New Family Of Polynomials Research Articles

Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

We study a delay-differential analogue of the first Painlevé equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlevé-I equation in terms of an affine Weyl group of type A1(1). As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas.

Δω-Laguerre based Appell polynomials and their properties associated with some special polynomials

In the present paper, we introduce Δω-Laguerre based Appell polynomials. The first main aim of the paper is to investigate their main properties including explicit representation, determinantal form, recurrence relation, lowering operators (LO), integro-partial raising operator (IPRO) and integro-partial difference equation (IPDE). The second aim is to give the corresponding results, which are known or new, to the usual Laguerre based Appell polynomials which can be obtain immediately from the limiting case ω→0. Furthermore, we introduce Δω-Laguerre-Carlitz Bernoulli, Δω-Laguerre-Carlitz Euler, Δω-Laguerre-Miller-Lee, Δω-Laguerre-Boole polynomials and Δω-Laguerre-Bernoulli polynomials of the second kind as particular cases of Δω-Laguerre based Appell polynomials. The corresponding main results have also been exhibited for these new families of polynomials.

On parametric types of Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials via Golden calculus

<abstract><p>This paper aims to give generating functions for the new family of polynomials, which are called parametric types of the Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci, and the Apostol Genocchi-Fibonacci polynomials by using Golden calculus. Numerous properties of these polynomials with their generating functions are investigated. These generating functions give us a generalization of some well-known generating functions for special polynomials such as Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials. Using the Golden differential operator technique, the functional equation method for generating function, we present some properties of these newly established polynomials.</p></abstract>

Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms.

A generalization of the Riemann–Siegel formula

The celebrated Riemann–Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable t. Siegel anticipated that this formula could be generalized to include the Hardy–Littlewood approximate functional equation, valid in any vertical strip. We give this generalization for the first time. The asymptotics contain Mordell integrals and an interesting new family of polynomials.

Certain Hybrid Matrix Polynomials Related to the Laguerre-Sheffer Family

The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to specific members of the matrix polynomial family are shown.

New Biparametric Families of Apostol-Frobenius-Euler Polynomials level-m

We introduce two biparametric families of Apostol-Frobenius-Euler polynomials of level-$m$. We give some algebraic properties, as well as some other identities which connect these polynomial class with the generalized $\lambda$-Stirling type numbers of the second kind, the generalized Apostol--Bernoulli polynomials, the generalized Apostol--Genocchi polynomials, the generalized Apostol--Euler polynomials and Jacobi polynomials. Finally, we will show the differential properties of this new family of polynomials.

Computational formulas and identities for new classes of Hermite‐based Milne–Thomson type polynomials: Analysis of generating functions with Euler's formula

The aim of this paper is to construct generating functions for a new family of polynomials, which are called parametric Hermite‐based Milne–Thomson type polynomials. Many properties of these polynomials with their generating functions are investigated. These generating functions give us generalization of some well‐known generating functions for special polynomials such as Hermite type polynomials, Milne–Thomson type polynomials, and Apostol type polynomials. Using the Euler formula, functional equation method for generating function, and differential operator technique, we give relations among parametric Hermite‐based Milne–Thomson type polynomials, the Bernoulli numbers, the Euler numbers, the Chebyshev polynomials, the Bernstein basis functions, homogeneous harmonic polynomials, and parametric kinds of Apostol type polynomials. Moreover, some computational formulas for these polynomials are derived. Finally, using Wolfram Mathematica version 12.0, some of these polynomials and their generating functions are illustrated by their plots under the special conditions. Potential relationships and connections of this paper's results with the results of previous and future research are pointed out.

Harmonic number identities via polynomials with r-Lah coefficients

In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We show that some of these polynomials and hyperharmonic numbers are closely related. Taking into account of these connections, we reach several identities for harmonic and hyperharmonic numbers.

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.

Supergroup OSP(2,2n) and super Jacobi polynomials

The coefficients of the super Jacobi polynomials of BC1,n-type are rational functions in three parameters k,p,q. At the point (−1,0,0) these coefficients may have poles. Let us set q=0 and consider a pair (k,p) as the point of A2. If we apply blow up procedure at the point (−1,0) then we get a new family of polynomials. This family depends on one parameter t∈P. If t=∞ then we get Euler supercharacters for Lie supergroup OSP(2,2n). The supercharacters of finite dimensional irreducible modules can be also obtained by a specialization of the parameter t. But in such a case the specialization depends on the highest weight of the corresponding irreducible module.

The multi-variable unified family of generalized Apostol-type polynomials

The aim of this paper is to investigate and give a new family of multi-variable Apostol-type polynomials. This family is related to Apostol-Euler, Apostol-Bernoulli, Apostol-Genocchi and Apostol-laguerre polynomials. Moreover, we derive some implicit summation formulae and general symmetry identities. The new family of polynomials introduced here, gives many interesting special cases.

