Polynomial Type Research Articles

On the new type of degenerate poly-Genocchi numbers and polynomials

Kim and Kim (J. Math. Anal. Appl. 487:124017, 2020) introduced the degenerate logarithm function, which is the inverse of the degenerate exponential function, and defined the degenerate polylogarithm function. They also studied a new type of the degenerate Bernoulli polynomials and numbers by using the degenerate polylogarithm function. Motivated by their research, we subdivide this paper into two parts. In Sect. 2, we construct a new type of degenerate Genocchi polynomials and numbers by using the degenerate polylogarithm function, called the degenerate poly-Genocchi polynomials and numbers, deriving several combinatorial identities related to the degenerate poly-Genocchi numbers and polynomials. Then, in Sect. 3, we also consider the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. In particular, we provide some new explicit computational identities of degenerate unipoly polynomials related to special numbers and polynomials.

The $1/k$-Eulerian Polynomials of Type $B$

In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.

On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.

A note on degenerate poly-Genocchi numbers and polynomials

Recently, some mathematicians have been studying a lot of degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects. Our research is also interested in this field. In this paper, we introduce a new type of the degenerate poly-Genocchi polynomials and numbers, based on Kim and Kim’s (J. Math. Anal. Appl. 487(2):124017, 2020) modified polyexponential function. The paper is divided into two parts. In Sect. 2, we consider a new type of the degenerate poly-Genocchi polynomials and numbers constructed from the modified polyexponential function. We also show several combinatorial identities related to the degenerate poly-Genocchi polynomials and numbers. Some of them include the degenerate and other special polynomials and numbers such as the Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, degenerate Euler polynomials, degenerate Bernoulli polynomials and Bernoulli numbers of order α, etc. In Sect. 3, we also introduce the degenerate unipoly Genocchi polynomials attached to an arithmetic function by using the degenerate polylogarithm function. We give some new explicit expressions and identities related to degenerate unipoly Genocchi polynomials and special numbers and polynomials.

A Unification of the Generalized Multiparameter Apostol-type Bernoulli, Euler, Fubini, and Genocchi Polynomials of Higher Order

Most unifications of the classical or generalized Bernoulli, Euler, and Genocchi polynomials involve unifying any two or all of the three special types of polynomials (see, [1, 4, 9, 18, 19,21, 24–26, 30, 31]). In this paper, we introduce a new class of multiparameter Fubini-type gener-alized polynomials that unifies four families of higher order generalized Apostol-type polynomials such as the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Fubini polynomials. Moreover, we obtain an explicit formula of these unified generalized polynomials in terms of the Gaussian hypergeometric function, and establish several symmetry identities.

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik–Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles.The purpose of this paper is to prove the algebraic counterpart of the Orlik–Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund–Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov, whose Euler matrix are calculated in terms of the “zeta function” of the inverse of the polarization.As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).

Some orthogonal polynomials on the finite interval and Gaussian quadrature rules for fractional Riemann‐Liouville integrals

Inspired with papers by Bokhari, Qadir, and Al‐Attas (2010) and by Rapaić, Šekara, and Govedarica (2014), in this paper we investigate a few types of orthogonal polynomials on finite intervals and derive the corresponding quadrature formulas of Gaussian type for efficient numerical computation of the left and right fractional Riemann‐Liouville integrals. Several numerical examples are included to demonstrate the numerical efficiency of the proposed procedure.

On the static deformation of FG sandwich beams curved in elevation using a new higher order beam theory

This article presents the static analysis of FG sandwich beams curved in elevation. Navier-type semi-analytical solutions are obtained based on polynomial type fifth order shear and normal deformation theory. The beam has FG skins and isotropic core. Material properties of FG skins are graded in z-direction according to the power-law distribution. The present theory accounts for a fifth-order distribution of axial displacement and fourth-order distribution of transverse displacement. The present theory considers the effect of thickness stretching and gives a realistic variation of transverse shear stress through the thickness of the beam. The governing equations are obtained within the framework of the principle of virtual work. Semi-analytical static solutions for the simply supported FG sandwich beams curved in elevation are obtained using Navier’s technique. The beam is subjected to uniformly distributed load. The non-dimensional numerical values for displacements and stresses are obtained for various power-law index and thickness of the core. The present results are found in good agreement with previously published results.

