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In this paper, we introduce the Heine binomial operators H$_{n}(bD_{q})$ based on $q$-differential operator $D_{q}$. The motivation for introducing the operators H$_{n}(bD_{q})$ is that their limit turns out to be the $q$-exponential operator T$(bD_{q})$ given by Chen. The Hahn polynomials $\Phi_{m}^{(q^n)}(b,x|q)$ can easily be represented by using the operators H$_{n}(bD_{q})$. Here, we derive $q$-exponential and ordinary generating function, Mehler's formula, Rogers formula, and other identities for the polynomials $\Phi_{m}^{(q^n)}(b,x|q)$.

This article introduces the deformed homogeneous $(s,t)$-Rogers-Szeg\"o polynomials h$_{n}(x,y;s,t,u,v)$. These polynomials are a generalization of the Rogers-Szeg\"o polynomials and the $(p,q)$-Rogers-Szeg\"o polynomials defined by Jagannathan. By using the deformed $(s,t)$-exponential operator based on operator T$_{a}D_{s,t}$ we find identities involving the polynomials h$_{n}(x,y;s,t,u,v)$, together with generalizations of the Mehler and Rogers formulas. In addition, a generating function for the polynomials h$_{n}(x,y;s,t,u,v)$ is found employing the deformed $\frac{\varphi}{u}$-commuting operators. A representation of deformed $(s,t)$-exponential function as the limit of a sequence of deformed $(s,t)$-Rogers-Szeg\"o polynomials is obtained.

GQ Lup B is one of the few substellar companions with a detected cicumplanetary disk, or CPD. Observations of the CPD suggest the presence of a cavity, possibly formed by an exo-satellite. Using the Keck Planet Imager and Characterizer (KPIC), a high contrast imaging suite that feeds a high resolution spectrograph (1.9-2.5 microns, R$\sim$35,000), we present the first dedicated radial velocity (RV) observations around a high-contrast, directly imaged substellar companion, GQ Lup B, to search for exo-satellites. Over 11 epochs, we find a best and median RV error of 400-1000 m/s, most likely limited by systematic fringing in the spectra due to transmissive optics within KPIC. With this RV precision, KPIC is sensitive to exomoons 0.6-2.8% the mass of GQ Lup B ($\sim 30 M_{\text{Jup}}$) at separations between the Roche limit and $65 R_{\text{Jup}}$, or the extent of the cavity inferred within the CPD detected around GQ Lup B. Using simulations of HISPEC, a high resolution infrared spectrograph planned to debut at W.M. Keck Observatory in 2026, we estimate future exomoon sensitivity to increase by over an order of magnitude, providing sensitivity to less massive satellites potentially formed within the CPD itself. Additionally, we run simulations to estimate the amount of material that different masses of satellites could clear in a CPD to create the observed cavity. We find satellite-to-planet mass ratios of $q > 2 \times 10^{-4}$ can create observable cavities and report a maximum cavity size of $\sim 51 \, R_{\text{Jup}}$ carved from a satellite., Comment: 15 pages, 5 figures

The Keck Planet Imager and Characterizer (KPIC) combines high contrast imaging with high resolution spectroscopy (R$\sim$35,000 in K band) to study directly imaged exoplanets and brown dwarfs in unprecedented detail. KPIC aims to spectrally characterize substellar companions through measurements of planetary radial velocities, spins, and atmospheric composition. Currently, the dominant source of systematic noise for KPIC is fringing, or oscillations in the spectrum as a function of wavelength. The fringing signal can dominate residuals by up to 10% of the continuum for high S/N exposures, preventing accurate wavelength calibration, retrieval of atmospheric parameters, and detection of planets with flux ratios less than 1% of the host star. To combat contamination from fringing, we first identify its three unique sources and adopt a physically informed model of Fabry-P\'{e}rot cavities to apply to post-processed data. We find this strategy can effectively model the fringing in observations of A0V/F0V stars, reducing the residual systematics caused by fringing by a factor of 2. Next, we wedge two of the transmissive optics internal to KPIC to eliminate two sources of fringing and confirm the third source as the entrance window to the spectrograph. Finally, we apply our previous model of the Fabry-P\'{e}rot cavity to new data taken with the wedged optics to reduce the amplitude of the residuals by a factor of 10., Comment: 13 pages, 6 figures

In this paper, we are mainly interested in using the $q$-exponential operator of Chen [8] in proving identities involving the generalized Rogers-Szeg\"o polynomials r$_{n}(x,b;q)$ of Saad [11], the $(s,t)$-derivatives of Partial Theta function $\Theta_{0}(x,q)$ of Ramanujan, the generalized simplicial $d$-polytopic numbers $\fibonomial{n+d}{d}_{s,t}$, and the deformed $(s,t)$-trinomial series, both introduced in this paper. In particular, we will give the deformed and non-deformed ordinary generating functions of the sequences $\rr_{n}(x,b;q)$, $\fibonomial{n+d}{d}_{s,t}$, $\qbinom{n+d}{d}_{q}^2$, and $\qbinom{n+d}{d}_{q}\rr_{n}(x,b;q)$. We also derive $(s,t)$-analogues of the alternating sums of squares and sums of cubes., Comment: 16 pages

Young, self-luminous super-Jovian companions discovered by direct imaging provide a challenging test of planet formation and evolution theories. By spectroscopically characterizing the atmospheric compositions of these super-Jupiters, we can constrain their formation histories. Here we present studies of the recently discovered HIP 99770 b, a 16 MJup high-contrast companion on a 17 au orbit, using the fiber-fed high-resolution spectrograph KPIC (R~35,000) on the Keck II telescope. Our K-band observations led to detections of H2O and CO in the atmosphere of HIP 99770 b. We carried out free retrieval analyses using petitRADTRANS to measure its chemical abundances, including the metallicity and C/O ratio, projected rotation velocity (vsini), and radial velocity (RV). We found that the companion's atmosphere has C/O=0.55(-0.04/+0.06) and [M/H]=0.26(-0.23/+0.24) (1{\sigma} confidence intervals), values consistent with those of the Sun and with a companion formation via gravitational instability or core accretion. The projected rotation velocity < 7.8 km/s is small relative to other directly imaged companions with similar masses and ages. This may imply a near pole-on orientation or effective magnetic braking by a circumplanetary disk. In addition, we added the companion-to-primary relative RV measurement to the orbital fitting and obtained updated constraints on orbital parameters. Detailed characterization of super-Jovian companions within 20 au like HIP 99770 b is critical for understanding the formation histories of this population., Comment: 18 pages, 5 figures, accepted to AJ

