protons around 1 GeV seem most important. This is in agreement with similar
studies done with XMM[16, 15].

Figure 8: RMS deviation from a linear fit to the ACIS high energy count rate
and AMS proton fluxes, each proton energy bin separately. The best fits,
lowest RMS values, are at the lower end of the energy scale around 1 GeV.

##  4 Conclusions and Future Work

In this proceedings paper, we have compared a particle background proxy
observed by ACIS on the Chandra X-ray Observatory with precision proton flux
measurements by AMS on the ISS. The ACIS background proxy shows a wide range
of variability behaviors, from the long-term solar cycle to shorter periods of
months or even days. The proton flux data also show variability, particularly
for lower energies below 10 GeV or so, where the heliosphere magnetic
shielding is important. For the proton data binned to the 27-day Bartels
rotation period, we find a good correlation between the temporal behavior of
the background proxy and the actual particle measurements. There is
substantial variation of the ACIS background proxy data within each Bartel
time bin, which may account for some of the scatter in the correlation. We
also find that the fits are proton energy dependent, with a best fit around 1
GeV and acceptable fits from the lowest energy bin at 0.5 GeV to a few GeV.

This work is exploratory, intended to determine if further investigation is
warranted. It is clear that AMS data are well suited for comparison to
particle background measurements from X-ray detectors. An obvious avenue to
pursue is comparison to the daily binned AMS proton fluxes, released late last
year. This potentially may explain some of the scatter in the 27 day Bartels
rotation binned data. It is also possible that some portion of the scatter is
due real spatial differences in the particle environment seen by AMS and by
Chandra. A question for future missions is whether in situ measurements by an
onboard particle monitor are more valuable than higher precision measurements
from a special purpose instrument like AMS. Further examination of the AMS
data may help answer this and guide decision-making in the future.

###### Acknowledgements.

We thank undergraduates Brian Avendano (MIT) and Alexander Yu (Oberlin) for
the initial work on the Jupyter notebook used for this comparison. We are
grateful to the AMS team for their thorough documentation and the public
availability of AMS data. This work was done as part of the Athena WFI
Background Working Group, a consortium including MPE, INAF/IASF-Milano, IAAT,
Open University, MIT, SAO, and Stanford. We gratefully acknowledge support
from NASA grant NNX17AB07G and NASA contracts NAS 8-37716 and NAS 8-38252.

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# Differential Operator Representation of
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ as Modules over Univariate and
Bivariate Hermite Polynomials

Manouchehr Amiri<EMAIL_ADDRESS>
###### Abstract

This paper presents the connections between univariate and bivariate Hermite
polynomials and associated differential equations with specific
representations of Lie algebra sl(2,R) whose Cartan sub-algebras coincide the
associated differential operators of these differential equations . Applying
the Baker-Campbell-Hausdorff formula to generators of these algebras, results
in new relation for one-variable and Bivariate Hermite polynomials. A general
form of sl(2,R) representation by differential operators and arbitrary
polynomial basis such as Laguerre and Legendre polynomials is introduced.

###### keywords:

Bivariate Hermite Polynomial,Lie Algebra,Baker-Campbell-Hausdorff formula,
generating function, sl(2,R) algebra

###### keywords:

Bivariate Hermite polynomials, Lie Algebra, Baker-Campbell-Hausdorff formula,
Rodrigues formula, sl(2,R)

##  1 Introduction

Hermite polynomials are among the most applicable orthogonal polynomials.
These polynomials arise in diverse fields of probability, physics, numerical
analysis, and signal processing. As an example, in quantum mechanics,
eigenfunction solutions to quantum harmonic oscillators are described in terms
of Hermite polynomials. Bivariate Hermite polynomials are useful in algebraic
geometry and two-dimensional quantum harmonic oscillators [1],[2],[3] and [4].
Concerning the field of applications, Hermite polynomials in one variable are
divided into probabilistic and physicist versions. In the present paper, we
focus on one and two-dimensional probabilistic Hermite polynomials which are
denoted as $H_{n}^{e}\left(x\right)$ and $H_{n,m}(x,y,\Lambda)$ respectively.
We prove that the symmetries of associated differential equations and
associated differential operators are compatible with
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ algebra. By introducing isomorphic
Lie algebras whose Cartan sub-algebras are the Hermite differential operators,
applying the Baker-Campbell-Hausdorff formula yields new relations for
univarite and bivariate Hermite polynomials in sections (2),(5). In section
(6) a general form of $\mathfrak{sl\left(\mathrm{2,R}\right)}$ representation
by certain differential operators and arbitrary polynomial basis including
Laguerre and Legendre polynomials is introduced.

