On the zero bounds of polynomials and regular functions of a quaternionic variable

We study the extensions of the classical Eneström-Kakeya theorem and its various generalizations regarding the distribution of zeros of polynomials from the complex to the quaternionic setting. We aim to build upon the previous work by various authors and derive zero-free regions of some special regular functions of a quaternionic variable with restricted coefficients, namely quaternionic coefficients whose real and imaginary components or moduli of the coefficients satisfy suitable inequalities. The obtained results for this subclass of polynomials and slice regular functions produce generalizations of a number of results known in the literature on this subject.

In the theory of bi-univalent functions,variety of special polynomials and special functions have been used. Using the q - analogue of Bessel functions, two families of regular and bi-univalent functions subordinate to (p, q) - Lucas Polynomials are introduced in this paper. For elements in these defined families, we derive estimates for |a2|, |a3| and for δ a real number we consider Fekete-Szegö problem |a3 − δa22|. We also provide relevent connection to existing result and discuss few interesting observations of the results investigated.

We had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units, called regularization networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known radial basis functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends radial basis functions (RBF) to hyper basis functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, some forms of projection pursuit regression, and several types of neural networks. We propose to use the term generalized regularization networks for this broad class of approximation schemes that follow from an extension of regularization. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In summary, different multilayer networks with one hidden layer, which we collectively call generalized regularization networks, correspond to different classes of priors and associated smoothness functionals in a classical regularization principle. Three broad classes are (1) radial basis functions that can be generalized to hyper basis functions, (2) some tensor product splines, and (3) additive splines that can be generalized to schemes of the type of ridge approximation, hinge functions, and several perceptron-like neural networks with one hidden layer.

Operational methods, fractional operators and special polynomials

We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials.

In the present paper we establish a result on teh representation of special monogenic functions by the product set of special monogenic polynomials. This generalises to the Clifford setting the analogue result in the complex case. We derive also bounds for the order of the product set of special monogenic polynomials. The constrcted examples show that the resulting bounds are attainable.

Scalar's quasibound states in cosmological black hole background

In this study, we examine integrals involving the incomplete H-function of several complex variables in combination with general class polynomials represented in product form. By employing definite integrals from well-known mathematical references, we derive closed-form results that highlight the interplay between special functions and polynomial structures. The incomplete H-function, known for its remarkable generality and applications in multivariate function theory, is here integrated with polynomial structures that unify many classical families. The integrals derived are expressed in a simplified and compact form, making them particularly suitable for applications in mathematical physics, probability theory, and statistical modeling. These results not only broaden the scope of incomplete special functions but also offer efficient tools for handling problems involving products of functions in multidimensional settings. Furthermore, the work provides a foundation for future investigations into more general classes of incomplete multivariable functions and their applications.

ABSTRACTThe aim of this paper is to derive many novel formulas involving the sum of powers of consecutive integers, the Bernoulli polynomials, the Stirling numbers and moments arise from conditional probability, moment generating functions and arithmetic functions by using the methods and techniques, which are used in discrete distributions in statistics such as uniform distribution, moment generating functions, and other probability distributions. Moreover, relations among the generalized Euler totient function, finite distributions containing special numbers and polynomials, discrete probability formula, and other special functions are given. By using the Riemann zeta function and the Liouville function, we derive a novel moment formula for probability distribution on the set positive integers. Finally, by using approximation formulas for certain family of finite sums, we derive formulas not only for the sum of powers of consecutive integers involving the Bernoulli polynomials, but also for the conditional probability involving the Laplace rule of succession.

The article considers the problem of a rectangular plate, supported by a cross system of stiffening ribs, bending. In addition to the transverse load, the plate is subjected to forces in its plane, transmitted through the ribs. Аn analytical solution to the boundary value problem for the resolving differential equation with respect to the normal deflection of the plate, describing the deformation of a rectangular plate supported by stiffeners, is obtained. The solution is presented in the form of series in combinations of regular and special discontinuous functions, which converge quickly and lead to a simple computational algorithm. The influence of the ribs is taken into account in the equation in the form of additional terms containing factors with a delta function. This approach allows us to get rid of a number of assumptions regarding the interaction of the plate with its reinforcing elements. The use of the apparatus of generalized functions when modeling objects of this type simplifies the boundary conditions (there are no conditions for conjugation of various structural elements), but at the same time the differential equations become more complicated. The problem is reduced to the so-called partially degenerate equations. Development of analytical methods that allow obtaining exact solutions of differential equations of this type, and their introduction into computational practice, is one of the urgent tasks of the mechanics of objects with disturbed regularity.

An analytical solution of a boundary problem for system of differential equations in displacements, describing a strain of the rectangular plate supported by ribs of rigidity is received. The solution is presented in the form of series with combination of regular and special singular functions which converge fast, and lead to simple computing algorithm.

On the zeros of a quaternionic polynomial with restricted coefficients

This works introduces several notions of subharmonicity for real-valued functions of one quaternionic variable. These notions are related to the theory of slice regular quaternionic functions introduced by Gentili and Struppa in 2006. The interesting properties of these new classes of functions are studied and applied to construct the analogs of Green's functions.

6 - q-Extensions of Some Special Functions and Polynomials

For D:={z∈C:|z|<1}, this paper derives refined conditions for the inclusion of special functions in lemniscate and nephroid domains focusing on solutions to the differential equations of the form znF″(z)+a(z)zn−1F′(z)+b(z)F(z)+d(z)=0,n∈{1,2},z∈D, with normalization F(0)=1, where a(z), b(z) and d(z) are analytic in D. Using advanced techniques from geometric function theory, we generalize and improve existing results, particularly for classes of functions defined by differential equations. Specific applications include generalized Bessel functions, regular Coulomb wave functions, and associated Laguerre polynomials, where we derive improved bounds for their inclusion in lemniscate domains. Additionally, we present open problems, supported by numerical experiments, to guide future research in this direction.