The 16 Myr-old A0V star HD 156623 in the Scorpius--Centaurus association hosts a high-fractional-luminosity debris disk, recently resolved in scattered light for the first time by the Gemini Planet Imager (GPI) in polarized intensity. We present new analysis of the GPI H-band polarimetric detection of the HD 156623 debris disk, with particular interest in its unique morphology. This debris disk lacks a visible inner clearing, unlike the majority of low-inclination disks in the GPI sample and in Sco-Cen, and it is known to contain CO gas, positioning it as a candidate ``hybrid'' or ``shielded'' disk. We use radiative transfer models to constrain the geometric parameters of the disk based on scattered light data and thermal models to constrain the unresolved inner radius based on the system's spectral energy distribution (SED). We also compute a measurement of the polarized scattering phase function, adding to the existing sample of empirical phase function measurements. We find that HD 156623's debris disk inner radius is constrained to less than 26.6 AU from scattered light imagery and less than 13.4 AU from SED modeling at a 99.7% confidence interval, and suggest that gas drag may play a role in retaining sub-blowout size dust grains so close to the star., Comment: 18 pages, 10 figures. Accepted for publication in AJ

In this paper, we define the deformed Euler $(s,t)$-numbers ${\rm e}_{s,t,u}$ Furthermore, we prove that ${\rm e}_{as,a^2t,u^{-1}}$ and ${\rm e}_{as,a^2t,u^{-1}}^{-1}$ are irrational numbers when $a,u\in\mathbb{Q}$ and $\vert au\vert>1$, thus providing a countable infinite family of irrational numbers. This is the first step in a program to study the irrationality of $(s,t)$-analog of known numbers., Comment: 24 pages

The Keck Planet Imager and Characterizer (KPIC), a series of upgrades to the Keck II Adaptive Optics System and Instrument Suite, aims to demonstrate high-resolution spectroscopy of faint exoplanets that are spatially resolved from their host stars. In this paper, we measure KPIC's sensitivity to companions as a function of separation (i.e., the contrast curve) using on-sky data collected over four years of operation. We show that KPIC is able to reach contrasts of $1.3 \times 10^{-4}$ at 90 mas and $9.2 \times 10^{-6}$ at 420 mas separation from the star, and that KPIC can reach planet-level sensitivities at angular separations within the inner working angle of coronagraphic instruments such as GPI and SPHERE. KPIC is also able to achieve more extreme contrasts than other medium-/high-resolution spectrographs that are not as optimized for high-contrast performance. We decompose the KPIC performance budget into individual noise terms and discuss limiting factors. The fringing that results from combining a high-contrast imaging system with a high-resolution spectrograph is identified as an important source of systematic noise. After mitigation and correction, KPIC is able to reach within a factor of 2 of the photon noise limit at separations < 200 mas. At large separations, KPIC is limited by the background noise performance of NIRSPEC., Comment: 16 pages, 6 figures, submitted to the proceedings of SPIE Astronomical Telescopes + Instrumentation 2024, 13096-69

Using Keck Planet Imager and Characterizer (KPIC) high-resolution ($R$~35000) spectroscopy from 2.29-2.49 $\mu$m, we present uniform atmospheric retrievals for eight young substellar companions with masses of ~10-30 $M_\textrm{Jup}$, orbital separations spanning ~50-360 au, and $T_\textrm{eff}$ between ~1500-2600 K. We find that all companions have solar C/O ratios, and metallicities, to within the 1-2$\sigma$ level, with the measurements clustered around solar composition. Stars in the same stellar associations as our systems have near-solar abundances, so these results indicate that this population of companions is consistent with formation via direct gravitational collapse. Alternatively, core accretion outside the CO snowline would be compatible with our measurements, though the high mass ratios of most systems would require rapid core assembly and gas accretion in massive disks. On a population level, our findings can be contrasted with abundance measurements for directly imaged planets with m<10 $M_\textrm{Jup}$, which show tentative atmospheric metal enrichment. In addition, the atmospheric compositions of our sample of companions are distinct from those of hot Jupiters, which most likely form via core accretion. For two companions with $T_\textrm{eff}$~1700-2000 K (kap And b and GSC 6214-210 b), our best-fit models prefer a non-gray cloud model with >3$\sigma$ significance. The cloudy models yield 2-3$\sigma$ lower $T_\textrm{eff}$ for these companions, though the C/O and [C/H] still agree between cloudy and clear models at the $1\sigma$ level. Finally, we constrain 12CO/13CO for three companions with the highest S/N data (GQ Lup b, HIP 79098 b, and DH Tau b), and report $v$sin($i$) and radial velocities for all companions., Comment: Accepted to ApJ, 52 pages, 20 figures

We used the Keck Planet Imager and Characterizer (KPIC) to obtain high-resolution (R$\sim$35,000) K-band spectra of kappa Andromedae b, a planetary-mass companion orbiting the B9V star, kappa Andromedae A. We characterized its spin, radial velocity, and bulk atmospheric parameters through use of a forward modeling framework to jointly fit planetary spectra and residual starlight speckles, obtaining likelihood-based posterior probabilities. We also detected H$_{2}$O and CO in its atmosphere via cross correlation. We measured a $v\sin(i)$ value for kappa And b of $38.42\pm{0.05}$ km/s, allowing us to extend our understanding of the population of close in bound companions at higher rotation rates. This rotation rate is one of the highest spins relative to breakup velocity measured to date, at close to $50\%$ of breakup velocity. We identify a radial velocity $-17.35_{-0.09}^{+0.05}$ km/s, which we use with existing astrometry and RV measurements to update the orbital fit. We also measure an effective temperature of $1700\pm{100}$ K and a $\log(g)$ of $4.7\pm{0.5}$ cgs dex., Comment: 14 pages, 7 figures, accepted for publication in AJ