Truncated-exponential-based Frobenius\u2013Euler polynomials

In this paper, we first introduce a new family of polynomials, which are called the truncated-exponential based Frobenius–Euler polynomials, based upon an exponential generating function. By making use of this exponential generating function, we obtain their several new properties and explicit summation formulas. Finally, we consider the truncated-exponential based Apostol-type Frobenius–Euler polynomials and their quasi-monomial properties.

Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

The aim of this paper is to construct and investigate some of the fundamental generalizations and unifications of new families of polynomials and numbers involving finite sums of higher powers of binomial coefficients and the Franel numbers by means of suitable generating functions and hypergeometric function. We derive several fundamental properties involving the generating functions, formulas, recurrence relations for these polynomials and numbers. These new families of polynomials and numbers are shown to generalize some known special polynomials such as the Legendre polynomials, the Michael Vowe polynomials, the Mirimanoff polynomials, Golombek type polynomials and also the Franel numbers. We give relations between the generalized Franel numbers, the Legendre polynomials, the Bernoulli numbers, the Stirling numbers, the Catalan numbers and other special numbers. Using the Riemann integral and p-adic integrals representations of these new polynomials, we derive combinatorial sums and identities related to sums of powers of binomial coefficients. Moreover, we introduce some revealing and historical remarks and observations on the finite sums of powers of binomial coefficients, special polynomials and numbers. Appropriate connections of identities, formulas, relations and results given in this paper with those in earlier and future studies are pointed out in detail. Special values of explicit formulas of our new numbers give solutions of the open problem 1 raised by Srivastava [58, p. 416, Open Problem 1]. Finally, we pose two open questions related to ordinary generating functions for these new numbers.

A Note on the Truncated-Exponential Based Apostol-Type Polynomials

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials.

Partial derivative formulas and identities involving $$\mathbf {2}$$2-variable Simsek polynomials

The 2-variable Simsek polynomials \(Y_n(x,y;\lambda , \delta )\) are introduced as the generalization of a new family of polynomials \(Y_n(x;\lambda )\). Certain partial derivatives formulas and identities for the 2-variable Simsek polynomials \(Y_n(x,y;\lambda , \delta )\) are established. A brief view of quasi-monomial approach establishing differential operators and equation is presented for these polynomials.

Asymptotic distributions of Wishart type products of random matrices

We study asymptotic distributions of large dimensional random matrices of the form $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using free probability and combinatorics of colored labeled noncrossing partitions. These matrices are taken from the set of off-diagonal blocks of the family $\mathcal{Y}$ of independent Hermitian random matrices which are asymptotically free, asymptotically free against the family of deterministic diagonal matrices, and whose norms are uniformly bounded almost surely. This class includes unitarily invariant Hermitian random matrices with limit distributions given by compactly supported probability measures $\nu$ on the real line. We express the limit moments in terms of colored labeled noncrossing pair partitions, to which we assign weights depending on even free cumulants of $\nu$ and on asymptotic dimensions of blocks (Gaussianization). For products of $p$ independent blocks, we show that the limit moments are linear combinations of a new family of polynomials called generalized multivariate Fuss-Narayana polynomials. In turn, the product of two blocks of the same matrix leads to an example with rescaled Raney numbers.

Higher order generalized geometric polynomials

According to the generalized Mellin derivative, we introduce a new family of polynomials called higher order generalized geometric polynomials and obtain some arithmetical properties of them. Then we investigate the relationship of these polynomials with degenerate Bernoulli, degenerate Euler, and Bernoulli polynomials. Finally, we evaluate several series and integrals in closed forms.

A new generalization of Apostol-type Laguerre–Genocchi polynomials

Many extensions and variants of the so-called Apostol-type polynomials have recently been investigated. Motivated mainly by those works and their usefulness, we aim to introduce a new class of Apostol-type Laguerre–Genocchi polynomials associated with the modified Milne–Thomson's polynomials introduced by Derre and Simsek and investigate its properties, including, for example, various implicit formulas and symmetric identities in a systematic manner. The new family of polynomials introduced here, being very general, contains, as its special cases, many known polynomials. So the properties and identities presented here reduce to yield those results of the corresponding known polynomials.

On encoding polynomials for strong algebraic manipulation detections codes

Algebraic manipulation detection codes were introduced in 2008 to protect data against a special type of a modification: an algebraic manipulation. One of the most effective ways of constructing such codes is based on polynomial encoding functions. In this paper a new family of polynomials is proposed, which may lead to higher detection capabilities and lower computational complexity of encoding and decoding procedures.