In this paper, we investigate the telephone polynomials T n ( x ) , an Appell-type generalization of the classical telephone (involution) numbers, which arise naturally in combinatorics, graph theory, and matching problems. Motivated by their relation to Hermite-type polynomial families, we study their algebraic and operational properties using generating function, differential, and umbral techniques. We derive several structural results for the telephone polynomials, including recurrence relations, explicit formulas, coefficient identities, exponential generating functions, and a Rodrigues-type formula. We also establish differentiation properties, derivative sequences, and connections with Euler and Changhee polynomials. Furthermore, we prove that T p ( x ) / x is irreducible over Q for every odd prime p, and obtain additional irreducibility results in several special cases using Eisenstein's criterion. Motivated by these results, we formulate a general irreducibility conjecture for the families T 2 k ( x ) and T 2 k + 1 ( x ) / x , which remains open in full generality. Finally, an umbral representation of the telephone polynomials is presented, leading to additional identities and relations.

Stieltjes moment property of the generalized Eulerian polynomials

ABSTRACT Cyclone separators are widely used in industry, where separation efficiency and pressure drop define cyclone performance. A common strategy for increasing cyclone efficiency is to use smaller cyclones. However, does reducing the cyclone size while maintaining geometric similarity continuously increase efficiency, or is there a limit to this approach? In this study, geometrically similar high efficiency Stairmand cyclones with body diameters of 10, 15, 20, 25, and 30 mm were investigated experimentally and theoretically at inlet velocities of 4–10 m/s using calcite particles at constant dust concentration. The highest separation efficiency (0.915) was obtained in the 30 mm cyclone at 10 m/s. The 25 and 20 mm cyclones showed similar efficiencies, whereas the 15 and 10 mm cyclones showed considerably lower efficiencies, decreasing to 0.14 in the 10 mm cyclone. Pressure loss coefficients decreased with cyclone size, with measured Euler numbers ranging from 3.4–4.4 (30 mm) to 1.0–2.0 (10 mm). Theoretical predictions were consistent with experimental trends, although deviations increased for smaller diameters due to relative roughness and transitional flow effects. The results indicate that for geometrically similar cyclones under constant inlet velocity, separation efficiency reaches a maximum at a certain cyclone diameter, beyond which further size reduction reduces efficiency.

In this paper, within the framework of unsigned degenerate r -Stirling numbers, we establish connections among Cauchy polynomials, degenerate Bernoulli polynomials, and degenerate hyperharmonic numbers. The key results include: (i) expressing the values of Cauchy polynomials at nonnegative integers using the unsigned degenerate r -Stirling numbers of the first kind, and inverting these expressions via the inversion relation with the degenerate r -Stirling numbers of the second kind; (ii) representing the values of Cauchy polynomials at nonnegative integers in terms of the values of fully degenerate Bernoulli polynomials, and vice versa, using either kind of degenerate r -Stirling numbers; (iii) introducing degenerate Cauchy polynomials and deriving an identity that connects them to degenerate hyperharmonic numbers.

In this paper, by considering higher order generalized Lehmer–Euler numbers, we introduce two families of polynomials involving a prime p, a rational p-adic integer α, non-negative integers n, r, s, and a variable x. We define M n ( α , p , x ) ( r , s ) = ∑ k = 0 n ( αp + k − p k ) r ( αp + n k ) s x k , M n ∗ ( α , x ) = ∑ k = 0 n ( α + k − 1 k ) ( α + n n − k ) x k . We then investigate several congruence properties associated with M n ( α , p , x ) ( r , s ) and M n ∗ ( α , x ) .

Summary The Eulerian numbers form a triangular array with many interesting properties. The numbers arise from various combinatorial and probabilistic interpretations, and have been studied in a variety of mathematical contexts. In this article we examine two distinct alternating sign formulas for the Eulerian numbers and show how they can be proved using a sign-reversing involution technique described by Benjamin and Quinn known as the “D.I.E.” method. Each of these arguments lends itself to a broad generalization, shedding light on different parts of mathematics such as Ehrhart Theory and Combinatorial Topology.