In this paper, we find solutions to first-order linear proportional difference equations via the $(s,t)$-integration factor method. The $(s,t)$-integration factor involves the $(s,t)$-Pantograph function, which is a generalization of the partial Theta function. Other equations are solved including the $(s,t)$-analog of the Bernoulli equation., Comment: 22 pages

We present the projected rotational velocity and molecular abundances for HD 33632 Ab obtained via Keck Planet Imager and Characterizer high-resolution spectroscopy. HD 33632 Ab is a nearby benchmark brown dwarf companion at a separation of $\sim$20 au that straddles the L/T transition. Using a forward-modeling framework with on-axis host star spectra, self-consistent substellar atmospheric and retrieval models for HD 33632 Ab, we derive a projected rotational velocity of 53 $\pm$ 3 km/s and carbon/water mass fractions of log CO = $-$2.3 $\pm$ 0.3 and log H$_2$O = $-$2.7 $\pm$ 0.2. The inferred carbon-to-oxygen ratio (C/O = 0.58 $\pm$ 0.14), molecular abundances, and metallicity ([C/H] = 0.0 $\pm$ 0.2 dex) of HD 33632 Ab are consistent with its host star. Although detectable methane opacities are expected in L/T transition objects, we did not recover methane in our KPIC spectra, partly due to the high $v\sin{i}$ and to disequilibrium chemistry at the pressures we are sensitive to. We parameterize the spin as the ratio of rotation over break-up velocity, and compare HD 33632 Ab to a compilation of >200 very low-mass objects (M$\lesssim$0.1 M$_{\odot}$) that have spin measurements in the literature. There appears to be no clear trend for the isolated field low-mass objects versus mass, but a tentative trend is identified for low-mass companions and directly imaged exoplanets, similar to previous findings. A larger sample of close-in gas giant exoplanets and brown dwarfs will critically examine our understanding of their formation and evolution through rotation and chemical abundance measurements., Comment: Accepted for publication in the Astrophysical Journal. 36 pages, 15 figures, 5 tables

We present the first on-sky segmented primary mirror closed-loop piston control using a Zernike wavefront sensor (ZWFS) installed on the Keck II telescope. Segment co-phasing errors are a primary contributor to contrast limits on Keck and will be necessary to correct for the next generation of space missions and ground-based extremely large telescopes (ELTs), which will all have segmented primary mirrors. The goal of the ZWFS installed on Keck is to monitor and correct primary mirror co-phasing errors in parallel with science observations. The ZWFS is ideal for measuring phase discontinuities such as segment co-phasing errors and is one of the most sensitive WFS, but has limited dynamic range. The vector-ZWFS at Keck works on the adaptive optics (AO) corrected wavefront and consists of a metasurface focal plane mask which imposes two different phase shifts on the core of the point spread function (PSF) to two orthogonal light polarizations, producing two pupil images. This design extends the dynamic range compared with the scalar ZWFS. The primary mirror segment pistons were controlled in closed-loop using the ZWFS, improving the Strehl ratio on the NIRC2 science camera by up to 10 percentage points. We analyze the performance of the closed-loop tests, the impact on NIRC2 science data, and discuss the ZWFS measurements., Comment: Accepted for publication in the Astrophysical Journal (ApJ). 17 pages, 16 figures

The 1RXS J034231.8+121622 system consists of an M dwarf primary and a directly imaged low-mass stellar companion. We use high resolution spectroscopic data from Keck/KPIC to estimate the objects' atmospheric parameters and radial velocities (RVs). Using PHOENIX stellar models, we find that the primary has a temperature of 3460 $\pm$ 50 K a metallicity of 0.16 $\pm$ 0.04, while the secondary has a temperature of 2510 $\pm$ 50 K and a metallicity of $0.13\substack{+0.12 \\ -0.11}$. Recent work suggests this system is associated with the Hyades, placing it an older age than previous estimates. Both metallicities agree with current $[Fe/H]$ Hyades measurements (0.11 -- 0.21). Using stellar evolutionary models, we obtain significantly higher masses for the objects, of 0.30 $\pm$ 0.15 $M_\odot$ and 0.08 $\pm$ 0.01 $M_\odot$ (84 $\pm$ 11 $M_{Jup}$) respectively. Using the RVs and a new astrometry point from Keck/NIRC2, we find that the system is likely an edge-on, moderately eccentric ($0.41\substack{+0.27 \\ -0.08}$) configuration. We also estimate the C/O ratio of both objects using custom grid models, obtaining 0.42 $\pm$ 0.10 (primary) and 0.55 $\pm$ 0.10 (companion). From these results, we confirm that this system most likely went through a binary star formation process in the Hyades. The significant changes in this system's parameters since its discovery highlight the importance of high resolution spectroscopy for both orbital and atmospheric characterization of directly imaged companions., Comment: 30 pages, 18 figures, accepted for publication in AJ

Vortex fiber nulling (VFN) is a technique for detecting and characterizing faint companions at small separations from their host star. A near-infrared ($\sim2.3 \mu$m) VFN demonstrator mode was deployed on the Keck Planet Imager and Characterizer (KPIC) instrument at the Keck Observatory and presented earlier. In this paper, we present the first VFN companion detections. Three targets, HIP 21543 Ab, HIP 94666 Ab, and HIP 50319 B, were detected with host-companion flux ratios between 70 and 430 at and within one diffraction beamwidth ($\lambda/D$). We complement the spectra from KPIC VFN with flux ratio and position measurements from the CHARA Array to validate the VFN results and provide a more complete characterization of the targets. This paper reports the first direct detection of these three M dwarf companions, yielding their first spectra and flux ratios. Our observations provide measurements of bulk properties such as effective temperatures, radial velocities, and v$\sin{i}$, and verify the accuracy of the published orbits. These detections corroborate earlier predictions of the KPIC VFN performance, demonstrating that the instrument mode is ready for science observations., Comment: 13 pages, 2 figures; Accepted to ApJ Letters