##  2  Associated Lie algebra of Hermite polynomials of one variable

Probabilistic Hermite polynomials, is presented as:

$H_{n}^{e}\left(x\right)=e^{-\frac{D^{2}}{2}}x^{n}$ (1)

$H_{n}^{e}\left(x\right)$ are probabilistic Hermite polynomials or
equivalently the solutions to Hermite differential equations:

$\mathbb{D_{\mathrm{H}}}H_{n}^{e}\left(x\right)=\left(xD-D^{2}\right)H_{n}^{e}\left(x\right)=nH_{n}^{e}\left(x\right)$
(2)

where:

$D=\frac{d}{dx}$ (3)

and $\mathbb{D_{\mathrm{H}}}=\left(xD-D^{2}\right)$ is denoted as Hermite
differential operator. This is an eigenvalue problem with positive integer
eigenvalues $n$. The equation (1) is the transformation of basis
$\left(1,x,x^{2},x^{3},...\right)$ under the action of operator
$e^{-\frac{D^{2}}{2}}$.This is compatible with Rodrigues’ formula and results
in probabilistic Hermite polynomials $H_{n}^{e}(x)$.

These monomials expand a polynomial vector space $\mathbb{V\mathfrak{}}$.
$\text{ The operator }e^{-\frac{D^{2}}{2}}$ changes the basis $x^{n}$ into the
basis $H_{n}^{e}(x)$. Let ${\mathfrak{gl\left(\mathbb{V}\right)}}$ denote the
linear transformation that maps $\mathbb{V}$ onto itself. We present
isomorphic Lie algebras $\mathfrak{sl\left(\mathrm{2,R}\right)}$ to be defined
by $\mathfrak{sl\left(\mathrm{2,R}\right)}$ modules on vector space
$\mathbb{V}$ which is a linear map $\phi$ defined by
$\phi:\mathfrak{sl\left(\mathrm{2,R}\right)}\to{\mathfrak{gl\left(\mathbb{V}\right)}}$
that preserves the commutator relations of
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ [5],[6].

$\phi\left[a,b\right]=\left[\phi\left(a\right),\phi\left(b\right)\right]$ (4)

This representation is $\mathfrak{sl\left(\mathrm{2,R}\right)}$ modules on
vector space $\mathbb{V}$. First, we review the structure of the irreducible
vector field representation of $\mathfrak{sl\left(\mathrm{2,R}\right)}$. The
generators of this algebra in matrix representation are as follows:

$H=\frac{1}{2}\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}\quad
X=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}\quad Y=\begin{bmatrix}0&1\\\
0&0\end{bmatrix}$ (5)

The commutation relations for this representation of
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ are:

$\left[X,Y\right]=2H\quad\left[H,X\right]=-X\quad\left[H,Y\right]=Y$ (6)

Let’s define a representation of $\mathfrak{sl\left(\mathrm{2,R}\right)}$ as
its module on $\mathbb{V}$ that preserves commutation relations of the
following differential operators as its generators [5]:

$\textbf{h = }xD-\frac{n}{2}\quad\textbf{e = }D=\frac{\partial}{\partial
x}\quad\textbf{f = }x^{2}D-nx$ (7)

With the same commutation relations

$\left[\textbf{e},\textbf{f}\right]=2\textbf{h}\quad\left[\textbf{h},\textbf{e}\right]=-\textbf{e}\quad\left[\textbf{h},\textbf{f}\right]=\textbf{f}$
(8)

The Cartan sub-algebra h produces a decomposition of representation space:

$\mathbb{V=}\oplus{V}_{j}$ (9)

${V}_{j}$ are the eigenspaces (eigenfunctions) of generator h as Cartan sub-
algebra of $\mathfrak{sl\left(\mathrm{2,R}\right)}$ and provides the solutions
to the related differential equation:

$\textbf{h}{V}_{j}=f\left(j\right){V}_{j}$ (10)

As an example, monomials $x^{n}$ are eigenfunctions or eigenspaces of
generator h realized as eigenspace ${V}_{j}$. The eigenvalues $f(j)$ in most
cases equals an integer $j$ or $j\left(j+1\right)$ as we observe in Hermite,
Laguerre, and Legendre differential equations. We search for a Lie algebra
$\mathfrak{L_{\mathrm{H}}}$ isomorphic to
$\mathfrak{sl\left(\mathrm{2,R}\right)})$ algebra that its generators to be
defined by Hermite differential operators. Here we apply the transformation
operator $e^{-\frac{D^{2}}{2}}$ as described in (1) for Hermite polynomials to
derive similarity transformations (conjugation) of
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ bases as follows:

$X_{1}=e^{-\frac{D^{2}}{2}}\textbf{h }e^{\frac{D^{2}}{2}}\quad
X_{2}=e^{-\frac{D^{2}}{2}}\textbf{e }e^{\frac{D^{2}}{2}}\quad
X_{3}=e^{-\frac{D^{2}}{2}}\textbf{f }e^{\frac{D^{2}}{2}}$ (11)

