arXiv subjects
Publications and source records attributed to Robert Reynolds.
In this work we derive and evaluate some infinite integrals involving the product of a generalized logarithm and polynomial functions in the denominator. These integrals are expressed in terms of finite series involving the Hurwitz-Lerch zeta function. We produce special cases of these integrals in terms of other special functions and fundamental constants.
We introduce UniversalCEFR, a large-scale multilingual and multidimensional dataset of texts annotated with CEFR (Common European Framework of Reference) levels in 13 languages. To enable open research in automated readability and language proficiency assessment, UniversalCEFR comprises 505,807 CEFR-labeled texts curated from educational and learner-oriented resources, standardized into a unified data format to support consistent processing, analysis, and modelling across tasks and languages. To demonstrate its utility, we conduct benchmarking experiments using three modelling paradigms: a) linguistic feature-based classification, b) fine-tuning pre-trained LLMs, and c) descriptor-based prompting of instruction-tuned LLMs. Our results support using linguistic features and fine-tuning pretrained models in multilingual CEFR level assessment. Overall, UniversalCEFR aims to establish best practices in data distribution for language proficiency research by standardising dataset formats, and promoting their accessibility to the global research community.
One-dimensional and two-dimensional integrals containing $E_b(-u)$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ are considered. $E_b(-u)$ is the Mittag-Leffler function and the integral is taken over the rectangle $0 \leq x < \infty, 0 \leq u < \infty$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ is the generalized Mittag-Leffler function and the integral is over $0\leq x \leq b$ with infinite intervals explored. A representation in terms of the Hurwitz-Lerch zeta function and other special functions are derived for the double and single integrals, from which special cases can be evaluated in terms of special function and fundamental constants.
Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two Bessel functions of the first kind $J_{u}(a z)$ $J_{v}(b z)$ with order $u,v$, studied by Landau et al. The method used in this work is contour integration.
Precision radial velocity (RV) spectrographs that use adaptive optics (AO) show promise to advance telescope observing capabilities beyond those of seeing-limited designs. We are building a spectrograph for the Large Binocular Telescope (LBT) named iLocater that uses AO to inject starlight directly into single mode fibers (SMF). iLocater's first acquisition camera system (the `SX' camera), which receives light from one of the 8.4m diameter primary mirrors of the LBT, was initially installed in summer 2019 and has since been used for several commissioning runs. We present results from first-light observations that include on-sky measurements as part of commissioning activities. Imaging measurements of the bright B3IV star 2 Cygni ($V=4.98$) resulted in the direct detection of a candidate companion star at an angular separation of only $\theta = 70$ mas. Follow-up AO measurements using Keck/NIRC2 recover the candidate companion in multiple filters. An $R\approx1500$ miniature spectrograph recently installed at the LBT named ``Lili'' provides spatially resolved spectra of each binary component, indicating similar spectral types and strengthening the case for companionship. Studying the multiplicity of young runaway star systems like 2 Cygni ($36.6 \pm 0.5$ Myr) can help to understand formation mechanisms for stars that exhibit anomalous velocities through the galaxy. This on-sky demonstration illustrates the spatial resolution of the iLocater SX acquisition camera working in tandem with the LBT AO system; it further derisks a number of technical hurdles involved in combining AO with Doppler spectroscopy.
In this work derivations of definite integrals listed in Prudnikov volume I, Gradshteyn and Ryzhik and a few other tables are produced. Special cases of these integrals in terms of fundamental constants are also evaluated. The method used in these derivations is contour integration.
In this work we derive a functional equation in terms of the Hurwitz-Lerch zeta function along with definite integrals in terms of the incomplete gamma and Hurwitz-Lerch zeta functions. The method used in these derivations is contour integration. Special cases in terms of fundamental constants are produced.
In this note we derive some interesting definite integrals involving Malmsten logarithm forms, reciprocal logarithm forms and K\"{o}lbig type integrals in terms of special functions.
A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.
The Large Binocular Telescope, with its expansive collecting area, angular resolving power, and advanced optical design, provides a robust platform for development and operation of advanced instrumentation for astronomical research. The LBT currently hosts a mature suite of instruments for spectroscopy and imaging at optical through mid-infrared wavelengths, supported by sophisticated adaptive optics systems. This contribution summarizes the current state of instrumentation, including upgrades to existing instruments and commissioning of second generation instruments now in progress. The LBT is soliciting proposals for next generation instrument concepts, with participation open to consortium members and others interested in participation in the Observatory.
This is an anthology of series involving rational, factorial, and power functions expressed in terms of special functions. New finite expansions involving quotient functions expressed in terms of the Hurwitz-Lerch zeta function are given. These results represent a new form of expressing this special function as a finite series where contour integration is required for derivation. Extended series previously known and derived are extended using differential equations and algebraic methods.
This is a conspectus of definite integrals, products and series. These formulae involve special functions in the integrand and summand functions and closed form solutions. Some of the special cases are stated in terms of fundamental constants.
An extension of two finite trigonometric series is studied to derive closed form formulae involving the Hurwitz-Lerch zeta function. The trigonometric series involves angles with a geometric series involving the powers of 3. These closed formulae are used to derive composite finite series involving special functions, trigonometric functions and fundamental constants. A short table summarizing some interesting results is produced.
By employing contour integration the derivation of a generalized double finite series involving the Hurwitz-Lerch zeta function is used to derive closed form formulae in terms of special functions. We use this procedure to find special cases of the summation and product formulae in terms of the Hurwitz-Lerch zeta function, trigonometric functions and the gamma function.
This is a compendium of generating functions involving single, double sums and definite integrals. These generating functions also involve special functions in both the summand function and closed form solution.
This paper introduces a set of finite summation formulas and utilize them to establish various functional relationships involving the multivariable Hurwitz-Lerch zeta function. Additionally, the paper examines several examples of these functional relationships.
Various product and sum relationships are established using special functions, specifically involving Special functions. These relationships are derived from formulas inspired by the finite sum that incorporates the Hurwitz-Lerch zeta function.
In this work we derive a bilateral generating function involving the product of an Appell-type product of the Bernoulli and Euler polynomials over independent indices and orders. This function is expressed in terms of the Hurwitz zeta function and special cases in terms of the finite sum of the Hurwitz zeta function and integral formula are derived.