Abstract We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with the help of the Hermite polynomials. Then we apply this result to deduce, under the Riemann hypothesis, a formula describing the local distribution of zeros of the Riemann zeta function lying along the critical line. For this application we make use of the Prime Number Theorem in conjunction with Kac’s formula for the distribution of values of trigonometric polynomials.

The interplay between the angular velocity of a revolving cone and the free flow of a chemically reactive ternary nanofluid results in enhanced shear-driven interaction, increased thermal conductivity, and faster reaction kinetics owing to the synergistic effects of rotation and nanoparticle distribution. Inspired by this, the present study investigates the influence of thermal radiation on the angular velocities of the free flow and the cone’s arbitrary temporal fluctuations, resulting in an unsteady stream over a rotating cone in a rotating ternary nanofluid. The flow and heat transfer processes influenced by thermal radiation are significant in scientific research because of their many applications. Moreover, thermal radiation-based heat transfer is essential in renewable energy systems. To solve the reduced equations, a physics-informed neural network integrated with the Hermite polynomial is utilized. The results of the Hermite polynomial neural network (H-PINN) algorithm demonstrate substantial consistency with the numerical finite difference method (FDM) results, with the absolute error falling between 10−4 and 10−6. As the ratio of the angular velocity of the cone to the angular velocity of the free-stream increases, the velocity profile decreases. Increasing the chemical reaction parameter decreases the concentration profile.

Abstract In the $$q^{-1}$$ q - 1 -symmetric Askey scheme, namely the $$q^{-1}$$ q - 1 -Askey–Wilson, continuous dual $$q^{-1}$$ q - 1 -Hahn, $$q^{-1}$$ q - 1 -Al-Salam–Chihara, continuous big $$q^{-1}$$ q - 1 -Hermite and continuous $$q^{-1}$$ q - 1 -Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a q -beta integral which comes from the continuous orthogonality relation for the $$q^{-1}$$ q - 1 -Askey–Wilson polynomials. In the $$q\rightarrow 1^{-}$$ q → 1 - limit, this q -beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for.

Introduction: Classical orthogonal polynomials such as the Legendre, Laguerre, Hermite, and Chebyshev polynomials have a wide range of applications across various domains of science and engineering. Aim: The aim of this research paper is to present selected recent developments in the theory of orthogonal polynomials, with particular emphasis on their analytical properties and their role in approximation theory. Methods: The study employs a combined quantitative-qualitative research methodology. It is based on the analysis of relevant scientific literature, from which data significant to the subject matter were collected, interpreted, and systematically examined. Results: The results indicate that contemporary research in probability theory, graph theory, coding theory, and related areas increasingly relies on the theory of orthogonal polynomials. This paper provides a theoretical analysis of the connection between orthogonal polynomials and their applications in specific computational problems. Fundamental properties of these polynomials are presented and illustrated through selected examples that contribute to a clearer understanding of their structure and practical relevance. Conclusion: The approximation of the transfer function of a low-pass filter can be achieved through a straightforward adaptation of orthogonal Jacobi polynomials. Furthermore, Gaussian quadrature represents a powerful numerical technique for the approximation of definite integrals, utilizing optimally chosen nodes and weight functions to achieve high accuracy with minimal computational effort. Orthogonal polynomials thus serve as a significant link between pure mathematics and engineering disciplines, highlighting the importance of further study on their properties and applications.

Abstract A primitive problem of predicting effective properties of composites is open boundary conditions. In this paper, Eshelby’s transformation field method (TFM) is developed to solve the open boundary problem of two-phase composites having arbitrary geometry of inclusion. The inhomogenous transformation fields are introduced in the composite system to cope with the complex interface boundary-value problem. Furthermore, the open boundary problem is solved by Hermite polynomial, which is used to express the transformation fields and the perturbation fields. As an example, the formulas of calculating effective dielectric property of two-dimensional isotropic dielectric composites having open boundary conditions are derived by TFM. The validity is verified by comparing the effective responses estimated by TFM with the exact solutions of dilute limit, and good agreements are obtained. The results show that TFM is valid to solve the open boundary problem of composites having complex geometric inclusions.