Concerning to a theorem in Lie algebra theory, these generators constitute an
isomorphic Lie algebra to $\mathfrak{sl\left(\mathrm{2,R}\right)})$ with
similar commutation relations. We call this algebra as “ Hermite operator Lie
algebra”. Due to the equation (1) that implies the change of basis $x^{n}$ to
$H_{n}^{e}$, the operator $xD$ with eigenfunctions $x^{n}$ corresponds an
operator $\mathbb{D_{\mathrm{H}}}$ with eigenfunctions $H_{n}^{e}$ and common
eigenvalues $n$ (we call $\mathbb{D_{\mathrm{H}}}$ as the Hermite differential
operator) through a similarity transformation described by

$\mathbb{D_{\mathrm{H}}}=e^{-\frac{D^{2}}{2}}\left(xD\right)e^{\frac{D^{2}}{2}}$
(12)

Therefore we have:

$X_{1}=e^{-\frac{D^{2}}{2}}\textbf{h
}e^{\frac{D^{2}}{2}}=e^{-\frac{D^{2}}{2}}\left(xD-\frac{n}{2}\right)e^{\frac{D^{2}}{2}}=\mathbb{D_{\mathrm{H}}}-\frac{n}{2}$
(13)

Generator $X_{2}$ simply be calculated as $X_{2}=D$.

######  Proposition 1.

For $X_{3}$ we have:

$X_{3}=\left(x-D\right)\left(\mathbb{D_{\mathrm{H}}}-n\right)$ (14)

###### Proof.

by equation (12) we have the identity:

$\mathbb{D_{\mathrm{H}}}=\left(e^{-\frac{D^{2}}{2}}xe^{\frac{D^{2}}{2}}\right)\left(e^{-\frac{D^{2}}{2}}De^{\frac{D^{2}}{2}}\right)=e^{-\frac{D^{2}}{2}}xe^{\frac{D^{2}}{2}}D$
(15)

Thus:

$\mathbb{D_{\mathrm{H}}}D^{-1}=e^{-\frac{D^{2}}{2}}xe^{\frac{D^{2}}{2}}$ (16)

By equation (2) we have:

$\left(xD-D^{2}\right)D^{-1}=\left(x-D\right)=e^{-\frac{D^{2}}{2}}xe^{\frac{D^{2}}{2}}$
(17)

Now for $X_{3}$ from (10) we have:

$X_{3}=e^{-\frac{D^{2}}{2}}\textbf{f
}e^{\frac{D^{2}}{2}}=e^{-\frac{D^{2}}{2}}(x^{2}D-nx)e^{\frac{D^{2}}{2}}=e^{-\frac{D^{2}}{2}}x^{2}De^{\frac{D^{2}}{2}}-ne^{-\frac{D^{2}}{2}}xe^{\frac{D^{2}}{2}}$
(18)

By equations (12) and (17) we get:

$X_{3}=[e^{-\frac{D^{2}}{2}}xe^{\frac{D^{2}}{2}}][e^{-\frac{D^{2}}{2}}xDe^{\frac{D^{2}}{2}}]-n(x-D)=(x-D)\mathbb{D_{\mathrm{H}}}-n(x-D)=(x-D)(\mathbb{D_{\mathrm{H}}}-n)$
(19)

∎

The commutation relations of these bases which coincide with the Lie algebra
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ are as follows

$\left[X_{1},X_{2}\right]=-X_{2}\quad\left[X_{1},X_{3}\right]=X_{3}\quad\left[X_{2},X_{3}\right]=2X_{1}$
(20)

######  Proposition 2.

Hermite polynomials satisfy the equation:

$e^{-\frac{D^{2}}{2}}(e^{-1}+x)^{n}=e^{(\mathbb{D\mathrm{{}_{H}}}-n-\frac{D}{1-e})}H_{n}^{e}$
(21)

###### Proof.

Due to a theorem for the BCH formula [7], [8], if $\left[X,Y\right]=sY$ for
$s\in\mathbb{R}$ we have:

$e^{X}e^{Y}=e^{X+\frac{s}{1-e^{-s}}Y}$ (22)

Respect to equation (20) for

$\left[X_{1},X_{2}\right]$, the BCH formula for $X_{1}$ and $X_{2}$ generators
gives:

$e^{(xD-D^{2})}e^{D}=e^{[(xD-D^{2})-\frac{D}{1-e}]}$ (23)

The term $\frac{n}{2}$ in $X_{1}$ was omitted because it has no role in
commutation relation of $\left[X_{1},X_{2}\right]$. Multiplying both sides by
$H_{n}^{e}(x)$ gives:

