Tootfinder

Playing with Sollya today, which is a tool for generating math function polynomials. I've managed to build a gamma function for 32, 64, 80 and 128 bits which doesn't totally suck. Range reduction to 1-2, then an 11th order polynomial (for 32-bits).
https://github.…

picolibc/newlib/libm/sollya/gamma_inc.h at gamma-accuracy · picolibc/picolibc
picolibc - a C library designed for embedded 32- and 64- bit systems. - picolibc/picolibc

Shear Particle Acceleration in Structured Gamma-Ray Burst Jets: III. The Radiation Physics of Bright Prompt Optical Flash
Zi-Qi Wang, Xiao-Li Huang, En-Wei Liang
https://arxiv.org/abs/2508.06798

Shear Particle Acceleration in Structured Gamma-Ray Burst Jets: III. The Radiation Physics of Bright Prompt Optical Flash
The radiation physics of bright prompt optical emission of gamma-ray bursts (GRBs) remains a puzzle. Assuming that the GRB ejecta is structured, we investigated this issue by characterizing the ejecta as an ultra-relativistic uniform jet core surrounded by a mild-relativistic cocoon. The mixed jet-cocoon (MJC) region can accelerate particles through the shear acceleration mechanism. Parameterizing the radial velocity profile of the MJC region with an exponential function and assuming a uniform …

Inclusive open charm photoproduction in ultraperipheral collisions at the LHC with G$\gamma$A-FONLL
Matteo Cacciari, Gian Michele Innocenti, Anna M. Sta\'sto
https://arxiv.org/abs/2506.09893

Inclusive open charm photoproduction in ultraperipheral collisions at the LHC with G$γ$A-FONLL
We compute the inclusive $D^{0}$ production cross section in ultraperipheral Pb-Pb collisions at the LHC as a function of the $D^{0}$ transverse momentum and rapidity. These calculations are carried out within the new G$γ$A-FONLL (Generalized Photon-Nucleus FONLL) framework, which can predict photonuclear cross sections for charm and beauty hadrons in electron-proton, electron-nucleus, and ultraperipheral heavy-ion collisions. The framework relies on FONLL (Fixed-Order Next-to-Leading Logarith…

Raabe's Formula For Gamma Function Via Riemann-Liouville Fractional Integrals And Generalized Glaisher Constants
Efe G\"urel
https://arxiv.org/abs/2505.22666

Raabe's Formula For Gamma Function Via Riemann-Liouville Fractional Integrals And Generalized Glaisher Constants
In this paper, we prove Raabe-type integral formulas for gamma function via left and right sided Riemann-Liouville fractional integrals. As corollaries, we give the left and right sided repeated integration formulas for the log-gamma and related functions. The relationship between the generalized Glaisher constants and aforementioned integrals are investigated.

HPGe-Compton Net: A Physics-Guided CNN for Fast Gamma Spectra Analysis via Compton Region Learning
Yanfeng Xie, Yiming Weng, Soo Hyun Byun
https://arxiv.org/abs/2508.01106 https…

HPGe-Compton Net: A Physics-Guided CNN for Fast Gamma Spectra Analysis via Compton Region Learning
High Purity Germanium (HPGe) detectors have been golden standard for gamma spectrometry in Low level radioactive waste (LLW) analysis; however, their notable shortcoming is prolonged measurement durations for weak radioactive waste materials. The present study aimed to develop the HPGe Compton Net, a 1D Physics Guided CNN to accelerate LLW analysis by taking advantage of the entire response function of a HPGe detector for each radionuclide of interest, in contrast to the traditional methods tha…

A Unifying Integral Representation of the Gamma Function and Its Reciprocal
Peter Reinhard Hansen, Chen Tong
https://arxiv.org/abs/2506.12112 https://

A Unifying Integral Representation of the Gamma Function and Its Reciprocal
We derive an integral expression $G(z)$ for the reciprocal gamma function, $1/Γ(z)=G(z)/π$, that is valid for all $z\in\mathbb{C}$, without the need for analytic continuation. The same integral avoids the singularities of the gamma function and satisfies $G(1-z)=Γ(z)\sin(πz)$ for all $z\in\mathbb{C}$.

