william ramirez quiroga | Universidad de la Costa (original) (raw)
Drafts by william ramirez quiroga
Lobachevskii J Math, 2024
In this paper, we introduce degenerate versions of the hypergeometric Bernoulli and Euler polynom... more In this paper, we introduce degenerate versions of the hypergeometric Bernoulli and Euler polynomials. We demonstrate that they form Δ λ-Appell sets and provide some of their algebraic properties, including inversion formulas, as well as the associated matrix formulation. Additionally, we focus our attention on the monomiality principle associated with them and determine the corresponding derivative and multiplicative operators.
In this article, we introduce Bell polynomials of two variables within the framework of generatin... more In this article, we introduce Bell polynomials of two variables within the framework of generating functions and explore various properties associated with them. Specifically, we delve into explicit representations, summation formulae, recurrence relations, and addition formulas. Additionally, we present the matrix form and product formula for these polynomials. Finally, we introduce the two-variable Bell-based Stirling polynomials of the second kind and outline their corresponding results. This study contributes to a deeper understanding of the properties and applications of Bell polynomials in mathematical analysis.
This paper introduces new families of Fubini-Euler type and Apostol Fubini-Euler type polynomials... more This paper introduces new families of Fubini-Euler type and Apostol Fubini-Euler type polynomials, providing expressions, recurrence relations, and identities. We also derive Fourier series, and integral representations, and present their rational argument representation.
This study explores the evolution and application of integral transformations, initially rooted i... more This study explores the evolution and application of integral transformations, initially rooted in mathematical physics but now widely employed across diverse mathematical disciplines. Integral transformations offer a comprehensive framework comprising recurrence relations, generating expressions, operational formalism, and special functions, enabling the construction and analysis of specialized polynomials. Specifically, the research investigates a novel extended family of Frobenius-Genocchi polynomials of the Hermite-Apostol-type, incorporating multivariable variables defined through fractional operators. It introduces an operational rule for this generalized family, establishes a generating connection, and derives recurring relations. Moreover, the study highlights the practical applications of this generalized family, demonstrating its potential to provide solutions for specific scenarios.
The algebraic approach based on Pascal matrices is important in many fields of mathematics, rangi... more The algebraic approach based on Pascal matrices is important in many fields of mathematics, ranging from algebraic geometry to optimization, matrix theory and combinatorics. The core of the proposed approach is to introduce a new family of Pascal-type matrices i, j,c,a [x, y], x, y ∈ R − {0} with parameters c, a ∈ R + − {1}. By employing the effective matrix algebra tools, certain algebraic properties including the product formula, inverse matrix, determinant and eigen values are determined for the Pascal matrix i, j,c,a [x, y]. Further, some new families of matrices like the Fibonacci F
In this paper, we introduce the U-Bernoulli, U-Euler, and U-Genocchi polynomials, their numbers, ... more In this paper, we introduce the U-Bernoulli, U-Euler, and U-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized U-Bernoulli, U-Euler and U-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the U-Bernoulli, U-Euler, and U-Genocchi polynomial matrices, which involves the generalized Pascal matrix.
In this article, the generalized F-Frobenius-Euler polynomials H (α) n,F (x; µ) are introduced, t... more In this article, the generalized F-Frobenius-Euler polynomials H (α) n,F (x; µ) are introduced, through their generating function, and properties are established for these generalized polynomials. In addition, we define the generalized polynomial Fibo-Frobenius-Euler matrix H (α) n (x, F, µ). Factorizations of the Fibo-Frobenius-Euler polynomial matrix are established with the generalized Fibo-Pascal matrix and the Fibonacci matrix. The inverse of the Fibo-Frobenius-Euler matrix is also found.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apost... more The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order α and level m in the variable x. Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.
The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apost... more The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order α and level m in the variable x. Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.