Spatially-resolved images of debris disks are necessary to determine disk morphological properties and the scattering phase function (SPF) which quantifies the brightness of scattered light as a function of phase angle. Current high-contrast imaging instruments have successfully resolved several dozens of debris disks around other stars, but few studies have investigated trends in the scattered-light, resolved population of debris disks in a uniform and consistent manner. We have combined Karhunen-Loeve Image Projection (KLIP) with radiative-transfer disk forward modeling in order to obtain the highest quality image reductions and constrain disk morphological properties of eight debris disks imaged by the Gemini Planet Imager at H-band with a consistent and uniformly-applied approach. In describing the scattering properties of our models, we assume a common SPF informed from solar system dust scattering measurements and apply it to all systems. We identify a diverse range of dust density properties among the sample, including critical radius, radial width, and vertical width. We also identify radially narrow and vertically extended disks that may have resulted from substellar companion perturbations, along with a tentative positive trend in disk eccentricity with relative disk width. We also find that using a common SPF can achieve reasonable model fits for disks that are axisymmetric and asymmetric when fitting models to each side of the disk independently, suggesting that scattering behavior from debris disks may be similar to Solar System dust., Comment: 23+5 pages, 12+6 figures, 15 pages of Online Supplemental Material included; Accepted for publication in MNRAS

M dwarfs are common host stars to exoplanets but often lack atmospheric abundance measurements. Late-M dwarfs are also good analogs to the youngest substellar companions, which share similar $T_{\rm eff}\sim2300-2800~K$. We present atmospheric analyses for the M7.5 companion HIP 55507 B and its K6V primary star with Keck/KPIC high-resolution ($R\sim35,000$) $K$ band spectroscopy. First, by including KPIC relative radial velocities between the primary and secondary in the orbit fit, we improve the dynamical mass precision by 60% and find $M_B=88.0_{-3.2}^{+3.4}$ $M_{\rm Jup}$, putting HIP 55507 B above the stellar-substellar boundary. We also find that HIP 55507 B orbits its K6V primary star with $a=38^{+4}_{-3}$ AU and $e=0.40\pm0.04$. From atmospheric retrievals of HIP 55507 B, we measure $\rm [C/H]=0.24\pm0.13$, $\rm [O/H]=0.15\pm0.13$, and $\rm C/O=0.67\pm0.04$. Moreover, we strongly detect $\rm ^{13}CO$ ($7.8\sigma$ significance) and tentatively detect $\rm H_2^{18}O$ ($3.7\sigma$ significance) in companion's atmosphere, and measure $\rm ^{12}CO/^{13}CO=98_{-22}^{+28}$ and $\rm H_2^{16}O/H_2^{18}O=240_{-80}^{+145}$ after accounting for systematic errors. From a simplified retrieval analysis of HIP 55507 A, we measure $\rm ^{12}CO/^{13}CO=79_{-16}^{+21}$ and $\rm C^{16}O/C^{18}O=288_{-70}^{+125}$ for the primary star. These results demonstrate that HIP 55507 A and B have consistent $\rm ^{12} C/^{13}C$ and $\rm ^{16}O/^{18}O$ to the $<1\sigma$ level, as expected for a chemically homogeneous binary system. Given the similar flux ratios and separations between HIP 55507 AB and systems with young, substellar companions, our results open the door to systematically measuring $\rm ^{13}CO$ and $\rm H_2^{18}O$ abundances in the atmospheres of substellar or even planetary-mass companions with similar spectral types., Comment: Accepted to ApJ, 28 pages, 14 figures

We present high-resolution $K$-band emission spectra of the quintessential hot Jupiter HD 189733 b from the Keck Planet Imager and Characterizer (KPIC). Using a Bayesian retrieval framework, we fit the dayside pressure-temperature profile, orbital kinematics, mass-mixing ratios of H$_2$O, CO, CH$_4$, NH$_3$, HCN, and H$_2$S, and the $\rm ^{13}CO/^{12}CO$ ratio. We measure mass fractions of $\rm \log H_2O = -2.0^{+0.4}_{-0.4}$ and $\rm \log CO = -2.2^{+0.5}_{-0.5}$, and place upper limits on the remaining species. Notably, we find $\rm \log CH_4 < -4.5$ at 99\% confidence, despite its anticipated presence at the equilibrium temperature of HD 189733 b assuming local thermal equilibrium. We make a tentative ($\sim3\sigma$) detection of $\rm ^{13}CO$, and the retrieved posteriors suggest a $\rm ^{12}C/^{13}C$ ratio similar to or substantially less than the local interstellar value. The possible $\rm ^{13}C$ enrichment would be consistent with accretion of fractionated material in ices or in the protoplanetary disk midplane. The retrieved abundances correspond to a substantially sub-stellar atmospheric $\rm C/O = 0.3\pm0.1$, while the carbon and oxygen abundances are stellar to slightly super-stellar, consistent with core-accretion models which predict an inverse correlation between C/O and metallicity. The specific combination of low C/O and high metallicity suggests significant accretion of solid material may have occurred late in the formation process of HD 189733 b., Comment: 17 pages, 7 figures, 2 tables, accepted in AJ

