A product of two hypergeometric series is generally not hypergeometric. However, there are a few cases when such a product does reduce to a single hypergeometric series. The oldest result of this type, beyond the obvious (1-x)a(1-x)b = (1-x)a+b, is Euler's transformation for the Gauss hypergeometric function 2F1. Another important one is the celebrated Clausen's identity, dating back to 1828, which expresses the square of a suitable 2F1 function as a single 3F2. By equating coefficients, each product identity corresponds to a special type of summation theorem for terminating series. Over the last two decades Euler's transformations and many summation theorems have been extended by introducing additional parameter pairs differing by positive integers. This amounts to multiplication of the power series coefficients by values of a fixed polynomial at nonnegative integers. The main goal of this paper is to present an extension of Clausen's identity obtained by such polynomial perturbation. To this end, we first reconsider the polynomial perturbations of Euler's transformations found by Miller and Paris around 2010. We propose new, simplified proofs of their transformations relating them to polynomial interpolation and exhibiting various new forms of the characteristic polynomials. We further introduce the notion of the Miller-Paris operators which play a prominent role in the construction of the extended Clausen's identity. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
In this paper, we explore (slightly generalized) f-Eulerian polynomials introduced by Stanley and frequently appearing in combinatorics. Notable special cases include the classical Eulerian polynomials, the generating polynomials of order polynomials for certain labeled posets, and the d-Narayana polynomials. We establish simple sufficient conditions for the reality (and sign) of their zeros and present implications for total positivity of sequences generated by values of polynomials at integers. We further relate these polynomials to generalized Euler's transformations for the generalized hypergeometric functions with integral parameter differences. Exploiting this and other hypergeometric connections, we provide purely hypergeometric proofs for various known and some new properties of d-Narayana polynomials. Another family encompassed by our definition of the generalized f-Eulerian polynomials is that of Jacobi-Piñeiro type II multiple orthogonal polynomials. Their zero location can thus be analyzed, for both canonical and non-canonical parameter values, without invoking orthogonality. Finally, we present several connection formulas relating d-Narayana polynomials to particular Jacobi-Piñeiro polynomials.
Generative AI presents an unprecedented challenge to our understanding of knowledge and its production. Unlike previous technological transformations, where engineering understanding preceded or accompanied deployment, generative AI operates through mechanisms whose epistemic character remains obscure, and without such understanding, its responsible integration into science, education, and institutional life cannot proceed on a principled basis. This paper argues that the missing account must begin with a paradigmatic break that has not yet received adequate philosophical attention. In the Turing-Shannon-von Neumann tradition, information enters the machine as encoded binary vectors, and semantics remains external to the process. Neural network architectures rupture this regime: symbolic input is instantly projected into a high-dimensional space where coordinates correspond to semantic parameters, transforming binary code into a position in a geometric space of meanings. It is this space that constitutes the active epistemic condition shaping generative production. Drawing on four structural properties of high-dimensional geometry concentration of measure, near-orthogonality, exponential directional capacity, and manifold regularity the paper develops an Indexical Epistemology of High-Dimensional Spaces. Building on Peirce semiotics and Papert constructionism, it reconceptualizes generative models as navigators of learned manifolds and proposes navigational knowledge as a third mode of knowledge production, distinct from both symbolic reasoning and statistical recombination.
Let G be a graph of order n. For a positive integer p, G is said to be a Wp graph if n >= p and every p pairwise disjoint independent sets of G are contained within p pairwise disjoint maximum independent sets. In this paper, we establish that every connected Wp graph G is p-quasi-regularizable if and only if n >= (p + 1) & centerdot; alpha, where alpha is the independence number of G and p =/ 2. This finding ensures that the independence polynomial of a connected Wp graph G is log-concave whenever (p + 1) & centerdot; alpha <= n <= p & centerdot; alpha + 2 root p & centerdot; alpha + p and alpha 2 <= p, or p & centerdot; alpha + 2 root p & centerdot; alpha + p 2+1)& centerdot;p+(alpha-1)2 alpha+1 <= p. Moreover, the clique corona graph G degrees Kp serves as an example of the Wp graph class. We further demonstrate that the independence polynomial of G degrees Kpis always log-concave for sufficiently large p. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The scaling law of deep learning, which governs the relationship between model size and performance, has led to critical concerns regarding efficiency and sustainability. To address these challenges, this study presents a computational approach using self-organized submillimeter-long tungsten disulfide nanotube cluster as a 3D very-large-scale physical reservoir. The reservoir, with its 0D van der Waals interfaces on the order of 108, or 1.0×1010 mm-3, matches the synaptic quantity and density of the fruit fly’s brain. The reservoir demonstrates the capability to perform a wide range of tasks from monomodal challenges to multimodal endeavors such as speech-to-image and medical image generation. The photosensitive mimetic synaptic connections in the very large scale reservoir emulate the optogenetic modulation of neuron circuits in in-vivo biological systems. By integrating the principles of the scaling law, multimodal task capabilities, and mimetic optogenetic mechanisms, this research paves a path toward advanced computing architectures tailored for next-generation energy-efficient artificial intelligence. Han et al. present a very-large-scale 3D computing reservoir self-assembled from 1D WS2 nanotubes, with dense mimetic optogenetic synapses matching a fruit fly’s brain. It efficiently handles monomodal and multimodal tasks, featuring optogenetics-inspired, light-tunable computing dynamics.
