New Class Of Polynomials Research Articles

Applying the monomiality principle to the new family of Apostol Hermite Bernoulli-type polynomials

Abstract In this article, we introduce a new class of polynomials, known as Apostol Hermite Bernoulli-type polynomials, and explore some of their algebraic properties, including summation formulas and their determinant form. The majority of our results are proven using generating function methods. Additionally, we investigate the monomiality principle related to these polynomials and identify the corresponding derivative and multiplicative operators.

Misiurewicz polynomials and dynamical units, part II

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The 2-successive partial Bell polynomials

In this paper, we discuss a new class of partial Bell polynomials. The first section gives an overview of partial Bell polynomials and their related 2-successive Stirling numbers. In the second section, we introduce the concept of 2-successive partial Bell polynomials. We give an explicit formula for computing these polynomials and establish their generating function. In addition, we derive several recurrence relations that govern the behaviour of these polynomials. Furthermore, we study specific cases to illustrate the applicability and versatility of this new class of polynomials.

An Algebraic Approach to the Δh-Frobenius–Genocchi–Appell Polynomials

In recent years, the generating function of mixed-type special polynomials has received growing interest in several fields of applied sciences and physics. This article intends to study a new class of polynomials, called the Δh-Frobenius–Genocchi–Appell polynomials. The generating function of Δh-Frobenius–Genocchi–Appell polynomials is constructed and some of their fundamental properties are studied. By making use of this generating function, we investigate some novel and interesting results, such as recurrence relations, explicit representations, and implicit formulas for the Δh-Frobenius–Genocchi–Appell polynomials. The quasi-monomiality and determinant form for these polynomials are established. The Δh-Genocchi–Appell polynomials are explored as a special case and several results for Δh-Genocchi–Appell polynomials are also obtained.

A New Numerical Approach for Solving the Fractional Nonlinear Multi-pantograph Delay Differential Equations

The numerical solution of the fractional nonlinear multi-pantograph delay differential equations is investigated by a new class of polynomials. These polynomials are equipped with an unknown auxiliary parameter a, which is obtained by using the collocation and least-squares methods. In this paper, the numerical solution of the fractional nonlinear multi-pantograph delay differential equation is displayed in the truncated series form. The existence and uniqueness of the solution and the error analysis are also investigated in this article. In four examples, the numerical results of the present method have been compared with other methods. For the first time, a-polynomials are used in this article to numerically solve the fractional nonlinear multi-pantograph delay differential equations, and accurate approximations have been displayed.

A note on partially degenerate Hermite-Bernoulli polynomials of the first kind

In this paper, we introduce a new class of partially degenerate Hermite-Bernoulli polynomials of the first kind and generalized Gould-Hopper-partially degenerate Bernoulli polynomials of the first kind and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould-Hopper, as well as the 1-d and 2-d holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the partially degenerate Hermite-Bernoulli polynomials of the first kind are also discussed.

Constructions of Goethals–Seidel Sequences by Using k-Partition

In this paper, we are devoted to finding Goethals–Seidel sequences by using k-partition, and based on the finite Parseval relation, the construction of Goethals–Seidel sequences could be transformed to the construction of the associated polynomials. Three different structures of Goethals–Seidel sequences will be presented. We first propose a method based on T-matrices directly to obtain a quad of Goethals–Seidel sequences. Next, by introducing the k-partition, we utilize two classes of 8-partitions to obtain a new class of polynomials still remaining the same (anti)symmetrical properties, with which a quad of Goethals–Seidel sequences could be constructed. Moreover, an adoption of the 4-partition together with a quad of four symmetrical sequences can also lead to a quad of Goethals–Seidel sequences.

Recurrence relation for the Appell sequences

In this paper, we present several explicit formulas for Appell polynomials in terms of the weighted Stirling numbers of the second kind. We also provide a unified approach to obtain a three-term recurrence formula for the computation of Appell polynomials. Several examples are given to illustrate our results. As an application, we define and study a new class of polynomials that we call SPI polynomials. They are related to sums of powers of integers.