Completely Integrable Gradient System on the bivariate beta statistical manifold
Prosper Rosaire Mama Assandje, Joseph Dongho, Thomas Bouetou Bouetou
https://arxiv.org/abs/2508.04523

Completely Integrable Gradient System on the bivariate beta statistical manifold
This paper investigates the geometry of a completely integrable gradient system defined on the three parameter bivariate beta statistical manifold of the first kind. We prove that the associated vector field is Hamiltonian and admits a Lax pair representation implying complete integrability. We show that the potential function derived from exponential family structure defines a Riemannian metric equivalent to the Fisher information metric. By applying Stirling's approximation to the gamma funct…

On the Efficiency of Producing Gamma-Ray Bursts from Isolated Population III Stars
Gibran Morales-Rivera, Ramandeep Gill, S. Jane Arthur, Paz Beniamini, Jonathan Granot
https://arxiv.org/abs/2508.03689

On the Efficiency of Producing Gamma-Ray Bursts from Isolated Population III Stars
The rate of long-duration gamma-ray bursts (GRBs) from isolated Pop III stars is not well known, as it depends on our poor understanding of their initial mass function (IMF), rotation rates, stellar evolution, and mass loss. A sub-population of massive ($M_{\rm ZAMS}\gtrsim20M_\odot$) Pop III stars is expected to suffer core-collapse and launch a relativistic jet that would power a GRB. In the collapsar scenario, a key requirement is that the pre-supernova star imparts sufficient angular moment…

Analysis of (p,$\gamma$) capture cross-sections relevant to p-process using TALYS for A=75-110
Satabdi Mondal, Enakshi Senapati, Deepak Pandit, Balaram Dey, Pampa Das, Alokkumar De, Srijit Bhattacharya
https://arxiv.org/abs/2508.05054

Analysis of (p,$γ$) capture cross-sections relevant to p-process using TALYS for A=75-110
The proton capture (p, $γ$) cross-sections for eight different atomic nuclei in the mass region A=75-110 were calculated within the nuclear reaction model code TALYS. For all the reactions, we tested different combinations of inputs for level density (l.d) parameter and gamma strength function ($γ_{sf}$). Finally, it was observed that application of hybrid input in TALYS (macroscopic l.d and microscopic or semi-microscopic $γ_{sf}$ or in abbreviation mac-mic) resulted successful agreement of…

Some two-parameter families of generalized Bernstein functions
Stamatis Koumandos, Henrik L. Pedersen
https://arxiv.org/abs/2507.03551 https://

Some two-parameter families of generalized Bernstein functions
We prove that certain functions involving ratios of Gamma functions and the Psi-function belong to generalized Bernstein classes and new properties of generalized Bernstein functions are given.

Constructing new probability distributions on the unit interval
Roberto Vila, Helton Saulo, Poliana Matos, Subhankar Dutta
https://arxiv.org/abs/2508.01154 https://

Constructing new probability distributions on the unit interval
In this paper, we introduce a novel method for constructing probability distributions on the unit interval by exploiting the non-injective transformation defined by the ratio of two positive random variables, $X$ and $Y$. For simplicity and tractability, we focus on independent random variables $X$ and $Y$ following gamma distributions. We derive some results including laws, density functions, quantile function and closed-form expressions for the moments, and discuss maximum likelihood estimati…

An elementary proof of Newman's eta-quotient theorem
David Savitt
https://arxiv.org/abs/2507.16225 https://arxiv.org/pdf/2507.162…

An elementary proof of Newman's eta-quotient theorem
Let eta(z) be the Dedekind eta function. Newman studied the modularity of eta-quotients, giving necessary and sufficient conditions for a function of the form \prod_{0 < m | N} eta(mz)^{r_m} to be a (weakly) holomorphic modular form of level N. We explain a proof of Newman's theorem, developed while teaching a class for talented high school students at Canada/USA Mathcamp. The key observation is that although Gamma_1(N) is not generated by its upper triangular and lower triangular subgroup…

On the Explicit Expression of an Extended Version of Riemann Zeta Function
Yushi Huang
https://arxiv.org/abs/2507.22961 https://arxiv.org/pdf/2507.22961

On the Explicit Expression of an Extended Version of Riemann Zeta Function
In this paper, we focus on the explicit expression of an extended version of Riemann zeta function. We use two different methods, Mellin inversion formula and Cauchy's residue theorem, to calculate a Mellin-Barnes type integral of the analytic function regarding $z$: $Γ(z)Γ(s-z)u^{-z}$ ($u\in (0,1)$, $s\in \mathbb{C}$). We provide the necessary background on the analytic properties of Gamma and Riemann zeta function to confirm the absolute convergence of this Mellin-Barnes integral. Next, we …

A Leptonic Interpretation of the UHE Gamma-ray Emission from V4641 Sgr
Su-Yu Wan, Jie-shuang Wang, Ruo-Yu Liu
https://arxiv.org/abs/2507.02763 https://