The Gemini Planet Imager (GPI) has excelled in imaging debris disks in the near-infrared. The GPI Exoplanet Survey (GPIES) imaged twenty-four debris disks in polarized $H$-band light, while other programs observed half of these disks in polarized $J$- and/or $K1$-bands. Using these data, we present a uniform analysis of the morphology of each disk to find asymmetries suggestive of perturbations, particularly those due to planet-disk interactions. The multi-wavelength surface brightness, the disk color and geometry permit identification of any asymmetries such as warps or disk offsets from the central star. We find that nineteen of the disks in this sample exhibit asymmetries in surface brightness, disk color, disk geometry, or a combination of the three, suggesting that for this sample, perturbations, as seen in scattered light, are common. The relationship between these perturbations and potential planets in the system are discussed. We also explore correlations among stellar temperatures, ages, disk properties, and observed perturbations. We find significant trends between the vertical aspect ratio and the stellar temperature, disk radial extent, and the dust grain size distribution power-law, $q$. We also confirm a trend between the disk color and stellar effective temperature, where the disk becomes increasingly red/neutral with increasing temperature. Such results have important implications on the evolution of debris disk systems around stars of various spectral types., Comment: 46 pages, 20 figures, 6 tables, accepted for publication in ApJ

Vortex fiber nulling (VFN) is a single-aperture interferometric technique for detecting and characterizing exoplanets separated from their host star by less than a diffracted beam width. VFN uses a vortex mask and single mode fiber to selectively reject starlight while coupling off-axis planet light with a simple optical design that can be readily implemented on existing direct imaging instruments that can feed light to an optical fiber. With its axially symmetric coupling region peaking within the inner working angle of conventional coronagraphs, VFN is more efficient at detecting new companions at small separations than conventional direct imaging, thereby increasing the yield of on-going exoplanet search campaigns. We deployed a VFN mode operating in K band ($2.0{-}2.5~\mu$m) on the Keck Planet Imager and Characterizer (KPIC) instrument at the Keck II Telescope. In this paper we present the instrument design of this first on-sky demonstration of VFN and the results from on-sky commissioning, including planet and star throughput measurements and predicted flux-ratio detection limits for close-in companions. The instrument performance is shown to be sufficient for detecting a companion $10^3$ times fainter than a $5^{\mathrm{th}}$ magnitude host star in 1 hour at a separation of 50 mas (1.1$\lambda/D$). This makes the instrument capable of efficiently detecting substellar companions around young stars. We also discuss several routes for improvement that will reduce the required integration time for a detection by a factor ${>}$3., Comment: 26 pages, 5 figures; Accepted to JATIS

The Keck Planet Imager and Characterizer (KPIC) is an instrument at the Keck II telescope that enables high-resolution spectroscopy of directly imaged exoplanets and substellar companions. KPIC uses single-mode fibers to couple the adaptive optics system to Keck's near-infrared spectrometer (NIRSPEC). However, KPIC's sensitivity at small separations is limited by the leakage of stellar light into the fiber. Speckle nulling uses a deformable mirror to destructively interfere starlight with itself, a technique typically used to reduce stellar signal on a focal-plane imaging detector. We present the first on-sky demonstration of speckle nulling through an optical fiber with KPIC, using NIRSPEC to collect exposures that measure speckle phase for quasi-real-time wavefront control while also serving as science data. We repeat iterations of measurement and correction, each using at least 5 exposures. We show a decrease in the on-sky leaked starlight by a factor of 2.6 to 2.8 in the targeted spectral order, at a spatial separation of 2.0 {\lambda}/D in K-band. This corresponds to an estimated factor of 2.6 to 2.8 decrease in the required exposure time to reach a given SNR, relative to conventional KPIC observations. The performance of speckle nulling is limited by instability in the speckle phase: when the loop is opened, the null-depth degrades by a factor of 2 on the timescale of a single phase measurement, which would limit the suppression that can be achieved. Future work includes exploring gradient-descent methods, which may be faster and thereby able to achieve deeper nulls. In the meantime, the speckle nulling algorithm demonstrated in this work can be used to decrease stellar leakage and improve the signal-to-noise of science observations., Comment: 18 pages, 4 figures

We define the deformed $(s,t)$-binomial formula and the deformed Newton $(s,t)$-binomial series, and we will use it to establish the generating functions of the generalized central binomial coefficients and the generalized Catalan numbers., Comment: arXiv admin note: text overlap with arXiv:2211.04450

We present Keck/KPIC high-resolution ($R\sim35,000$) $K$-band thermal emission spectroscopy of the ultra-hot Jupiter WASP-33b. The use of KPIC's single-mode fibers greatly improves both blaze and line-spread stabilities relative to slit spectrographs, enhancing the cross-correlation detection strength. We retrieve the dayside emission spectrum with a nested sampling pipeline which fits for orbital parameters, the atmospheric pressure-temperature profile, and molecular abundances.We strongly detect the thermally-inverted dayside and measure mass-mixing ratios for CO ($\log\rm CO_{MMR} = -1.1^{+0.4}_{-0.6}$), H$_2$O ($\log\rm H_2O_{MMR} = -4.1^{+0.7}_{-0.9}$) and OH ($\log\rm OH_{MMR} = -2.1^{+0.5}_{-1.1}$), suggesting near-complete dayside photodissociation of H$_2$O. The retrieved abundances suggest a carbon- and possibly metal-enriched atmosphere, with a gas-phase C/O ratio of $0.8^{+0.1}_{-0.2}$, consistent with the accretion of high-metallicity gas near the CO$_2$ snow line and post-disk migration or with accretion between the soot and H$_2$O snow lines. We also find tentative evidence for $\rm ^{12}CO/^{13}CO \sim 50$, consistent with values expected in protoplanetary disks, as well as tentative evidence for a metal-enriched atmosphere (2--15$\times$ solar). These observations demonstrate KPIC's ability to characterize close-in planets and the utility of KPIC's improved instrumental stability for cross-correlation techniques., Comment: Accepted in AJ, 26 pages, 12 figures

