In this paper, we investigate some properties between q-Bernstein polynomials and q-Euler numbers by using fermionic p-adic q-integrals on Z p . From these properties, we derive some interesting identities related to the q-Euler numbers.1 q , (see [2]).

The first purpose of this paper is to present a systemic study of some families of multiple q-Bernoulli numbers and polynomials by using multivariate q-Volkenborn integral (= p-adic q-integral) on Z p . From the studies of these q-Bernoulli numbers and polynomials of higher order we derive some interesting q-analogs of Stirling number identities.

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Context. The Spitzer Survey of Stellar Structure in Galaxies (S4G) is a detailed study of over 2300 nearby galaxies in the near-infrared (NIR), which has been critical to our understanding of the detailed structures of nearby galaxies. Because the sample galaxies were selected only using radio-derived velocities, however, the survey favored late-type disk galaxies over lenticulars and ellipticals. Aims. A follow-up Spitzer survey was conducted to rectify this bias, adding 465 early-type galaxies (ETGs) to the original sample, to be analyzed in a manner consistent with the initial survey. We present the data release of this ETG extension, up to the third data processing pipeline (P3): surface photometry. Methods. We produce curves of growth and radial surface brightness profiles (with and without inclination corrections) using reduced and masked Spitzer IRAC 3.6 μm and 4.5 μm images produced through Pipelines 1 and 2, respectively. From these profiles, we derive the following integrated quantities: total magnitudes, stellar masses, concentration parameters, and galaxy size metrics. We showcase NIR scaling relations for ETGs among these quantities. Results. We examine general trends across the whole S4G and ETG extension among our derived parameters, highlighting differences between ETGs and late-type galaxies (LTGs). The latter are, on average, more massive and more concentrated than LTGs, and subtle distinctions are seen among ETG morphological subtypes. We also derive the following scaling relations and compare them with previous results in visible light: mass-size (both half-light and isophotal), mass-concentration, mass-surface brightness (central, effective, and within 1 kpc), and mass-color. Conclusions. We find good agreement with previous works, though some relations (e.g., mass-central surface brightness) will require more careful multicomponent decompositions to be fully understood. The relations between mass and isophotal radius and between mass and surface brightness within 1 kpc, in particular, show notably small scatter. The former provides important constraints on the limits of size growth in galaxies, possibly related to star formation thresholds, while the latter–particularly when paired with the similarly tight relation for LTGs–showcases the striking self-similarity of galaxy cores, suggesting they evolve little over cosmic time. All of the profiles and parameters described in this paper will be provided to the community via the NASA/IPAC database on a dedicated website.