$e^{(xD-D^{2})}e^{D}H_{n}^{e}(x)=e^{[(xD-D^{2})-\frac{D}{1-e}]}H_{n}^{e}(x)$
(24)

For $e^{D}H_{n}^{e}(x)$ we obtain:

$e^{D}H_{n}^{e}(x)=(1+D+\frac{D^{2}}{2}+...)H_{n}^{e}(x)=H_{n}^{e}(x)+nH^{e}_{n-1}(x)+\frac{n(n-1))}{2!}H^{e}_{n-2}(x)+...+1$
(25)

Thus:

$e^{D}H_{n}^{e}(x)=\sum_{k=0}^{n}\binom{n}{k}H^{e}_{n-k}(x)$ (26)

Substituting in equation (24) and replacing Hermite differential
operator$(xD-D^{2})$ with $\mathbb{D\mathrm{{}_{H}}}$ gives:

$e^{\mathbb{D_{\mathrm{H}}}}\sum_{k=0}^{n}\binom{n}{k}H^{e}_{n-k}(x)=e^{\left[\mathbb{D_{\mathrm{H}}}-\frac{\mathrm{D}}{1-e}\right]}H^{e}_{n}(x)$
(27)

Then we have:

$\sum_{k=0}^{n}\binom{n}{k}e^{\mathbb{D}_{\mathrm{H}}}H^{e}_{n-k}(x)=\sum_{k=0}^{n}\binom{n}{k}e^{n-k}H^{e}_{n-k}(x)=e^{\left[\mathbb{D_{\mathrm{H}}}-\frac{\mathrm{D}}{1-e}\right]}H^{e}_{n}(x)$
(28)

By identity $H_{n}^{e}\left(x\right)=e^{-\frac{D^{2}}{2}}x^{n}$ we get:

$\sum_{k=0}^{n}\binom{n}{k}e^{n-k}e^{-\frac{D^{2}}{2}}x^{n-k}=e^{\left[\mathbb{D_{\mathrm{H}}}-\frac{\mathrm{D}}{1-e}\right]}H^{e}_{n}(x)$
(29)

$e^{-\frac{D_{2}}{2}}\sum_{k=0}^{n}\binom{n}{k}(e^{-1})^{k}x^{n-k}=e^{\left[\mathbb{D_{\mathrm{H}}}-n-\frac{\mathrm{D}}{1-e}\right]}H^{e}_{n}(x)$
(30)

$e^{-\frac{D_{2}}{2}}(e^{-1}+x)^{n}=e^{\left[\mathbb{D_{\mathrm{H}}}-n-\frac{\mathrm{D}}{1-e}\right]}H^{e}_{n}(x)$
(31)

∎

Thus a new version of Hermite polynimial equation is as follows:

$e^{-\left[\mathbb{D_{\mathrm{H}}}-n-\frac{\mathrm{D}}{1-e}\right]}e^{-\frac{D_{2}}{2}}(e^{-1}+x)^{n}=H^{e}_{n}(x)$
(32)

##  3 Bivariate Hermite Polynomials

An ordinary definition for bivariate Hermite polynomials is as follows
[1],[4],[3],[9]:

$H_{n,m}(s,t,\Lambda)=(-1)^{m+n}e^{\varphi(s,t)/2}\frac{\partial^{m+n}}{\partial
x^{m}{\partial x^{n}}}e^{-\varphi(s,t)/2}$ (33)

With definition $\varphi(s,t)=as^{2}+2bst+ct^{2}$ as a positive definite
quadratic form and $\Lambda=\begin{bmatrix}a&b\\\ b&c\end{bmatrix}$ , the
equivalent equation for $H_{n,m}(s,t,\Lambda)$ reads as [1],[4],[3],[9]:

$H_{n,m}(s,t,\Lambda)=\sum_{k=0}^{min(n,m)}(-1)^{k}k!\binom{m}{k}\binom{n}{k}a^{\frac{n-k}{2}}b^{k}c^{\frac{m-k}{2}}H^{e}_{n-k}(\frac{as+bt}{\sqrt{a}}H^{e}_{m-k}(\frac{bs+ct}{\sqrt{c}})$
(34)

With $ac-b^{2}>0$ , $a,c>0$ .