On the zeros of certain composite polynomials and an operator preserving inequalities

If all the zeros of nth degree polynomials f(z) and $$g(z) = \sum _{k=0}^{n}\lambda _k\left( {\begin{array}{c}n\\ k\end{array}}\right) z^k$$ respectively lie in the cricular regions $$|z|\le r$$ and $$|z| \le s|z-\sigma |$$ , $$s>0$$ , then it was proved by Marden (Geometry of polynomials, Math Surveys, No. 3, American Mathematical Society, Providence, 1949, p. 86) that all the zeros of the polynomial $$h(z)= \sum _{k=0}^{n}\lambda _k f^{(k)}(z) \frac{(\sigma z)^k}{k!}$$ lie in the circle $$|z| \le r ~ \max (1,s)$$ . In this paper, we relax the condition that f(z) and g(z) are of the same degree and instead assume that f(z) and g(z) are polynomials of arbitrary degree n and m, respectively, $$m\le n,$$ and obtain a generalization of this result. As an application, we also introduce a linear operator which preserves Bernstein type polynomial inequalities.

Generalized attenuated ray transforms and their integral angular moments

In this article generalized attenuated ray transforms (ART) and integral angular moments are investigated. Starting from the Radon transform, the attenuated ray transform and the longitudinal ray transform, we derive the concept of ART-operators of order k over functions defined on the phase space and depending on time. The ART-operators are generalized for complex-valued absorption coefficient as well as weight functions of polynomial and exponential type. Connections between ART operators of various orders are established by means of the application of the linear part of a transport equation. These connections lead to inhomogeneous differential equations of order (k+1) for the ART of order k. Uniqueness theorems for the corresponding boundary-value and initial boundary-value problems are proved. Properties of integral angular moments of order p are considered and connections between the moments of different orders are deduced. A close connection of the considered operators with mathematical models for tomography, physical optics and integral geometry allows to treat the inversion of ART of order k as an inverse problem of determining the right-hand side of a corresponding differential equation.

Intermolecular Interactions in 1,6-Diaminohexane + Water Mixtures at 293.15 to 333.15 K

The density (ρ) and viscosity (η) of the mixtures of 1,6-diaminohexane (DAH) + water binary system were determined over the whole composition range at p = 101 kPa from 293.15 to 333.15 K. The measured density and viscosity values decrease with the increase of temperature. In addition, the viscosity of the binary system shows a maximum value along with the molar fraction of DAH. According to the experimental data, excess molar volume (VE), apparent molar volume Vϕ, viscosity deviation (Δη), and excess Gibbs free energy (ΔG*E) of the activation of viscous flow were obtained and correlated by the well-known Redlich–Kister type polynomial equation by using the least square method and regression parameters were calculated. It was found that all of the VE values are negative while $$\Delta \eta $$ and ΔG*E values are positive. The intermolecular interactions between DAH and water were characterized by the averaged non-covalent interaction index (aNCI) and thermal fluctuation index (TFI) method. The obtained results proved that N···H–O and O···H–N hydrogen bonds which lightly affected by thermal motion, and some of the flexible van der Waals interactions are the reason for the specific variation of the viscosity and excess properties of the studied binary system.

Associated prime ideals over skew PBW extensions

In this article, we continue the study of ideals of the noncommutative rings of polynomial type known as skew Poincaré-Birkhoff-Witt extensions. More exactly, we focus on the associated prime ideals of these extensions. Our purpose here is to describe how the associated primes of a ring of coefficients behave under passage to a skew PBW extension over this ring. As we show, all results established here coincide and generalize those corresponding in the commutative case and the Ore extensions.

Mirror Symmetry for Nonabelian Landau-Ginzburg Models

We consider Landau-Ginzburg models stemming from groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group $G^\star$, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials., we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials.

A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type

In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.

Blow-Up of Solutions for a System Viscoelastic Equation with Balakrishnan–Taylor Damping and Nonlinear Source of Polynomial Type

In this paper, we study the initial-boundary value problem for a system of nonlinear viscoelastic equations with Balakrishnan–Taylor damping terms and nonlinear source of polynomial type. We demonstrate that the nonlinear source of polynomial type is able to force solutions to blow up infinite time even in presence of stronger damping with non positive initial energy combined with a positive initial energy.