In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomial... more In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials H [m−1,α] n (x; c, a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix U [m−1,α] (x; c, a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix U [m−1,α] (c, a; λ; u), we deduce a product formula for U [m−1,α] (x; c, a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix U [m−1,α] (x; c, a; λ; u), which involving the generalized Pascal matrix.
New biparametric families of Apostol-Frobenius-Euler polynomials of level m, Mat. Stud. 55 (2021)... more New biparametric families of Apostol-Frobenius-Euler polynomials of level m, Mat. Stud. 55 (2021), 10-23. 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 λ-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. 1. Introduction. Throughout this paper, we use the following standard notions: N = {1, 2,. . .}, N 0 = {0, 1, 2,. . .}, Z, R and C denotes the set of integers numbers, the set of real numbers and the set of complex numbers, respectively. Furthermore, (λ) 0 = 1 and (λ) k = λ(λ + 1)(λ + 2). .. (λ + k − 1), where k ∈ N, λ ∈ C. For the complex logarithm, we consider the principal branch, and w = z α we denote the single branch of the a multiple-valued function w = z α such that 1 α = 1. We take also 0 0 = 1 and 0 n = 0 if n ∈ N. The generating functions for the special polynomials are important from different view points and help in finding connection formulas, recursive relations, difference equations, and in solving problems in combinatorics and encoding their solutions. In particular, the Frobenius-Euler polynomials appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials (see [10, 13, 16-20, 23]). Also, these polynomials play an important role in the number of theories and classical analysis. In this paper, we focus our attention on introducing two new biparametric class of Apostol-Frobenius-Euler polynomials of level-m considering the works of [9,14]. Then, we can prove that such a new polynomial class preserves some similar algebraic and differential properties as the generalized Apostol-type polynomials, that as an immediate consequence, we recover many known algebraic and differential properties of such polynomials. For parameters λ, u ∈ C and a, b, c ∈ R + , with a ̸ = b, b > 1 and a ≥ 1; the Apostol type Frobenius-Euler polynomials H n (x; λ; u), n ≥ 0, and the generalized Apostol-type Frobenius-Euler polynomials H
2020
The main purpose of this paper is to investigate the Fourier series representation of the general... more The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius-Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius-Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz-Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius-Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius-Genocchi polynomials.
This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E. Ta... more This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of E. We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices.
This paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials B [... more This paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials B [m−1] n (x; q) of level m, as well as some others identities which connect this polynomial class with the q-generalized Bernoulli polynomials of level m, as well as the q-gamma function, and the q-Stirling numbers of the second kind and the q-Bernstein polynomials.
The aim of this paper is to introduce the generalized Apostol-type polynomial matrix W [m−1,α] (x... more The aim of this paper is to introduce the generalized Apostol-type polynomial matrix W [m−1,α] (x; c, a; λ; µ; ν) and the generalized Apos-tol-type matrix W [m−1,α] (c, a; λ; µ; ν). Using some properties of the generalized Apostol-type polynomials and numbers, we deduce a product formula for W [m−1,α] (x; c, a; λ; µ; ν) and provide some factorizations of the Apostol-type polynomial matrix W [m−1] (x; c, a; λ; µ; ν), involving the generalized Pascal matrix, Fibonacci and Lucas matrices, respectively.
Through a modification on the parameters associated with generating function of the í µí±-extens... more Through a modification on the parameters associated with generating function of the í µí±-extensions for the Apostol type polynomials of order í µí»¼ and level í µí±, we obtain some new results related to a unified presentation of the í µí±-analog of the generalized Apostol type polynomials of order í µí»¼ and level í µí±. In addition, we introduce some algebraic and differential properties for the í µí±-analog of the generalized Apostol type polynomials of order í µí»¼ and level í µí± and the relation of these with the í µí±-Stirling numbers of the second kind, the generalized í µí±-Bernoulli polynomials of level í µí±, the generalized í µí±-Apostol type Bernoulli polynomials, the generalized í µí±-Apostol type Euler polynomials, the generalized í µí±-Apostol type Genocchi polynomials of order í µí»¼ and level í µí±, and the í µí±-Bernstein polynomials.
