Multinomial random combinatorial structures and r-versions of Stirling, Eulerian and Lah numbers

In this paper, we mainly prove some congruences involving harmonic numbers and Bernoulli polynomials modulo prime p>3 via hypergeometric transformations. To prove these congruences, we first demonstrate some congruences modulo p 2 . These congruences have already been used in another article.

In this work, we employ a spectral collocation approach to numerically solve pantograph-type Volterra integro-differential equations subject to given initial conditions. The scheme combines Bernoulli polynomials with Gauss quadrature for numerical integration. By leveraging this Bernoulli-Gauss framework, the original integro-differential problem is transformed into a solvable system of algebraic equations. Accurate approximations are achieved using only a modest number of collocation points. The convergence behavior of the method is illustrated through graphical analysis, revealing an exponential rate of convergence. To validate the proposed technique, we present several test problems whose numerical solutions are compared against exact results and those reported by alternative methods. These comparisons are summarized in tables and figures to highlight the accuracy and efficiency of the approach.

ABSTRACT In this paper, we introduce diverse new central special polynomials and numbers utilizing two types of ‐exponential functions. We first consider ‐central factorial numbers and polynomials of the second kind and investigate some of their properties and formulas, such as addition formulas, summation formulas, ‐derivative properties, and Jackson integral representations. As part of our main content, we define trivariate central ‐Bell polynomials and acquire several identities and relations, such as some summation and addition formulas, three ‐derivative properties, two Jackson integral representations, two implicit summation formulas, and a symmetric identity. We investigate an important correlation between these two new ‐polynomials and provide some of its consequences. Then, we provide many correlations between new and old ‐polynomials and ‐numbers, such as the ‐Stirling numbers and polynomials of the second kind, the ‐combinatorial Simsek polynomials and numbers of the first kind, the newly defined ‐polynomials and ‐numbers, ‐Euler polynomials, and ‐Bernoulli polynomials. Also, we consider a new ‐extension of Stirling numbers of the second kind and type 2 ‐Bernoulli polynomials and derive some mixed correlations related to the other new and old ‐polynomials and ‐numbers. Furthermore, we compute two ‐operator formulas for trivariate central ‐Bell polynomials and two ‐operator formulas for bivariate and one‐variable ‐central factorial polynomials of the second kind. In the end, we present graphical illustrations and zero distribution patterns of these newly introduced ‐polynomials, which exhibit a striking and structured scattering in the real and complex planes, offering both aesthetic appeal and deep analytical significance.

In this paper, we introduce and investigate a new class of analytic functions generated by Euler polynomials through a suitable normalization. Using classical tools from geometric function theory, including coefficient monotonicity, Fejér-type inequalities, MacGregor’s criteria, and Ozaki’s close-to-convexity condition, we establish sufficient conditions for the univalence, starlikeness, convexity, and close-to-convexity of the proposed Euler-polynomial-based normalized function. Sharp radius results for starlikeness, convexity, and close-to-convexity in the disk D1/2 are derived by exploiting refined coefficient bounds involving higher-order Euler polynomial terms. Several illustrative examples and graphical demonstrations are provided to verify the theoretical findings. The results obtained extend the known geometric properties of special function-based analytic classes and offer a new perspective on the geometric behavior of Euler polynomials in the unit disk.

This study introduces two- and three-dimensional optimal control problems characterised by the ψ-tempered fractional derivative and proposes a novel numerical methodology based on deep fractional-order Bernoulli optimisation to achieve their efficient solution. To this end, the problems under consideration are first reformulated as equivalent variational problems. Subsequently, a deep neural network employing the fractional-order Bernoulli functions and sinh as activation functions is utilised to approximate the state variable. To facilitate the effective implementation of the proposed method, we construct several integral operators of both integer and ψ-tempered fractional orders based on the basis functions derived from the deep neural network. These operators are discretised via the Fejér quadrature rule adapted to the ψ-tempered fractional calculus, ensuring stability and high-precision integration. The proposed approach converts the 2D/3D ψ-tempered fractional optimal control problem into an algebraic system using deep neural networks with integral operators and Gauss–Legendre integration, which is efficiently solved via Newton's method. The proposed method combines simple implementation, low computational cost, and high accuracy. Its effectiveness and robustness are validated through several representative 2D and 3D examples, confirming the method's applicability to complex fractional optimal control problems.