On λ-Changhee–Hermite polynomials

Abstract In this paper, we introduce a new class of λ-analogues of the Changhee–Hermite polynomials and generalized Gould–Hopper–Appell type λ-Changhee polynomials, and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould–Hopper, as well as the 1D and 2D holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the Appell type λ-Changhee polynomials are also discussed.

Numerical Approach for Solving the Fractional Pantograph Delay Differential Equations

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a, which is obtained using the collocation and least‐squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in this study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, a‐polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed.

Polynomials, numerical triangles and sequences associated with the probability distribution of the hyperbolic cosine type for odd values of the parameter

The article introduces a new class of polynomials that first appeared in the probability distribution density function of the hyperbolic cosine type. With an integer change in one of the parameters of this distribution, polynomials in the form of a product of positive factors are written out with an increasing degree. Earlier, the author found a connection between the distribution of the hyperbolic cosine type and numerical sets, in particular, in the simplest cases with the triangle of coefficients of Bessel polynomials, the triangle of Stirling numbers, sequences of coefficients in the expansion of various functions, etc. Also from the distribution formed numerous numerical sequences, both new and widely known. Consideration of polynomials separately from the density function made it possible to reconstruct numerical sets of coefficients, ordered in the form of numerical triangles and numerical sequences. The connections between the elements of the sets are established. Among the sequences obtained, in the simplest cases, there are those known from others, for example, physical problems. However, the overwhelming majority of the found number sets have not been encountered earlier in the literature. The obvious applications of this research are number theory and algebra. And the interdisciplinarity of the results indicates the possibility of applications and enhances their practical significance in other areas of knowledge.

Polynomials and number sets associated with the probability distribution of the hyperbolic cosine type for even values of the parameter

The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

A new class of generalized polynomials involving Laguerre and Euler polynomials

Motivated by their importance and potential for applications in a variety of research fields, recently, numerous polynomials and their extensions have been introduced and investigated. In this paper, we modify the known generating functions of polynomials, due to both Milne-Thomsons and Dere-Simsek, to introduce a new class of polynomials and present some involved properties. As obvious special cases of the newly introduced polynomials, we also introduce power sum-Laguerre-Hermite polynomials and generalized Laguerre and Euler polynomials and give certain involved identities and formulas. We point out that our main results, being very general, are specialised to yield a number of known and new identities involving relatively simple and familiar polynomials.

A new class of polynomials from the spectrum of a graph, and its application to bound the k-independence number

The k-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k. A graph is called k-partially walk-regular if the number of closed walks of a given length l≤k, rooted at a vertex v, only depends on l. In particular, a distance-regular graph is also k-partially walk-regular for any k. In this paper, we introduce a new family of polynomials obtained from the spectrum of a graph, called minor polynomials. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the k-independence number of a k-partially walk-regular graph. With some examples and infinite families of graphs whose bounds are tight, we also show that the odd graph Oℓ with ℓ odd has no 1-perfect code. Moreover, we show that our bound coincides, in general, with the Delsarte's linear programming bound and the Lovász theta number θ, the best ones to our knowledge. In fact, as a byproduct, it is shown that the given bounds also apply for θ and Θ, the Shannon capacity of a graph. Moreover, apart from the possible interest that the minor polynomials can have, our approach has the advantage of yielding closed formulas and, so, allowing asymptotic analysis.