This article analyzes Legendre, Laguerre, and Hermite polynomials as eigenfunctions of self-adjoint differential operators defined on Hilbert spaces of the type , with the goal of strengthening their conceptual understanding in advanced educational settings. By reformulating their differential equations as Sturm–Liouville problems, we identify the structural components that allow for a spectral interpretation: weight functions, domains, eigenvalues, and orthogonality. This unified framework presents these polynomials not only as formal solutions but also as pedagogical tools to explain key concepts in functional analysis, such as orthonormality, discrete spectra, and complete bases. The approach combines mathematical rigor with visual and comparative resources to support their integration into the teaching of upper-level mathematics and physics courses. It is concluded that the spectral interpretation of these polynomial systems can significantly enhance students' conceptual comprehension by connecting topics from linear algebra, differential equations, and functional spaces. The study proposes their inclusion as didactic resources in university-level mathematics education, promoting meaningful learning around abstract structures.

In-band full-duplex (IBFD) communication systems offer a promising means of improving spectral efficiency by enabling simultaneous transmission and reception on the same frequency channel. Despite this advantage, self-interference (SI) remains a major challenge to their practical deployment. Among the different SI cancellation (SIC) techniques, this paper focuses on digital SIC methodologies tailored for multiple-input multiple-output (MIMO) wireless transceivers operating under digital beamforming architectures. Two distinct digital SIC approaches are evaluated, employing a generalized memory polynomial (GMP) model augmented with Itô–Hermite polynomial basis functions and a phase-normalized neural network (PNN) to effectively model the nonlinearities and memory effects introduced by transmitter and receiver hardware impairments. The robustness of the SIC is further evaluated under both single off-line training and closed-loop real-time adaptation, employing estimation techniques such as least squares (LS), least mean squares (LMS), and fast Kalman (FK) for model coefficient estimation. The performance of the proposed digital SIC techniques is evaluated through detailed simulations that incorporate realistic power amplifier (PA) characteristics, channel conditions, and high-order modulation schemes. Metrics such as error vector magnitude (EVM) and total bit error rate (BER) are used to assess the quality of the received signal after SIC under different signal-to-interference ratio (SIR) and signal-to-noise ratio (SNR) conditions. The results show that, for time-variant scenarios, a low-complexity adaptive SIC can be realized using a GMP model with FK parameter estimation. However, in time-invariant scenarios, an open-loop SIC approach based on PNN offers superior performance and maintains robustness across various modulation schemes.

Deformation of polynomials is a kind of operation where we add a new variable to the original polynomial. In our case, suppose P is a monic polynomial of degree n with complex coefficients. We evolve P with respect to time by heat flow, creating a function P(t,z) of two variables with given initial dataP(0,z)=P(z)for whichtP(t,z)=zzP(t,z). In this paper, we focus on the deformed polynomial P(t,z). First, we proved the Taylor series representation of deformed polynomial. Then we apply the results to the classical Hermite polynomials and extend to the case of matrix-valued polynomials. From the inspiration of deformed polynomials roots movement, we proved the behavior of Hermite polynomials after heat flow deformation and got an explicit formula. For further work, similar to what we have done in this paper, we want to have an explicit formula for deformed matrix Hermite polynomials and give a proof.