In this work, we construct a new class of Appell-type polynomials generated through extended truncated and truncated exponential kernels, and we analyze their core algebraic and operational features. In particular, we establish a suitable recurrence scheme and obtain the associated multiplicative and differential operators. By confirming the quasi-monomial structure, we further deduce the governing differential equation for the proposed family. In addition, we present both a series expansion and a determinant formulation, providing complementary representations that are useful for symbolic manipulation and computation. As special cases, we introduce and study subfamilies arising from this setting, namely, extended truncated exponential versions of the Bernoulli, Euler, and Genocchi polynomials, and discuss their structural identities and operational behavior. Overall, these developments broaden the theory of special polynomials and furnish tools relevant to problems in mathematical physics and differential equations.

In previous papers, we attempted to analyze the complete loop counting functions that count all loops in an infinite random walk, represented by the digits of a real number. In this paper, the consideration will be restricted to the partial loop counting functions V that count the returns to the origin only. This simplification allows us to find closed-form expressions for various integrals related to V. Some applications to the complete loop counting functions, in particular, their connections with Bernoulli polynomials, are also provided.

Generalized Euler numbers and ordered set partitions

Enumerative proof of a curious congruence for Eulerian numbers

Abstract Derived from the half hyperbolic secant distribution, new expressions and bounds for Catalan’s constant are presented. They are obtained by using expressions for the Lorenz curve, the Gini and Theil indices, convolutions, and a mixture of distributions, among other approaches. The new expressions are presented both in terms of integral (simple and double) representation and also as an interesting series representation. In addition, some integral representations of Apéry’s constant and Euler numbers are obtained.

For two-dimensional topological band models, the Euler number is a topological invariant to characterize the non-trivial interband topology for adjacent two bands with real eigenstates in the presence of the combined parity and time-reversal symmetry, which is different from the Chern number to characterize the single-band topology of complex eigenstates in the absence of time reversal symmetry. In this work, we study a three-orbital square lattice model that hosts topological Euler insulators, clearly manifested by the bulk band structures, the Wilson loop spectra and the non-trivial Euler numbers. And we observe a bulk-edge correspondence of topological Euler insulators in stark contrast to Chern insulators. We also present various topological phase diagrams based on the Euler numbers and related calculations for different sets of model parameters. Such results enrich our understanding of intriguing Euler class topological phases and may also pave the way for material realizations.

Abstract We give an algorithm for computing weighted Ehrhart functions of lattice polytopes with polynomial weights on its lattice points using Lagrange interpolation. We show how to compute generating functions of polynomials using those of unit cubes and Eulerian numbers, and apply integer programming to study the algebraic properties of the Ehrhart ring of the d -th unit cube. We then present some applications to weighted Ehrhart functions and enumeration problems using linear functions and homogeneous polynomials as weights.

In quark­-gluon plasma (QGP), at high temperatures T the spontaneous generation of color magnetic fields, b3(T ), b8(T ) ̸= 0 (3, 8 are color indexes), and usual magnetic field b(T ) ̸= 0 happens. Also, the Polyakov loop and related to it the A0(T) condensate, which is solution to Yang­-Mills imaginary time equations, create. Recently, with the new type two-­loop effective potential, which generalizes the known integral represen­tation for the Bernoulli polynomials and takes into consideration the magnetic background, these effects were derived. The corresponding effective potential W(T, b3 , b8 , b, A0 ) was calculated either in SU(2) gluodynam­ics or full quantum chromodynamics (QCD). The values of magnetic field strengths at different temperatures were calculated and the mechanism for stabilizing the background due to A0(T) was also discovered. In present paper, we concentrate on the one­loop quark contributions. In particular, we derive the effective vertexes, which couple magnetic fields and A0. The vertexes result in new specific effects signalling the creation of QGP in heavy ion collision experiments.

This paper studies higher-order Eulerian numbers based on Stirling permutations and utilizing Eulerian triangles. It primarily focuses on the chain of higher-order Eulerian numbers, higher-order Eulerian polynomials, and higher-order Eulerian fractions, especially their computation. Many results for Eulerian numbers and second-order Eulerian numbers are generalized to higher-order Eulerian numbers. More specifically, we present recurrence relations of high-order Eulerian numbers, row-generating functions, and row sums of higher-order Eulerian triangles. Furthermore, we investigate the higher-order Eulerian fraction and its alternative form. Some properties of higher-order Eulerian fractions are expressed using differentiation and integration. We derive the inversion relations between second-order Eulerian numbers and Stirling numbers of the second and first kinds. Finally, we provide exact expressions and a computational method for higher-order Eulerian numbers.