By changing variables $x=\frac{as+bt}{\sqrt{a}}$ and
$y=\frac{bs+ct}{\sqrt{c}}$

we have:

$\hat{H}_{n,m}(x,y,\Lambda)=\sum_{k=0}^{min(n,m)}(-1)^{k}k!\binom{m}{k}\binom{n}{k}a^{\frac{n-k}{2}}b^{k}c^{\frac{m-k}{2}}H^{e}_{n-k}(x)H^{e}_{m-k}(y)$
(35)

These polynomials satisfy the partial differential equation [1], [10]:

$\left[(x\frac{\partial}{\partial x}-\frac{\partial^{2}}{\partial
x^{2}})+(y\frac{\partial}{\partial y}-\frac{\partial^{2}}{\partial
y^{2}})-2\frac{b}{\sqrt{ac}}\frac{\partial^{2}}{\partial x\partial
y}\right]\hat{H}_{n,m}(x,y,\Lambda)=(m+n)\hat{H}_{n,m}(x,y,\Lambda)$ (36)

Let denote $\mathfrak{D}$ as the differential operator in equation (36)

$\mathfrak{D}=(x\frac{\partial}{\partial x}-\frac{\partial^{2}}{\partial
x^{2}})+(y\frac{\partial}{\partial y}-\frac{\partial^{2}}{\partial
y^{2}})-2\frac{b}{\sqrt{ac}}\frac{\partial^{2}}{\partial x\partial y}$ (37)

If we denote $\partial_{x}=\frac{\partial}{\partial x}$ and
$\partial_{y}=\frac{\partial}{\partial y}$ ,with the identities:

$e^{-\frac{{\partial^{2}_{x}}}{2}}x^{m-k}=H^{e}_{m-k}(x),e^{-\frac{{\partial^{2}_{y}}}{2}}y^{n-k}=H^{e}_{n-k}(y)$
(38)

Equation (35) changes to :

$\hat{H}_{n,m}(x,y,\Lambda)=e^{-\frac{{\partial^{2}_{x}}}{2}}e^{-\frac{{\partial^{2}_{y}}}{2}}\sum_{k=0}^{min(n,m)}(-1)^{k}k!\binom{m}{k}\binom{n}{k}a^{\frac{n-k}{2}}b^{k}c^{\frac{m-k}{2}}x^{m-k}y^{n-k}$
(39)

We denote the new polynomials as $u_{n,m}(x,y)$ defined by:

$u_{n,m}(x,y)=\sum_{k=0}^{min(n,m)}(-1)^{k}k!\binom{m}{k}\binom{n}{k}a^{\frac{n-k}{2}}b^{k}c^{\frac{m-k}{2}}x^{m-k}y^{n-k}$
(40)

If these polynomials are assumed as the linearly independent basis, the
transformation from these bases to $\hat{H}_{n,m}(x,y,\Lambda)$ is as follows:

$\hat{H}_{n,m}(x,y,\Lambda)=e^{-\frac{{\partial^{2}_{x}}}{2}}e^{-\frac{{\partial^{2}_{y}}}{2}}u_{n.m}(x,y)=e^{-\frac{{\partial^{2}_{x}}+\partial^{2}_{y}}{2}}u_{n,m}(x,y)$
(41)

Therfore, the corresponding differential operator with $u_{n,m}(x,y)$ as its
eigenfunctions could be derived by similarity transformation:

$\mathfrak{D^{\prime}}=e^{\mathfrak{}\frac{{\partial^{2}_{x}}+\partial^{2}_{y}}{2}}\mathfrak{D}e^{-\frac{{\partial^{2}_{x}}+\partial^{2}_{y}}{2}}$
(42)

$\mathfrak{D}$ is denoted as the differential operator given in the eigenvalue
equation (36).Thus:

$\mathfrak{D^{\prime}}=e^{\frac{{\partial^{2}_{x}}+\partial^{2}_{y}}{2}}\left[\left(x\partial_{x}-\partial_{x}^{2}\right)+\left(y\partial_{y}-\partial_{y}^{2}\right)-2\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}\right]e^{-\frac{{\partial^{2}_{x}}+\partial^{2}_{y}}{2}}$
(43)

######  Proposition 3.

The reduced form of $\mathfrak{D^{\prime}}$ is:

$\mathfrak{D^{\prime}}=x{\partial_{x}}+y{\partial_{y}}-2\frac{b}{\sqrt{ac}}{\partial_{x}}{\partial_{y}}$
(44)

###### Proof.

Respect to equation (36) and the commutativity of $\partial^{2}_{x}$ and
$\partial^{2}_{y}$ we get:

$\mathfrak{D^{\prime}}=e^{\frac{{\partial^{2}_{x}}}{2}}\left(x{\partial}_{x}-{\partial}_{x}^{2}\right)e^{-\frac{{\partial^{2}_{x}}}{2}}+e^{\frac{{\partial^{2}_{y}}}{2}}\left(x{\partial}_{y}-{\partial}_{y}^{2}\right)e^{-\frac{{\partial^{2}_{y}}}{2}}-2\frac{b}{\sqrt{ac}}\left(e^{\frac{{\partial^{2}_{x}}}{2}}{\partial}_{x}e^{-\frac{{\partial^{2}_{x}}}{2}}\right)\left(e^{\frac{{\partial^{2}_{y}}}{2}}{\partial}_{y}e^{-\frac{{\partial^{2}_{y}}}{2}}\right)$
(45)