Analyses of replicated spectrophotometric data by using soft computing methods

Soft computing based chemometric studies are needed to model and to optimize complex processes, such as the extraction stage. In this study, it was aimed to obtain optimal values of ethanol (EtOH) concentration, extraction time, extraction temperature, solvent/solid ratio for the ultrasound-assisted extraction (UAE) of grape seed polyphenols to maximize the total phenolic content (TPC) and total antioxidant activity (TAA) in multi-objective perspective by using soft computing methods. The experimental data set was composed with replicated response measures to see the behavior of the responses. The replicated response measured (RRM) data was recently modeled by using fuzzy linear regression in which the replicated responses were considered as triangular linear fuzzy numbers (TLFNs). Also, polynomial type fuzzy linear regression model parameters were dealt as TLFNs whereas the experiment conditions were crisp. The predicted fuzzy linear models were obtained by using least square (LS) approach. The predicted fuzzy models were optimized through the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) and Pareto set was obtained. The obtained Pareto solution set was experimentally verified. A compromise solution was chosen among many non-dominated solutions of experimental conditions by using a distance function based on root mean square of fuzzy errors. According to the results, it is possible to say that the proposed soft computing based modeling and multi-objective optimization (MOO) approaches can be used as flexible analyses tools in chemometry.

Some properties of generalized hypergeometric Appell polynomials

Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N}}_0$, $k \in {\mathbb{N}},$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg| \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k)}}{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k)}} \frac{z^k}{k!}$ as follows $$ A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k}}, {\displaystyle -\frac{n-1}{k}}, {\ldots}, {\displaystyle-\frac{n-k+1}{k}}\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg| \: \displaystyle \frac{m}{x^k} \end{bmatrix} $$ and, at the same time, the polynomials from this family are Appell-type polynomials.
 The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.

A parametric type of Bernoulli polynomials with level 3

In this paper, we introduce a parametric type of Bernoulli polynomials with level 3 and study their characteristic and combinatorial properties. In particular, we show some determinant expressions of these polynomials. We also give determinant expressions of a parametric type of Bernoulli polynomials with level 2.

Design of Reinforced Fuzzy Radial Basis Function Neural Network Classifier Driven With the Aid of Iterative Learning Techniques and Support Vector-Based Clustering

In this article, a reinforced fuzzy radial basis function neural network (R-FRBFNN) classifier is proposed. It focuses on the development of methodologies of reinforced architecture to improve classification accuracy and enhance the robust capability based on two learning strategies. The two learning strategies are summarized: 1) R-FRBFNN designed via support vector (SV)-based fuzzy C-means (FCM) clustering and softmax-based iterative reweighted least square (IRLS), which concentrate on improving the classification performance of R-FRBFNN; and 2) R-FRBFNN designed via SV-based FCM and softmax-based iterative quadratic programming (IQP), which focus on improving the robust abilities of the R-FRBFNN and reducing the effects of noise and outliers. The essential points of the proposed R-FRBFNN classifier are summarized as follows. a) The proposed R-FRBFNN consists of three phases: condition, conclusion, and inference. b) An SV-based FCM is considered for prioritizing the classification boundary and improving the classification performance of the proposed classifier. c) Three types of polynomials construct the conclusion phase. Two learning techniques are designed to update the coefficients of the polynomials. Softmax-based IRLS is a type of iterative learning technique based on Newton's method. Softmax-based IQP is more robust and avoids the degradation of generalization capabilities caused by outliers and noisy data. d) In the concept of reinforced architecture, SV-based FCM imposes compensation (membership degrees) on learning techniques according to the data characteristics encountered in the inference phase. Experimental results reported for benchmark data and outliers/noisy datasets demonstrate that the proposed classifier shows improved classification performance compared with other previously studied methods.

Скалярная задача равновесия и предельное распределение нулей полиномов Эрмита-Паде II типа

С помощью скалярной задачи равновесия, поставленной на двулистной римановой поверхности, доказано существование предельного распределения нулей полиномов Эрмита-Паде II типа для пары функций, образующих систему Никишина. Обсуждается связь полученных результатов с результатами Г. Шталя 1988 г. Приводятся результаты численных экспериментов. Показано, что результаты, полученные в настоящей работе и в более ранних работах второго автора, вполне согласуются как с результатами Г. Шталя, так и с результатами численных экспериментов.