This paper presents a wavelet-based numerical method for solving time–space fractional advection equations involving Caputo derivatives. The governing equation is given by \[d_1 \frac{\partial^{\beta} W}{\partial z^{\beta}} + d_2 \frac{\partial^{\gamma} W}{\partial u^{\gamma}} = h(z, u),\]where \( 0 < \beta, \gamma \leq 1 \) denote the fractional orders in the Caputo sense, and \( h(z,u) \) is a known source function. The proposed scheme uses a collocation approach based on Euler wavelets—compactly supported bases constructed from shifted and scaled Euler polynomials. This structure enables exact symbolic evaluation of fractional derivatives and facilitates the accurate enforcement of boundary conditions. The numerical framework builds the solution through coefficient matrices and vector terms derived from a symbolic system, ensuring consistency with the governing equation at carefully selected collocation points. A central result shows that, when the exact solution is polynomial and symbolic computation is used, the method reproduces the solution exactly at all collocation nodes.Numerical experiments support the theoretical findings, demonstrating high accuracy and computational efficiency, particularly for smooth solutions where rapid convergence is observed. Compared to existing approaches, the method offers enhanced precision and broader applicability, especially for problems involving coupled space–time nonlocality. This work expands the use of Euler wavelets in the context of fractional partial differential equations and provides a mathematically rigorous framework suitable for future extensions to nonlinear and multidimensional problems.

This paper investigates the application of the Bernoulli operational matrices (BOM) method for solving second-order singular nonlinear differential equations. Unlike traditional implementations of operational matrices with Chebyshev, Legendre, or Bernstein polynomials, this work emphasizes the efficiency of the Bernoulli basis in achieving high accuracy with relatively few expansion terms. The proposed approach expands the solution in shifted Bernoulli polynomials and uses corresponding operational matrices of differentiation, integration, and product operations to reduce the problem to a system of algebraic equations. The method was implemented in Python and tested on benchmark models such as the Lane–Emden equation. Numerical experiments show that with only basis functions, the BOM achieves a maximum error of order closely matching reference solutions. The convergence analysis confirms the spectral accuracy of the method, with errors decreasing rapidly as increases. A comparison with Chebyshev collocation highlights that the Bernoulli approach reaches the same accuracy with fewer basis terms and lower computational effort. These results establish the Bernoulli operational matrices method as a reliable and computationally efficient tool for singular nonlinear models. Future extensions may include fractional-order systems and higher-dimensional applications.

Abstract The increase in FO frequency due to the use of BFS is accompanied by an increase in pressure losses. The study was conducted using the URANS governing equation and the SST k-ω turbulence model. Double BFS exhibited the highest frequency, with an average increase of 25.78% over the prototype. In contrast, the average frequency increases of single and triple BFS were 20.29% and 19.6%, respectively. The frequency increase is influenced by the momentum of the backflow in the feedback channel. Double BFS had a lower pressure loss than the prototype model, with 4.54% reduction. The average pressure loss of the single BFS model was 24.9% higher than that of the prototype model, whereas the triple BFS model showed a 0.039% increase. The pressure loss is influenced by the recirculation bubble in the FO chamber. Nondimensional analysis using Strouhal and Euler numbers also showed that double BFS exhibited the best performance. The prototype model and single BFS had a velocity profile shape that is closer to a homogeneous shape. The double and triple BFS exhibited a velocity profile shape that is closer to the bifurcated jet shape. Bifurcated jets, which exhibit a wider spread, are characteristic of oscillatory flows. Thus, it can be concluded that the double BFS FO is more recommended.

ABSTRACT In this article, we introduce a new class of truncated exponential‐based Appell polynomials and investigate their key properties and identities. The study further explores their connection with the monomiality principle, providing a deeper algebraic structure. Extensions of this framework lead to the construction of truncated exponential‐based Bernoulli, Euler, and Genocchi polynomials, along with their respective properties. Further, the generating relation and operational identity for these polynomials are established using fractional operators. Numerical evaluations of these special cases are presented, and the distribution of their zeros is visualized using Mathematica. The paper concludes with a summary of results and suggestions for future research directions.

<p>This paper studies the Pell-Narayana sequence modulo <span class="math inline">\(m\)</span>. It starts by defining the Pell-Narayana numbers and examining their combinatorial relationships with well-known sequences and functions, including Eulerian, Catalan, and Delannoy numbers. Building on this, the concept of a Pell-Narayana orbit is introduced for a 2-generator group with generating pair <span class="math inline">\((x, y) \in G\)</span>, which allows the analysis of the periods of these orbits. The results include explicit calculations of the Pell-Narayana periods for polyhedral and binary polyhedral groups, depending on the choice of generating pair <span class="math inline">\((x, y)\)</span>, along with a discussion of their properties. Furthermore, the paper determines the periodic lengths of Pell-Narayana orbits for the groups <span class="math inline">\(Q_8, Q_8 \times \mathbb{Z}_{2m},\)</span> and <span class="math inline">\(Q_8 \times_\varphi \mathbb{Z}_{2m}\)</span> for all <span class="math inline">\(m \geq 3\)</span>.</p>