Search Results (20)

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related to Riordan arrays, Sheffer matrices, and their q analogs. Full article

This paper explores the operational principles and monomiality principles that significantly shape the development of various special polynomial families. We argue that applying the monomiality principle yields novel results while remaining consistent with established findings. The primary focus of this study is the introduction of degenerate multidimensional Hermite-based Appell polynomials (DMHAP), denoted as ${}_{\mathbb{H}}\mathbb{A}_n^{\left[r\right]}(l_1,l_2,l_3,\dots ,l_r;\vartheta )$. These DMHAP forms essential families of orthogonal polynomials, demonstrating strong connections with classical Hermite and Appell polynomials. Additionally, we derive symmetric identities and examine the fundamental properties of these polynomials. Finally, we establish an operational framework to investigate and develop these polynomials further. Full article

This research aims to introduce and examine a new type of polynomial called the ${\Delta }_h$ Legendre–Appell polynomials. We use the monomiality principle and operational rules to define the ${\Delta }_h$ Legendre–Appell polynomials and explore their properties. We derive the generating function and recurrence relations for these polynomials and their explicit formulas, recurrence relations, and summation formulas. We also verify the monomiality principle for these polynomials and express them in determinant form. Additionally, we establish similar results for the ${\Delta }_h$ Legendre–Bernoulli, Euler, and Genocchi polynomials. Full article

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in ${\mathbb{C}}^s$ for some positive integer $s>1.$ We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials. Full article

Microflora play an important role in the fermentation of blueberry wine, influencing the flavor and nutrient formation. Commercial yeasts give blueberry wines an average flavor profile that does not highlight the specific aroma and origin of the blueberry. In the present study, ITS1-ITS2 region sequencing analysis was performed using Illumina MiSeq high-throughput technology to sequence fermented blueberry wine samples of three Vaccinium ashei varieties, Gardenblue, Powderblue, and Britewell, from the Majiang appellation in Guizhou province to analyze the trends of fungal communities and the diversity of compositional structures in different periods of blueberry wine fermentation. The study’s results revealed that 114 genera from seven phyla were detected in nine samples from different fermentation periods of blueberry wine. The main fungal phyla were Ascomycota, Basidiomycota, Kickxellomycota, Chytridiomycota, and Olpidiomycota. The main fungal genera were Hanseniaspora, Saccharomyces, unidentified, Aureobasidium, Penicillium, Mortierella, Colletotrichum, etc. Hanseniaspora was dominant in the pre-fermentation stage of blueberry wine, accounting for more than 82%; Saccharomyces was the dominant genera in the middle and late fermentation stages of blueberry wine, with Saccharomyces accounting for more than 72% in the middle of fermentation and 93% in the late fermentation stage. This study screened indigenous flora for the natural fermentation of blueberry wine in the Majiang production area of Guizhou, improved the flavor substances of the blueberry wine, highlighted the characteristics of the production area, and made the blueberry wine have the characteristic flavor of the production area. Full article

The study presented in this paper follows the line of research created by the fact that by employing the monomiality principle, new outcomes are produced. This article deals with the inducement of ${\Delta }_h$ tangent-based Appell polynomials and derivation of certain of its characterizations such as explicit form, determinant form, monomiality principle, etc. These polynomials are designed to exhibit certain symmetries themselves or to capture and describe symmetrical patterns in mathematical structures. Further, certain members of ${\Delta }_h$ Appell polynomials such as ${\Delta }_h$ Bernoulli, Euler, and Genocchi polynomials are taken, and their corresponding results are obtained. Full article

This study follows the line of research that by employing the monomiality principle, new outcomes are produced. Thus, in line with prior facts, our aim is to introduce the ${\mathrm{\Delta }}_h$ multi-variate Hermite Appell polynomials ${}_{{\mathrm{\Delta }}_h}{}_{\mathfrak{H}}{\mathfrak{A}}_m^{\left[r\right]}(q_1,q_2,\cdots ,q_r;h)$. Further, we obtain their recurrence sort of relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families and thus operational principle satisfied by these polynomials is derived and will prove beneficial for future observations. Further, a few members of the ${\mathrm{\Delta }}_h$ Appell polynomial family are considered and their corresponding results are derived accordingly. Full article

The development of certain aspects of special polynomials in line with the monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, and their differential equation and series representation is obtained, which are important in several branches of mathematics and physics. Thus, in line with prior facts, our aim is to introduce the ${\Delta }_h$ hybrid special polynomials associated with Hermite polynomials denoted by ${}_{{\Delta }_{h}H}{Q}_m(u,v,w;h)$. Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials. Full article

In this paper, we introduce a generalization of the Kantrovich–Stancu-type Szasz operator asymmetry with hybrid families of special polynomials. Additionally, we construct certain positive linear operators together with the Sheffer–Appell polynomial sequences and then obtain the properties of convergence and the order of convergence, which is symmetric to these operators. For applications, we consider certain explicit examples including mixed-type special polynomials. Full article

In this paper, as a generalization of Szasz operators, a brand-new sequence of operators including the Appell polynomials of class $A^{\left(2\right)}$ is introduced. First, the convergence of this new sequence of operators is obtained, and then, some approximation results are presented by using the tools of approximation theory. In addition, an explicit example for this kind of sequence of operators containing Gould–Hopper polynomials is introduced. The error of the approximation of this new sequence of operators to a function is established. Full article

An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and known examples of bivariate Appell polynomial sequences are given. Full article

In this paper, binomial convolution in the frame of quantum calculus is studied for the set ${\mathcal{A}}_q$ of q-Appell sequences. It has been shown that the set ${\mathcal{A}}_q$ of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure ${\mathcal{A}}_q$ have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences. Full article

A remarkably large of number of polynomials have been presented and studied. Among several important polynomials, Legendre polynomials, Gould-Hopper polynomials, and Sheffer polynomials have been intensively investigated. In this paper, we aim to incorporate the above-referred three polynomials to introduce the Legendre-Gould Hopper-based Sheffer polynomials by modifying the classical generating function of the Sheffer polynomials. In addition, we investigate diverse properties and formulas for these newly introduced polynomials. Full article

Degenerate versions of polynomial sequences have been recently studied to obtain useful properties such as symmetric identities by introducing degenerate exponential-type generating functions. As part of our continued work in degenerate versions of generating functions, we subsequently present our study on degenerate complex Appell polynomials by considering a partially degenerate version of the generating functions of ordinary complex Appell polynomials in this paper. We only consider partially degenerate generating functions to retain the crucial properties of the Appell sequence, and we present useful identities and general properties by splitting complex values into their real and imaginary parts; moreover, we provide several explicit examples. Additionally, the differential equations satisfied by degenerate complex Bernoulli and Euler polynomials are derived by the quasi-monomiality principle using Appell-type polynomials. Full article