The detection of satellites around extrasolar planets, so called exomoons, remains a largely unexplored territory. In this work, we study the potential of detecting these elusive objects from radial velocity monitoring of self-luminous directly imaged planets. This technique is now possible thanks to the development of dedicated instruments combining the power of high-resolution spectroscopy and high-contrast imaging. First, we demonstrate a sensitivity to satellites with a mass ratio of 1-4% at separations similar to the Galilean moons from observations of a brown-dwarf companion (HR 7672 B; Kmag=13; 0.7" separation) with the Keck Planet Imager and Characterizer (KPIC; R~35,000 in K band) at the W. M. Keck Observatory. Current instrumentation is therefore already sensitive to large unresolved satellites that could be forming from gravitational instability akin to binary star formation. Using end-to-end simulations, we then estimate that future instruments such as MODHIS, planned for the Thirty Meter Telescope, should be sensitive to satellites with mass ratios of ~1e-4. Such small moons would likely form in a circumplanetary disk similar to the Jovian satellites in the solar system. Looking for the Rossiter-McLaughlin effect could also be an interesting pathway to detecting the smallest moons on short orbital periods. Future exomoon discoveries will allow precise mass measurements of the substellar companions that they orbit and provide key insight into the formation of exoplanets. They would also help constrain the population of habitable Earth-sized moons orbiting gas giants in the habitable zone of their stars., Comment: Accepted to AJ (Jan 10, 2023)

In this paper, we study the existence of solutions of the functional difference equations with proportional delay on deformed generalized Fibonacci polynomials via successive approximation method and Bell polynomials. First, we introduce the deformed generalized Fibonacci polynomials and show that the $q$-numbers can be viewed as "bifurcation" of deformed $(s,t)$-numbers. These deformations are closely related to proportional delay. Second, a differential and integral calculus on deformed generalized Fibonacci polynomials is introduced. The main reason for introducing this calculation is to have a framework for solving proportional functional equations and thus obtain the Pell calculus, Jacobsthal calculus, Chebysheff calculus, and Mersenne calculus, among others. We study the convergence of $(s,t)$-exponential type series and its dependence on the deformation parameter. We define the deformed $(s,t)$-exponential functions and we give its analytic and algebraic properties. In addition, we study the $(1,u)$-deformed $(s,t)$-exponential function and use it to prove the existence of functional difference equations with proportional delay. The solution is not unique when it is related to $q$-periodic functions.

The Keck Planet Imager and Characterizer (KPIC) is a series of upgrades for the Keck II Adaptive Optics (AO) system and the NIRSPEC spectrograph to enable diffraction-limited, high-resolution ($R>30,000$) spectroscopy of exoplanets and low-mass companions in the K and L bands. Phase I consisted of single-mode fiber injection/extraction units (FIU/FEU) used in conjunction with an H-band pyramid wavefront sensor. Phase II, deployed and commissioned in 2022, adds a 1000-actuator deformable mirror, beam-shaping optics, a vortex coronagraph, and other upgrades to the FIU/FEU. The use of single-mode fibers provides a gain in stellar rejection, a substantial reduction in sky background, and an extremely stable line-spread function on the spectrograph. In this paper we present the results of extensive system-level laboratory testing and characterization showing the instrument's Phase II throughput, stability, repeatability, and other key performance metrics prior to delivery and during installation at Keck. We also demonstrate the capabilities of the various observing modes enabled by the new system modules using internal test light sources. Finally, we show preliminary results of on-sky tests performed in the first few months of Phase II commissioning along with the next steps for the instrument. Once commissioning of Phase II is complete, KPIC will continue to characterize exoplanets at an unprecedented spectral resolution, thereby growing its already successful track record of 23 detected exoplanets and brown dwarfs from Phase I. Using the new vortex fiber nulling (VFN) mode, Phase II will also be able to search for exoplanets at small angular separations less than 45 milliarcseconds which conventional coronagraphs cannot reach., Comment: 13 pages; 6 figures; to appear in Proceedings of the SPIE, Ground-based and Airborne Instrumentation for Astronomy IX, Vol. 12184

The formation and evolution pathway for the directly-imaged multi-planetary system HR 8799 remains mysterious. Accurate constraints on the chemical composition of the planetary atmosphere(s) are key to solving the mystery. We perform a detailed atmospheric retrieval on HR 8799~c to infer the chemical abundances and abundance ratios using a combination of photometric data along with low- and high-resolution spectroscopic data (R$\sim$20-35,000). We specifically retrieve [C/H], [O/H], and C/O and find them to be 0.55$^{+0.36}_{-0.39}$, 0.47$^{+0.31}_{-0.32}$, and 0.67$^{+0.12}_{-0.15}$ at 68\% confidence. The super-stellar C and O abundances, yet a stellar C/O ratio, reveal a potential formation pathway for HR 8799~c. Planet c, and likely the other gas giant planets in the system, formed early on (likely within $\sim$1 Myr), followed by further atmospheric enrichment in C and O through the accretion of solids beyond the CO iceline. The enrichment either preceded or took place during the early phase of the inward migration to the planet current locations., Comment: 19 pages, 6 figures, 3 tables, accepted to AAS journals

Direct imaging studies have mainly used low-resolution spectroscopy ($R\sim20-100$) to study the atmospheres of giant exoplanets and brown dwarf companions, but the presence of clouds has often led to degeneracies in the retrieved atmospheric abundances (e.g. C/O, metallicity). This precludes clear insights into the formation mechanisms of these companions. The Keck Planet Imager and Characterizer (KPIC) uses adaptive optics and single-mode fibers to transport light into NIRSPEC ($R\sim35,000$ in $K$ band), and aims to address these challenges with high-resolution spectroscopy. Using an atmospheric retrieval framework based on petitRADTRANS, we analyze KPIC high-resolution spectrum ($2.29-2.49~\mu$m) and archival low-resolution spectrum ($1-2.2~\mu$m) of the benchmark brown dwarf HD 4747 B ($m=67.2\pm1.8~M_{\rm{Jup}}$, $a=10.0\pm0.2$ au, $T_{\rm eff}\approx1400$ K). We find that our measured C/O and metallicity for the companion from the KPIC high-resolution spectrum agree with that of its host star within $1-2\sigma$. The retrieved parameters from the $K$ band high-resolution spectrum are also independent of our choice of cloud model. In contrast, the retrieved parameters from the low-resolution spectrum are highly sensitive to our chosen cloud model. Finally, we detect CO, H$_2$O, and CH$_4$ (volume mixing ratio of log(CH$_4$)=$-4.82\pm0.23$) in this L/T transition companion with the KPIC data. The relative molecular abundances allow us to constrain the degree of chemical disequilibrium in the atmosphere of HD 4747 B, and infer a vertical diffusion coefficient that is at the upper limit predicted from mixing length theory., Comment: 33 pages, 16 figures, Accepted to ApJ