With respect to equations (2) and (12)ans (16) we have :

$\mathbb{D_{\mathrm{H}}}(x)=e^{\frac{{-\partial^{2}_{x}}}{2}}x\partial_{x}e^{\frac{{\partial^{2}_{x}}}{2}}$
(46)

$e^{\frac{{\partial^{2}_{x}}}{2}}\mathbb{D_{\mathrm{H}}}(x)e^{\frac{{-\partial^{2}_{x}}}{2}}=e^{\frac{{\partial^{2}_{x}}}{2}}(x\partial_{x}-\partial^{2}_{x})e^{\frac{{-\partial^{2}_{x}}}{2}}=x\partial_{x}$
(47)

Repeating for $\mathbb{D_{\mathrm{H}}}(y)$ yields :

$e^{\frac{{\partial^{2}_{y}}}{2}}(y\partial_{y}-\partial^{2}_{y})e^{\frac{{-\partial^{2}_{y}}}{2}}=y\partial_{y}$
(48)

We have the identities:
$e^{\frac{{-\partial^{2}_{x}}}{2}}\partial_{x}e^{\frac{{\partial^{2}_{x}}}{2}}=\partial_{x}$
,
$e^{\frac{{-\partial^{2}_{y}}}{2}}\partial_{y}e^{\frac{{\partial^{2}_{y}}}{2}}=\partial_{y}$

Then equation (37) reduces to:

$\mathfrak{D^{\prime}}=x{\partial_{x}}+y{\partial_{y}}-2\frac{b}{\sqrt{ac}}{\partial_{x}}{\partial_{y}}$
(49)

∎

Therefore the differential operator $\mathfrak{D^{\prime}}$ satisfies the
differential equation:

$\mathfrak{D^{\prime}}u_{n,m}(x,y)=(m+n)u_{n,m}(x,y)$ (50)

The eigenvalues of $\mathfrak{D^{\prime}}$ are the same as the differential
equation (36), because $\mathfrak{D}$ and $\mathfrak{D^{\prime}}$ are related
by the similarity relation (42).

##  4 sl(2,R) Modules on Bivariate Hermite Polynomials

In this section we introduce an associated Lie algebra of bivariate Hermite
differential operator. First, we search for the compatible
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ algebra in terms of differential
operators of two variables. with respect to equations (6) and (7) the Cartan
sub-algebra of sl(2,R) can be taken as:

$\textbf{h}=\frac{1}{2}(x{\partial_{x}}+y{\partial_{y}}+1)+\alpha{\partial_{x}}{\partial_{y}}$
(51)

The additional term $\alpha{\partial_{x}}{\partial_{y}}$ has been chosen to
satisfy the required commutation relations. The other generators are proposed
as

$\textbf{e}=\alpha{\partial_{x}}{\partial_{y}}$ (52)

$\textbf{f}=\frac{1}{2\alpha}xy+\frac{1}{2}(x\partial_{x}+y\partial_{y})+\frac{\alpha}{4}\partial_{x}\partial_{y}$
(53)

These generators satisfy the commutation relations of
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ as described in equation (7). By
substituting $\alpha=\frac{-b}{\sqrt{ac}}$ the differential operator h
satisfies the differential equation:

$\textbf{h}u_{n,m}(x,y)=\left[(x\partial_{x}+y\partial_{y}+1)-\frac{2b}{\sqrt{c}}\partial_{x}\partial_{y}\right]u_{n,m}(x,y)$
(54)

Respect to equations (43),(44) and (45) we have:

$\textbf{h}u_{n,m}(x,y)=\frac{1}{2}(m+n+1)u_{n,m}(x,y)$ (55)

Thus $u_{n.m}(x,y)$ are eigenfunctions or weight vectors of h as the Cartan
sub-algebra. According to the equation (39), With respect to equations (12)
and (13), the similarity transformation of generators $\textbf{h},$e and f by
operator $e^{-\mathfrak{}\frac{{\partial^{2}_{x}}+\partial^{2}_{y}}{2}}$
yields:

$\textbf{h}^{\prime}=\frac{1}{2}(\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)+1)-\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}$
(56)

$\textbf{e}^{\prime}=-\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}$ (57)
$\textbf{f}^{\prime}=-\frac{\sqrt{ac}}{2b}(x-{\partial_{x}})(y-{\partial_{y}})+\frac{1}{2}(\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)+1)-\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}$
(58)

$\mathbb{D}_{\mathrm{H}}(x)$ and $\mathbb{D}_{\mathrm{H}}(y)$ are defined in
equation (2) and (46). The bivariate Hermite polynomials $H_{n,m}(x,y)$ are
eigenfunctions of $\textbf{h}^{\prime}$ with eigenvalues $\frac{1}{2}(m+n+1)$.