An operational matrix method based on generalized Bernoulli polynomials of level m is introduced ... more An operational matrix method based on generalized Bernoulli polynomials of level m is introduced and analyzed in order to obtain numerical solutions of initial value problems. The novelty of our method comes essentially from the incorporation of the generalized Bernoulli polynomials of level m, which generalize the classical Bernoulli polynomials. Also, a comparison between the numerical solutions of initial value problems associated to dierent levels is done. Keywords. Bernoulli polynomials Generalized Bernoulli polynomials of level m Operational matrix Galerkin method.
Papers by william ramirez quiroga
CONSTRUCTIVE MATHEMATICAL ANALYSIS, 2024
The primary objective of this paper is to introduce and examine the new class of discrete orthogo... more The primary objective of this paper is to introduce and examine the new class of discrete orthogonal
polynomials called U–Bernoulli Korobov-type polynomials. Furthermore, we derive essential recurrence relations and
explicit representations for this polynomial class. Most of the results are proven through the utilization of generating
function methods. Lastly, we place particular emphasis on investigating the orthogonality relation associated with
these polynomials.
CarpathianMath. Publ., 2024
This article investigates the properties and monomiality principle within Bell-based Apostol- Ber... more This article investigates the properties and monomiality principle within Bell-based Apostol- Bernoulli-type polynomials. Beginning with the establishment of a generating function, the study proceeds to derive explicit expressions for these polynomials, providing insight into their structural characteristics. Summation formulae are then derived, facilitating efficient computation and manipulation. Implicit formulae are also examined, revealing underlying patterns and relationships.
Through the lens of themonomiality principle, connections between various polynomial aspects are elucidated, uncovering hidden symmetries and algebraic properties. Moreover, connection formulae are derived, enabling seamless transitions between different polynomial representations. This analysis contributes to a comprehensive understanding of Bell-based Apostol-Bernoulli-type polynomials,
offering valuable insights into their mathematical nature and applications.
Lobachevskii J Math, 2024
In this paper, we introduce degenerate versions of the hypergeometric Bernoulli and Euler polynom... more In this paper, we introduce degenerate versions of the hypergeometric Bernoulli and Euler polynomials. We demonstrate that they form Δ λ-Appell sets and provide some of their algebraic properties, including inversion formulas, as well as the associated matrix formulation. Additionally, we focus our attention on the monomiality principle associated with them and determine the corresponding derivative and multiplicative operators.
In this article, we introduce Bell polynomials of two variables within the framework of generatin... more In this article, we introduce Bell polynomials of two variables within the framework of generating functions and explore various properties associated with them. Specifically, we delve into explicit representations, summation formulae, recurrence relations, and addition formulas. Additionally, we present the matrix form and product formula for these polynomials. Finally, we introduce the two-variable Bell-based Stirling polynomials of the second kind and outline their corresponding results. This study contributes to a deeper understanding of the properties and applications of Bell polynomials in mathematical analysis.
This paper introduces new families of Fubini-Euler type and Apostol Fubini-Euler type polynomials... more This paper introduces new families of Fubini-Euler type and Apostol Fubini-Euler type polynomials, providing expressions, recurrence relations, and identities. We also derive Fourier series, and integral representations, and present their rational argument representation.
This study explores the evolution and application of integral transformations, initially rooted i... more This study explores the evolution and application of integral transformations, initially rooted in mathematical physics but now widely employed across diverse mathematical disciplines. Integral transformations offer a comprehensive framework comprising recurrence relations, generating expressions, operational formalism, and special functions, enabling the construction and analysis of specialized polynomials. Specifically, the research investigates a novel extended family of Frobenius-Genocchi polynomials of the Hermite-Apostol-type, incorporating multivariable variables defined through fractional operators. It introduces an operational rule for this generalized family, establishes a generating connection, and derives recurring relations. Moreover, the study highlights the practical applications of this generalized family, demonstrating its potential to provide solutions for specific scenarios.