A benchmark brown dwarf (BD) is a BD whose properties (e.g., mass and chemical composition) are precisely and independently measured. Benchmark BDs are valuable in testing theoretical evolutionary tracks, spectral synthesis, and atmospheric retrievals for sub-stellar objects. Here, we report results of atmospheric retrieval on a synthetic spectrum and a benchmark BD -- HR 7672~B -- with \petit. First, we test the retrieval framework on a synthetic PHOENIX BT-Settl spectrum with a solar composition. We show that the retrieved C and O abundances are consistent with solar values, but the retrieved C/O is overestimated by 0.13-0.18, which is $\sim$4 times higher than the formal error bar. Second, we perform retrieval on HR 7672~B using high spectral resolution data (R=35,000) from the Keck Planet Imager and Characterizer (KPIC) and near infrared photometry. We retrieve [C/H], [O/H], and C/O to be $-0.24\pm0.05$, $-0.19\pm0.04$, and $0.52\pm0.02$. These values are consistent with those of HR 7672~A within 1.5-$\sigma$. As such, HR 7672~B is among only a few benchmark BDs (along with Gl 570~D and HD 3651~B) that have been demonstrated to have consistent elemental abundances with their primary stars. Our work provides a practical procedure of testing and performing atmospheric retrieval, and sheds light on potential systematics of future retrievals using high- and low-resolution data., Comment: 29 pages, 17 figures, 5 tables, resubmitted to AAS journals after first revision

The Keck Planet Imager and Characterizer (KPIC) is a purpose-built instrument to demonstrate new technological and instrumental concepts initially developed for the exoplanet direct imaging field. Located downstream of the current Keck II adaptive optic system, KPIC contains a fiber injection unit (FIU) capable of combining the high-contrast imaging capability of the adaptive optics system with the high dispersion spectroscopy capability of the current Keck high resolution infrared spectrograph (NIRSPEC). Deployed at Keck in September 2018, this instrument has already been used to acquire high resolution spectra ($R > 30,000$) of multiple targets of interest. In the near term, it will be used to spectrally characterize known directly imaged exoplanets and low-mass brown dwarf companions visible in the northern hemisphere with a spectral resolution high enough to enable spin and planetary radial velocity measurements as well as Doppler imaging of atmospheric weather phenomena. Here we present the design of the FIU, the unique calibration procedures needed to operate a single-mode fiber instrument and the system performance., Comment: 31 pages, 17 figures, submitted to JATIS

Using the Keck Planet Imager and Characterizer (KPIC), we obtained high-resolution (R$\sim$35,000) $K$-band spectra of the four planets orbiting HR 8799. We clearly detected \water{} and CO in the atmospheres of HR 8799 c, d, and e, and tentatively detected a combination of CO and \water{} in b. These are the most challenging directly imaged exoplanets that have been observed at high spectral resolution to date when considering both their angular separations and flux ratios. We developed a forward modeling framework that allows us to jointly fit the spectra of the planets and the diffracted starlight simultaneously in a likelihood-based approach and obtained posterior probabilities on their effective temperatures, surface gravities, radial velocities, and spins. We measured $v\sin(i)$ values of $10.1^{+2.8}_{-2.7}$~km/s for HR 8799 d and $15.0^{+2.3}_{-2.6}$~km/s for HR 8799 e, and placed an upper limit of $< 14$~km/s of HR 8799 c. Under two different assumptions of their obliquities, we found tentative evidence that rotation velocity is anti-correlated with companion mass, which could indicate that magnetic braking with a circumplanetary disk at early times is less efficient at spinning down lower mass planets., Comment: 31 pages, 12 figures, Accepted to AJ

In this paper a Ward-Fonten\'e differential universal algebra is constructed. In this algebra it is possible to obtain a product $\psi$-rule and a general $\psi$-rule of Leibniz for any calculus on sequences. In particular, the simplicial polytopic calculus and the calculus on Bell numbers are introduced., Comment: 29 pages

In this paper special values of Bell polynomials are given by using the power series solution of the equation $y^{(k)}=e^{ay}$. In addition, complete and partial exponential autonomous functions, exponential autonomous polynomials, autonomous polynomials and $(k,a)$-autonomous coefficients are defined. Finally, we show the relationship between various numbers counting combinatorial objects and binomial coefficients, Stirling numbers of the second kind and autonomous coefficients., Comment: 23 p\'aginas

In this paper, autonomous differential equations type delta of order one on integral domains are defined. For this we will use the autonomous ring defined on the Hurwitz expansion ring of exponential generating functions with coefficients in an integral domain. We will also use delta operators, which behave like derivatives when acting on polynomials, along with Umbral calculus. As a particular example of a delta operator we have the forward difference operator that defines the difference equations. Then the delta-type equations generalize to the ordinary equations and to the difference equations., Comment: In Spanish, 19 pages

In this article, the ring of flows of autonomous differential equations of order one on integral domains is constructed. First, we build the autonomous ring $\Opa(\hurw_{R}[[x]])$ and then its structure is studied. Next, we build the ring of formal exponential generating series of the ring $\Opa(\hurw_ {R} [[x]]) $, where it is possible to find solutions of differential equations of order one when the vector field of the system can be decomposed in sum or product of functions., Comment: 22 pages