Interpolation of exponential-type functions on a uniform grid by shifts of a basis function

<p style='text-indent:20px;'>In this paper, we present a new approach to solving the problem of interpolating a continuous function at <inline-formula><tex-math id="M1">\begin{document}$ (n+1) $\end{document}</tex-math></inline-formula> equally-spaced points in the interval <inline-formula><tex-math id="M2">\begin{document}$ [0, 1] $\end{document}</tex-math></inline-formula>, using shifts of a kernel on the <inline-formula><tex-math id="M3">\begin{document}$ (1/n) $\end{document}</tex-math></inline-formula>-spaced infinite grid. The archetypal example here is approximation using shifts of a Gaussian kernel. We present new results concerning interpolation of functions of exponential type, in particular, polynomials on the integer grid as a step en route to solve the general interpolation problem. For the Gaussian kernel we introduce a new class of polynomials, closely related to the probabilistic Hermite polynomials and show that evaluations of the polynomials at the integer points provide the coefficients of the interpolants. Finally we give a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval (assuming knowledge of the discrete moments of the Gaussian).

Quasi-subfield Polynomials and the Elliptic Curve Discrete Logarithm Problem

AbstractWe initiate the study of a new class of polynomials which we call quasi-subfield polynomials. First, we show that this class of polynomials could lead to more efficient attacks for the elliptic curve discrete logarithm problem via the index calculus approach. Specifically, we use these polynomials to construct factor bases for the index calculus approach and we provide explicit complexity bounds. Next, we investigate the existence of quasi-subfield polynomials.

On a new Sheffer class of polynomials related to normal product distribution

Consider a generic random element $F_\infty= \sum_{\text{finite}} \lambda_k (N^2_k -1)$ in the second Wiener chaos with a finite number of non-zero coefficients in the spectral representation where $(N_k)_{k \ge 1}$ is a sequence of i.i.d $\mathscr{N}(0,1)$. Using the recently discovered (see Arras et al. \cite{a-a-p-s-stein}) stein operator $\RR_\infty$ associated to $F_\infty$, we introduce a new class of polynomials $$\PP_\infty:= \{ P_n = \RR^n_\infty \textbf{1} \, : \, n \ge 1 \}.$$ We analysis in details the case where $F_\infty$ is distributed as the normal product distribution $N_1 \times N_2$, and relate the associated polynomials class to Rota's {\it Umbral calculus} by showing that it is a \textit{Sheffer family} and enjoys many interesting properties. Lastly, we study the connection between the polynomial class $\PP_\infty$ and the non-central probabilistic limit theorems within the second Wiener chaos.

A new class of irreducible pentanomials for polynomial-based multipliers in binary fields

We introduce a new class of irreducible pentanomials over {mathbb F}_{2} of the form f(x) = x^{2b+c} + x^{b+c} + x^b + x^c + 1. Let m=2b+c and use f to define the finite field extension of degree m. We give the exact number of operations required for computing the reduction modulo f. We also provide a multiplier based on Karatsuba algorithm in mathbb {F}_2[x] combined with our reduction process. We give the total cost of the multiplier and found that the bit-parallel multiplier defined by this new class of polynomials has improved XOR and AND complexity. Our multiplier has comparable time delay when compared to other multipliers based on Karatsuba algorithm.

A new class of generalized Laguerre–Euler polynomials

A variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential of applications in diverse research areas. In this sequel, we introduce a new class of generalized Laguerre–Euler polynomials and present certain potentially useful formulas and identities such as implicit summation formulae and symmetry identities. The new class of polynomials introduced and the results presented here, being very general, are shown to be specialized to reduce to various known polynomials and to yield some known and new formulas and identities, respectively.

Modified Euler-Frobenius Polynomials With Application to Sampled Data Modelling

The broad class of polynomials generally known as Eulerian or Euler-Frobenius polynomials has a rich history in Mathematics and Engineering. They have been generalised in several directions and find application in diverse areas ranging from sampled-data modelling and polynomial interpolation to wavelets. Here we describe a modified class of Euler-Frobenius polynomials which are functions of two independent variables. We show that the sampling zero (dynamics) of general linear and nonlinear systems can be described using this new class of polynomials when considering several non-standard sampled-data modelling problems, including when generalised hold functions are used to generate the input and when the continuous system has a time delay.