Abstract The transport characteristics of collective agent motion, governed by a combination of anomalous drift and Brownian motion, are investigated. The anomalous drift is modeled as a power-law function of the magnitude of the agents’ peculiar velocity. In the mean-field limit, the Fokker–Planck equation incorporates both anomalous drift and Brownian motion. Assuming that the velocity distribution function is weakly nonequilibrium, transport coefficients are derived by applying the first Maxwellian iteration to Grad’s 13-moment equations, which are obtained from the Fokker–Planck equation. However, when anomalous drift is included, the fluctuation amplitude–i.e. the diffusion coefficient–depends on the nonequilibrium state in order to conserve energy. Consequently, the expansion of the velocity distribution function using modified Hermite polynomials becomes inadequate under strongly nonequilibrium conditions, because the diffusion coefficient cannot be expressed solely in terms of temperature. As the drift becomes super-drift, the collective motion of agents asymptotically approaches a perfect fluid. Finally, the convergence of the velocity distribution function toward the equilibrium distribution, as governed by the Fokker–Planck equation, is confirmed, numerically. Transport coefficients obtained via their Green–Kubo expression using the direct simulation Monte Carlo method are compared with their corresponding analytical results.

Hermite polynomials facilitating on-line learning analysis of layered neural networks with arbitrary activation function

ABSTRACT Diffusion processes and more generally, stochastic differential equations (SDEs), are widely used to model natural and financial systems. However, accurately simulating them remains challenging due to the limitations of discretization methods. We propose a recursive algorithm to approximate the transition density of scalar diffusion processes using Hermite polynomial expansions. Unlike standard numerical schemes, our method uses an expansion in Hermite polynomials to approximate the transition density without requiring an arbitrarily small discretization step. This approximation is then used to simulate diffusion paths with high fidelity. Numerical experiments, including the Vasicek and CIR processes, confirm the effectiveness and efficiency of the method.

ABSTRACT The ‐Dunkl‐classical symmetric orthogonal ‐polynomials provide a unification and generalization of the ‐analogue of the generalized Hermite polynomials and the ‐analogue of the generalized Gegenbauer polynomials. In this paper, we characterize these polynomials by means of the so‐called structure relation. The paper concludes with an open problem based upon the findings of this paper.

Abstract Improving the efficiency of electron acceleration in a vacuum presents a significant challenge in sophisticated laser-driven acceleration methodologies. This research examines the combined effects of Hermite-Sinh-Gaussian (HShG) laser beams (laser electric field amplitude, Hermite polynomial mode index, Decentered parameter of Sinh function) and magnetic wigglers (wiggler magnetic field amplitude, wiggler field propagation constant) on electron dynamics to facilitate high-energy acceleration. The distinct intensity and phase distribution of HShG beams alter the trajectory of electrons, facilitating effective energy transfer. The periodic magnetic field of the wiggler enhances resonance conditions, thereby sustaining prolonged interactions between electrons and the laser field. The numerical analysis indicates that the integration of these two mechanisms markedly enhances the relativistic factor ( γ ) of electrons, resulting in greater energy gains. The findings demonstrate that customized laser beam configurations and external magnetic fields can enhance laser-driven acceleration in vacuum, presenting a viable strategy for future compact accelerators. This study offers insights into optimizing acceleration efficiency for high-energy physics applications, free-electron lasers, and advanced radiation sources.

A new paradigm for deriving the higher-order 2D Hermite polynomial basis: Part II - DQM matrix formulation for the LaDQM and SQEM and WQEM GLL for some fourth-order systems

Abstract We establish precise relationships among four distinct positivity conditions for bihomogeneous polynomials. We consider various one-parameter families $$r_t$$ r t of metrics on the line bundle $$U^{m}$$ U m over projective space $${\mathbb P}^1$$ P 1 and give precise results about how these positivity conditions depend on t and m.