We report the results of a ${\sim}4$-year direct imaging survey of 104 stars to resolve and characterize circumstellar debris disks in scattered light as part of the Gemini Planet Imager Exoplanet Survey. We targeted nearby (${\lesssim}150$ pc), young (${\lesssim}500$ Myr) stars with high infrared excesses ($L_{\mathrm{IR}} / L_\star > 10^{-5}$), including 38 with previously resolved disks. Observations were made using the Gemini Planet Imager high-contrast integral field spectrograph in $H$-band (1.6 $\mu$m) coronagraphic polarimetry mode to measure both polarized and total intensities. We resolved 26 debris disks and three protoplanetary/transitional disks. Seven debris disks were resolved in scattered light for the first time, including newly presented HD 117214 and HD 156623, and we quantified basic morphologies of five of them using radiative transfer models. All of our detected debris disks but HD 156623 have dust-poor inner holes, and their scattered-light radii are generally larger than corresponding radii measured from resolved thermal emission and those inferred from spectral energy distributions. To assess sensitivity, we report contrasts and consider causes of non-detections. Detections were strongly correlated with high IR excess and high inclination, although polarimetry outperformed total intensity angular differential imaging for detecting low inclination disks (${\lesssim} 70 \deg$). Based on post-survey statistics, we improved upon our pre-survey target prioritization metric predicting polarimetric disk detectability. We also examined scattered-light disks in the contexts of gas, far-IR, and millimeter detections. Comparing $H$-band and ALMA fluxes for two disks revealed tentative evidence for differing grain properties. Finally, we found no preference for debris disks to be detected in scattered light if wide-separation substellar companions were present., Comment: arXiv resubmission with typographical corrections. Accepted for publication in AJ. 19 figures, 7 tables

We present new $H$-band scattered light images of the HD 32297 edge-on debris disk obtained with the Gemini Planet Imager (GPI). The disk is detected in total and polarized intensity down to a projected angular separation of 0.15", or 20au. On the other hand, the large scale swept-back halo remains undetected, likely a consequence of its markedly blue color relative to the parent body belt. We analyze the curvature of the disk spine and estimate a radius of $\approx$100au for the parent body belt, smaller than past scattered light studies but consistent with thermal emission maps of the system. We employ three different flux-preserving post-processing methods to suppress the residual starlight and evaluate the surface brightness and polarization profile along the disk spine. Unlike past studies of the system, our high fidelity images reveal the disk to be highly symmetric and devoid of morphological and surface brightness perturbations. We find the dust scattering properties of the system to be consistent with those observed in other debris disks, with the exception of HR 4796. Finally, we find no direct evidence for the presence of a planetary-mass object in the system., Comment: Accepted for publication in the Astronomical Journal

In this paper the circulant Hadamard conjecture is proved., Comment: 23 pages

In this paper the relationship between $S$-subgroups in $\Z_{2}^{n}$ and binary codes is shown. If the codes used are both $P(T)$-codes and $G$-codes, then the $S$-subgroup is free. The codes constructed are cyclic, decimated or symmetric and the $S$-subgroups obtained are free under the action the cyclic permutation subgroup, invariants under the action the decimated permutation subgroup and symmetric under the action of symmetric permutation subgroup, respectively. Also it is shows that there is no codes generating whole $\Z_{2}^{n}$ in any $\G_{n}(a)$-complete $S$-set of the $S$-ring $\mathfrak{S}(\Z_{2}^{n},S_{n})$., Comment: 22 pages

In this paper we use Faa di Bruno's formula to associate Bell polynomial values to differential equations of the form $y^{\prime}=f(y)$. That is, we use partial Bell polynomials to represent the solution of such an equation and use the solution to compute special values of partial Bell polynomials., Comment: 13 pages

We construct a ring of flows where we can decompose autonomous nonlinear dynamical systems into smaller parts, then solve each part and finally put everything together to obtain the exact solution of these systems., Comment: 20 pages, in Spanish

In this paper three Schur ring are discussed, namenly: Hamming, circulant orbists and decimated circulant orbits Schur ring. By using autocorrelation function and the run structure of binary sequences we proof the relation between this Schur ring and combinatorial structures such as Hadamard matrices, periodic compatible binary sequences and perfect binary sequences. Cai proved for binary sequences that the autocorrelation function is in fact completely determined by its run structure. Also, he characterised the structure of the circulant Hadamard matrices. We characterise a more general structure, called periodic compatible binary sequences ($PComS$ for brevety), which generalises Hadamard matrices, periodic complementary binary sequences and binary sequences with $2$-level autocorrelation. Families of periodic compatibles binary sequences are presented. Also, we compute a bounds on familias $PComS$ in Hamming Schur ring. The results obtained are applied to families of $PComS$ such as circulant, with one and two circulant cores, Goethals-Seidel type and partial Hadamard matrices and perfect binary sequences., Comment: 23 pages

Turyn prove that if a circulant Hadamard matrix of order $n$ exists then $n$ must be of the form $n=4m^{2}$ for some odd integer $m$. In this paper we use the structure constant of Schur ring of $\Z_{2}^{4m^{2}}$ to prove that there is no circulant Hadamard matrix in $\Z_{2}^{4m^{2}}$ except possibly for sequences with Hamming weight $a+b$, with $\frac{m^{2}-m}{2}\leq a\leq\frac{3m^{2}-m}{2}$ and $b=2m^{2}-m-a$ and with $\frac{m^{2}+m}{2}\leq 2m^{2}-a\leq\frac{3m^{2}+m}{2}$ and $b=m+a$., Comment: 7 pages, substantial text overlap eliminated

In this paper a variety of issues are discussed, Schur ring, $S$-sets, circulant orbits, decimation operator and Hadamard matrices and their relation between them is shown. Firstly we define the complete $S$-sets. Next, we study the structure of Schur ring with circulant basic sets over $\Z_{2}^{n}$ and we define the free and non-free circulant $S$-sets, the symmetric, non-symmetric and antisymmetric circulant $S$-sets. We prove that all this $S$-sets are invariants under decimation. Finally, we prove that if a Hadamard matrix exist then this is contained in a complete $S$-set. Also, we prove that can't exist circulant and with one core Hadamard matrices with some particular structure. These theorems include a result known on symmetric circulant Hadamard matrices of order $4n$ only when $n$ is an odd number., Comment: 23 pages