The lowering operator in this algebra is given by:

$A^{-}=e^{\prime}=-\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}$ (59)

$\textbf{h}^{\prime}$ represents the Cartan subalgebra of related Lie algebra.
One of the commutator relations is

$\left[\textbf{h}^{\prime},A^{-}\right]=-A^{-}=-\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}$
(60)

##  5  BCH Formula and sl(2,R) Generators of Bivariate Hermite Polynomial

######  Theorem 4.

If in equation (35), $b$ be replaced by $b^{\prime}=2b(1-e)$ and
$\mathfrak{D}$ is defined by equation (37), the resultant bivariate Hermite
polynomial $\hat{H}_{n,m}(x,y,\Lambda^{\prime})$ satisfies the equation :

$e^{\mathfrak{D}}\hat{H}_{n,m}(x,y,\Lambda^{\prime})=a^{n/2}c^{m/2}e^{m+n}H_{n}^{e}(x)H_{m}^{e}(y)$
(61)

###### Proof.

Due to a theorem for the BCH formula [7] , [8], if $\left[X,Y\right]=sY$ then
we have:

$e^{X}e^{Y}=e^{X+\frac{s}{1-e^{-s}}Y}$ (62)

we choose $X$ and $Y$ in such a way that the BCH formula is simplified to
equations that gives rise to new relations of bivariate Hermite polynomials.
Let assume $X$ and $Y$ in the form of bases introduced in equations (56) and
(57) :

$X=\mathfrak{D}=\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)-2\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}$
(63)

$Y=-\frac{2(1-e)b}{\sqrt{ac}}\partial_{x}\partial_{y}$ (64)

Where we used equation (46) for variables $X$ and $Y$. With respect to the
commutation relation

$\left[X,Y\right]=-Y=\frac{2(1-e)b}{\sqrt{ac}}\partial_{x}\partial_{y}$ (65)

Then, we have

$exp(X)exp(Y)=exp(X-\frac{Y}{1-e})=exp(X+\frac{2b}{\sqrt{ac}}\partial_{x}\partial_{y})=exp(\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)$
(66)

$exp[\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)-2\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}]exp[-\frac{2(1-e)b}{\sqrt{ac}}\partial_{x}\partial_{y}]=exp[\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)]$
(67)

Multiplying both sides of (67) by $H_{n}^{e}(x)H_{m}^{e}(y)$ yields:

$e^{\left[\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)-2\frac{b}{\sqrt{ac}}\partial_{x}\partial_{y}\right]}e^{\left[-\frac{2(1-e)b}{\sqrt{ac}}\partial_{x}\partial_{y}\right]}H_{n}^{e}(x)H_{m}^{e}(y)=e^{\left[\mathbb{D}_{\mathrm{H}}(x)+\mathbb{D}_{\mathrm{H}}(y)\right]}H_{n}^{e}(x)H_{m}^{e}(y)$
(68)

$e^{\mathfrak{D}}\sum_{k=0}^{min(m,n)}(-1)^{n}k!\binom{n}{k}\binom{m}{k}\left[\frac{2b(1-e)}{\sqrt{ac}}\right]^{k}H_{n-k}^{e}(x)H_{m-k}^{e}(y)=e^{m+n}H_{n}^{e}(x)H_{m}^{e}(y)$
(69)

Comparing this with equation (35) shows the replacement $b\to
b^{\prime}=2b(e-1)$ yields a new bivariate Hermite polynomial
$\hat{H}_{n,m}(x,y,\Lambda^{\prime})$ :

$\hat{H}_{n,m}(x,y,\Lambda)=\sum_{k=0}^{min(n,m)}(-1)^{k}k!\binom{m}{k}\binom{n}{k}a^{\frac{n-k}{2}}b^{k}c^{\frac{m-k}{2}}H^{e}_{n-k}(x)H^{e}_{m-k}(y)$
(70)

$\hat{H}_{n,m}(x,y,\Lambda^{\prime})=a^{n/2}c^{m/2}\sum_{k=0}^{min(n,m)}(-1)^{k}k!\binom{m}{k}\binom{n}{k}(\frac{b^{\prime}}{\sqrt{ac}})^{k}H^{e}_{n-k}(x)H^{e}_{m-k}(y)$
(71)

Substitution of equation (69) into (71) gives:

$e^{\mathfrak{D}}\hat{H}_{n,m}(x,y,\Lambda^{\prime})=a^{n/2}c^{m/2}e^{m+n}H_{n}^{e}(x)H_{m}^{e}(y)$
(72)