In this paper, we extend the notions of complex C-R-calculus and complex Hermite polynomials to the bicomplex setting and compare the bicomplex polyanalytic function theory with the classical complex case. Specifically, we introduce two kinds of bicomplex Hermite polynomials and present some of their basic properties, such as the Rodrigues formula and generating functions. We also define three bicomplex Landau operators and calculate their action on the bicomplex Hermite polynomials of the first kind.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

. Many models in engineering, industry, physics, environmental science, and finance are driven by uncertain differential equations. However, the parameters, or the terms in the equations, are sometimes difficult to estimate. In this article, we propose a new non parametric method based on Hermite polynomial approximation to estimate the drift and diffusion terms of homogeneous uncertain differential equations when the structures of these terms are unknown. Three numerical examples are provided to demonstrate our method. Moreover, we use an uncertain hypothesis test, which is based on the residuals analysis technique, to certify the efficiency of our results. Our article also incorporates an empirical study on modeling the USD-CNY exchange rate using uncertain differential equations. Compared with other methods, especially parametric estimation, the method obtained in this article is very feasible to implement for uncertain differential equations without information about the structures of the drift and diffusion terms in particular.

The current research investigates, for the first time, the multidimensional transport process of solute in a hydromagnetic, viscous, incompressible, unidirectional, steady, fully developed, third-grade fluid flowing through a channel saturated by a porous medium under the influence of a constant pressure gradient. First-order heterogeneous boundary reactions are applied at both channel walls. A regular perturbation method is applied to derive an approximate steady velocity profile for the third-grade fluid. Aris's method of moments is employed on the governing time-dependent advection–diffusion equation, followed by an implicit finite-difference scheme to study the dispersion process of solute through the channel. The first four central moments are used on the Hermite polynomial representation to ascertain the axial distribution of the solute's mean concentration. The concentration profiles of the solute in both longitudinal and transverse directions are obtained using Aris's method of moments, extending beyond one-dimensional axial dispersion. The study demonstrates how solute dispersion in a third-grade fluid is influenced by Darcy number, Hartmann number, magnetic field inclination, and absorption parameter. It is observed that the flow velocity significantly drops across the channel as the third-grade parameter, Hartmann number, and angle of inclination of the magnetic field increase. However, the fluid velocity increases as the Darcy number rises. The dispersion coefficient decreases with lower values of the third-grade parameter and Darcy number, but the opposite scenario is observed for the increment of the Hartmann number, inclination angle of the magnetic field, and reaction parameter. It is evident that the dispersion of solute enhances by 177.89% when the Darcy number rises from 0.1 to 0.2, and by 54.93% when it increases from 0.2 to 0.3. The dispersion of solute decreases by 1.15% when the third-grade parameter increases from 0.1 to 0.5. As Darcy number and absorption parameter raise, the amplitude of the mean concentration distribution of the solute sharply declines. On the other hand, when Hartmann number, angle of inclination of the magnetic field, and the third-grade parameter enhance, the peak of the tracer's mean concentration distribution increases. It is also seen that the mean concentration distribution of solute decreases by 63.91% when the Darcy number rises from 0.1 to 0.2, and by 24.44% when it increases from 0.2 to 0.3. Also, the mean concentration distribution of solute increases by 0.019% when third-grade parameter advances from 0.1 to 0.5, and by 0.19% when it improves from 0.5 to 1. The results provide insights into complex transport mechanisms relevant to petrochemical engineering, pharmaceutical processes, lubricant manufacturing, and food industries.

Revisiting moment-based hermite polynomial model in estimation of extreme wind loads

Purpose In this paper applications to information theory are presented. Generalized majorization theorem is presented in term of different entropies and divergences. So that obtained results are generalized and comprehensive. Design/methodology/approach The analytical approach is used to prove the results. i.e. Different n-convex functions are used to prove results. Generalized majorization theorem and Peanos’s representation of Hermite polynomial are also imply. Findings The results related to Csiszár f divergence using generalized majorization theorem via Peanos’s representation of Hermite polynomial are proved. In seek of applications, generalized results are established using Shannon entropy, Kullback-Leibler distance. Moreover, the unique cases of obtained results are derived in terms of Bhattacharyya coefficient, Jeffrey’s distance and Triangular discrimination. Originality/value The results are new and never submitted before.