A Leptonic Interpretation of the UHE Gamma-ray Emission from V4641 Sgr
Recently, the microquasar V4641 Sgr and its surrounding is detected at TeV-PeV gamma-ray band. Interestingly, the spectrum follows a power-law function continuing up to 0.8 PeV as reported by LHAASO, and the morphology of the emission appears a puzzling elongated structure. In this work, we propose that the elongated UHE emission from V4641 Sgr could originate from the inverse Compton radiation of electrons with a very hard spectrum, which may result from shear acceleration mechanism in the jet…

PET Tracer Separation Using Conditional Diffusion Transformer with Multi-latent Space Learning
Bin Huang, Feihong Xu, Xinchong Shi, Shan Huang, Binxuan Li, Fei Li, Qiegen Liu
https://arxiv.org/abs/2506.16934

PET Tracer Separation Using Conditional Diffusion Transformer with Multi-latent Space Learning
In clinical practice, single-radiotracer positron emission tomography (PET) is commonly used for imaging. Although multi-tracer PET imaging can provide supplementary information of radiotracers that are sensitive to physiological function changes, enabling a more comprehensive characterization of physiological and pathological states, the gamma-photon pairs generated by positron annihilation reactions of different tracers in PET imaging have the same energy, making it difficult to distinguish t…

The Event Rate and Luminosity Function of Fermi/GBM Gamma-Ray Bursts
Yang Liu, Zhi-Bin Zhang, Xiao-Fei Dong, Long-Biao Li, Xiu-Yun Du
https://arxiv.org/abs/2507.16595

The Event Rate and Luminosity Function of Fermi/GBM Gamma-Ray Bursts
Luminosity function and event rate of Gamma-Ray Bursts (GRBs) are easily biased by the instrument and selection effects. We select 115 Fermi/GBM GRBs with good spectra fitted by a smoothly broken power-law function. The $τ$-statistic method is used to describe how the luminosity evolves with redshift. The non-parametric Lynden-Bell's c$^{-}$ method has been applied to get the cumulative luminosity function and event rate which is compared with the star formation history. How the selection and …

On fractional differential equations, dimensional analysis, and the double gamma function
J. Vaz, E. Capelas de Oliveira
https://arxiv.org/abs/2506.00026 h…

On fractional differential equations, dimensional analysis, and the double gamma function
In this paper we discuss some issues that arise in the process of writing a fractional differential equation (FDE) by replacing an integer order derivative by a fractional order derivative in a given differential equation. To address these issues, we propose a dimensional regularization of the Caputo fractional derivative, ensuring consistency in physical dimensions. Then we solve some FDEs using this proposed dimensional regularization. We show that the solutions of these FDEs are most conveni…

Replaced article(s) found for math.CA. https://arxiv.org/list/math.CA/new
[1/1]:
- Hyperelliptic values of the Gamma function
Jan L\"ugering
https://

Systematic Biases from ARM-Based Sensitivity and Imaging in Compton Cameras
Tomonori Ikeda, Tatsuya Sawano, Naomi Tsuji, Yoshitaka Mizumura
https://arxiv.org/abs/2506.18659

Systematic Biases from ARM-Based Sensitivity and Imaging in Compton Cameras
The angular resolution measure (ARM) is widely used in Compton camera analyses to characterize angular uncertainty and to define event selection criteria. However, the ARM distribution is fundamentally different from the point spread function (PSF), which represents the true spatial imaging response of the system on the celestial sphere. In this study, motivated by applications in gamma-ray astrophysics, we investigated the consequences of using ARM-based definitions in sensitivity estimation a…

On the dual valued generalized hypergeometric function and its special cases
Ravi Dwivedi, Juan Carlos Cort\'es
https://arxiv.org/abs/2507.19501 https://

On the dual valued generalized hypergeometric function and its special cases
This paper explores the calculus of dual-valued functions and investigates the gamma function, beta function and generalized hypergeometric functions by incorporating dual numbers as parameters and variables. We examine its fundamental properties, including regions of convergence, differential equations, and integral representations. Furthermore, we provide an in-depth discussion on the various properties of the dual confluent and Gauss hypergeometric functions.

Broadband Spectral Modeling of Blazars: Constraining the Lorentz Factor and Gamma-Ray Emission Site
Ajay Sharma, Aishwarya Sarath, Sakshi Chaudhary, Debanjan Bose
https://arxiv.org/abs/2506.12832

Broadband Spectral Modeling of Blazars: Constraining the Lorentz Factor and Gamma-Ray Emission Site
We present a comprehensive temporal and spectral analysis of a few blazars using multi-wavelength observations. Rapid flux variations are quantified via the doubling/halving timescale method, revealing the shortest variability timescales of a few hours in $γ$-ray emissions. The broadband fractional variability is systematically computed and examined as a function of frequency, displaying a characteristic double-hump structure akin to the typical spectral energy distribution (SED) of blazars. T…