∎

##  6  A General form of sl(2,R) Representation By Differential Operators and
Polynomial Basis

Rodrigues’ formula for some orthogonal polybnomials such as Hermite, Laguerre
and Legendre polynomilas are defined by by the action of specific differential
operators on the n-th integer power of some function $B(x)$ , with
$n\in\mathbb{N}\cup\left\\{0\right\\}$:

$P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}}(x^{2}-1)^{n}$ (73)

$L_{n}(x)=\frac{1}{n!}\left(\frac{d}{dx}-1\right)^{n}x^{n}$ (74)

The $B(x)$ in these equations are $B(x)=(x^{2}-1)$ and $B(x)=x$ for Legendre
and Laguerre polynomials respectively. Let $B(x)$ and its integer powers form
a set $\left\\{1,B(x),B^{2}(x),...\right\\}=\left\\{B^{n}(x)\right\\}_{n\geq
0}$ of independent basis in the infinite dimensional polynomial space. One can
interpret these formulas as transformations from polynomial space basis
$\left\\{1,B(x),B^{2}(x),...\right\\}$ to new basis i.e Legendre, and Laguerre
polynomials. The definition of $B(x)$ in Rodrigues’ formula limited to a
polynomial with degree at most 2. However in this section we use $B(x)$ with
no limitation and as any kind of smooth functions.

######  Proposition 5.

Let $\left\\{1,B(x),B^{2}(x),...\right\\}=\left\\{B^{n}(x)\right\\}_{n\geq 0}$
constitutes a basis for polynomial space. A differential operator
representation of $\mathfrak{sl\left(\mathrm{2,R}\right)}$ modules on these
polynomials is represented as

$\textbf{h}=\frac{B}{B^{\prime}}D-\frac{n}{2}\quad\textbf{e}=\frac{D}{B^{\prime}}\quad\textbf{f}=\frac{B^{2}}{B^{\prime}}D-nB$
(75)

Where $B^{\prime}(x)$ is the derivative of $B(x)$ and $D$ is derivative
operator respect to $x$.

###### Proof.

It is straightforward to prove that these bases satisfy the commutation
relations of $\mathfrak{sl\left(\mathrm{2,R}\right)}$ in equation (8). ∎

The polynomials $B^{n}(x)$ are eigenfunctions of the operator h as Cartan sub-
algebra with integer eigenvalues and as the author proved [11], similarity
transformation of h with operators defined in equations (74) and (75) yields
the Legendre and Laguerre differential operators and equations respectively.
Solutions of these differential equations are the corresponding
eigenfunctions. We showed that the BCH formula mentioned in equation (62)
holds when the commutation relation is $\left[X,Y\right]=sY$ . Therefor this
formula also holds for bases h and e in equation (75) :

$\left[\textbf{h},\textbf{e}\right]=-\textbf{e}$ (76)

Let $\textbf{e}^{\prime}=(1-e)\textbf{e}$ ,where $e$ is Euler number and
$\textbf{h}^{\prime}=\textbf{h}+\textbf{e}$. Then the commutation relation
becomes
$\left[\textbf{h}^{\prime},\textbf{e}^{\prime}\right]=-(1-e)\textbf{e}=-\textbf{e}^{\prime}$
and the BCH formula reads as:

$exp(\textbf{h}^{\prime})exp(\textbf{e}^{\prime})=exp\left(\textbf{h}^{\prime}-\frac{\textbf{e}^{\prime}}{1-e}\right)=exp(\textbf{h})$
(77)

$exp\left(\frac{B}{B^{\prime}}D+\frac{D}{B^{\prime}}\right)exp\left((1-e)\frac{D}{B^{\prime}}\right)=exp\left(\frac{B}{B^{\prime}}D\right)$
(78)

Substituting $B$ in equation (78) by any polynomials of the form of Laguerre
and Legendre polynomial that is compatible with operator relations in
equations (73) and (74), results in new relations.

##  7 Conclusion

By introducing isomorphic Lie algebras to
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ whose generators are defined by
differential operators involved in univariate and bivariate Hermite
differential equations, we showed that these Hermite polynomials, are
eigefunctions of Cartan suba-lgebras of
$\mathfrak{sl\left(\mathrm{2,R}\right)}$ . By using the commutation relations
of these algebras and the BCH formula, new relations for Hermite polynomials
are obtained. A general form of sl(2,R) representation by differential
operators and polynomial basis such as Laguerre, and Legendre differential
polynomials, is introduced.

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## Declarations

  * •

The author confirms sole responsibility for the study conception and
manuscript preparation.

  * •

No funding was received for conducting this study.

  * •

The author declares that he has no conflict of interest.