Abstract The purpose of this paper is to give moment formulas with the aid of Milovanović [16]. Other aims are to establish new integral formulas in order to define new Apostol-type splines in terms of the Apostol-type polynomials. By the aid of these integral formulas, we derive a novel class of moment-type expressions arising from integrals of these polynomials. By applying generating function techniques and moment computations, we derive explicit representations and approximation formulas for Apostol- Bernoulli, Euler, and Frobenius spline polynomials. Closed-form expansions are established using Goldman’s formula and symbolic moment identities. The connection between cardinal B-splines $$\phi _n(x)$$ ϕ n ( x ) and uniform B-splines $$N_{0,n-1}(x) $$ N 0 , n - 1 ( x ) is given. We compute integrals using beta-type representations and provide recurrence relations for numerical implementation. Furthermore, we develop a comparative numerical table that confirms the validity of the approximation.

In this article, our aim is to extend the research conducted by Kurokawa and Wakayama in 2003, particularly focusing on the q-analogue of the Hurwitz zeta function. Our specific emphasis lies in exploring the coefficients in the Laurent series expansion of a q-analogue of the Hurwitz zeta function around s=1. We establish the closed-form expressions for the first two coefficients in the Laurent series of the q-Hurwitz zeta function. Additionally, utilizing the reflection formula for the digamma function and the identity of Bernoulli polynomials, we explore transcendence results related to γ 0 ( q , x ) for q>1 and 0<x<1, where γ 0 ( q , x ) is the constant term which appears in the Laurent series expansion of q-Hurwitz zeta function around s=1. Furthermore, we put forth a conjecture about the linear independence of special values of γ 0 ( q , x ) along with 1 at rational arguments with co-prime conditions, over the field of rational numbers. Finally, we show that at least one more than half of the numbers are linearly independent over the field of rationals.

Fluid mechanics in disordered structures gives rise to rich multiscale dynamics through the interplay of topology, symmetry breaking, and fluid–structure interactions. Heterogeneous networks encode mechanical responses, regulate flow organization, and shape energy dissipation, enabling memory effects and emergent collective behaviors across both natural and engineered systems. These principles operate across vast scales: from seamounts with characteristic scales of L≈103m and Froude numbers Fr≈10−2–10−1 generating deep-ocean turbulent mixing, to marine tidal turbines operating at Reynolds numbers Re≈107–108 and Euler numbers Eu≈10−1–100, where inertial forces dominate flow dynamics. Although the dominant physical forces may vary across scales—for example, planetary rotation and stratification in large-scale oceanic flows versus viscous or interfacial effects in microscale systems—the comparison of dimensionless parameters provides a useful framework for discussing similarities in flow organization and scaling behavior. Empirical observations, network-based descriptions, and multiscale simulations collectively demonstrate how topological features constrain symmetry, organize transport pathways, and support predictive reconstruction and inverse design. These principles underpin applications ranging from engineered systems that exploit broken symmetries to rectify chaotic transport, to biological architectures where flows mediate information transfer, locomotion, and structural self-organization. In this Review, we synthesize recent advances to propose a unifying physical paradigm: fluid flows actively interact with disorder, reorganize dissipation, and convert structural asymmetry into functional mechanical performance across scales.

The Real Stability of the Multivariate Eulerian Polynomials for Restricted Excedance Statistics

Abstract In this article, we capture Phillips type operators based on adjoint Bernoulli polynomials and their connection to other operators. We derive their characteristic function and provide pointwise convergence and estimate errors using various types of modulus of continuity. Then we establish the theorems based on the difference of operators with their decomposed parts. Additionally, we modify these operators in order to preserve e Ax and e 2Ax and give asymptotic formula and a Korovkin-type result for modified operators.

Comprehensive Subclasses of Bi-univalent Functions Specified by Liouville–Caputo-Type Fractional Derivatives and Euler Polynomials

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.