The algebraic approach based on Pascal matrices is important in many fields of mathematics, rangi... more The algebraic approach based on Pascal matrices is important in many fields of mathematics, ranging from algebraic geometry to optimization, matrix theory and combinatorics. The core of the proposed approach is to introduce a new family of Pascal-type matrices i, j,c,a [x, y], x, y ∈ R − {0} with parameters c, a ∈ R + − {1}. By employing the effective matrix algebra tools, certain algebraic properties including the product formula, inverse matrix, determinant and eigen values are determined for the Pascal matrix i, j,c,a [x, y]. Further, some new families of matrices like the Fibonacci F
In this paper, we introduce the U-Bernoulli, U-Euler, and U-Genocchi polynomials, their numbers, ... more In this paper, we introduce the U-Bernoulli, U-Euler, and U-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized U-Bernoulli, U-Euler and U-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the U-Bernoulli, U-Euler, and U-Genocchi polynomial matrices, which involves the generalized Pascal matrix.
In this article, the generalized F-Frobenius-Euler polynomials H (α) n,F (x; µ) are introduced, t... more In this article, the generalized F-Frobenius-Euler polynomials H (α) n,F (x; µ) are introduced, through their generating function, and properties are established for these generalized polynomials. In addition, we define the generalized polynomial Fibo-Frobenius-Euler matrix H (α) n (x, F, µ). Factorizations of the Fibo-Frobenius-Euler polynomial matrix are established with the generalized Fibo-Pascal matrix and the Fibonacci matrix. The inverse of the Fibo-Frobenius-Euler matrix is also found.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apost... more The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order α and level m in the variable x. Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.
The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apost... more The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order α and level m in the variable x. Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.
In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomial... more In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials H [m−1,α] n (x; c, a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix U [m−1,α] (x; c, a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix U [m−1,α] (c, a; λ; u), we deduce a product formula for U [m−1,α] (x; c, a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix U [m−1,α] (x; c, a; λ; u), which involving the generalized Pascal matrix.
New biparametric families of Apostol-Frobenius-Euler polynomials of level m, Mat. Stud. 55 (2021)... more New biparametric families of Apostol-Frobenius-Euler polynomials of level m, Mat. Stud. 55 (2021), 10-23. 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 λ-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. 1. Introduction. Throughout this paper, we use the following standard notions: N = {1, 2,. . .}, N 0 = {0, 1, 2,. . .}, Z, R and C denotes the set of integers numbers, the set of real numbers and the set of complex numbers, respectively. Furthermore, (λ) 0 = 1 and (λ) k = λ(λ + 1)(λ + 2). .. (λ + k − 1), where k ∈ N, λ ∈ C. For the complex logarithm, we consider the principal branch, and w = z α we denote the single branch of the a multiple-valued function w = z α such that 1 α = 1. We take also 0 0 = 1 and 0 n = 0 if n ∈ N. The generating functions for the special polynomials are important from different view points and help in finding connection formulas, recursive relations, difference equations, and in solving problems in combinatorics and encoding their solutions. In particular, the Frobenius-Euler polynomials appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials (see [10, 13, 16-20, 23]). Also, these polynomials play an important role in the number of theories and classical analysis. In this paper, we focus our attention on introducing two new biparametric class of Apostol-Frobenius-Euler polynomials of level-m considering the works of [9,14]. Then, we can prove that such a new polynomial class preserves some similar algebraic and differential properties as the generalized Apostol-type polynomials, that as an immediate consequence, we recover many known algebraic and differential properties of such polynomials. For parameters λ, u ∈ C and a, b, c ∈ R + , with a ̸ = b, b > 1 and a ≥ 1; the Apostol type Frobenius-Euler polynomials H n (x; λ; u), n ≥ 0, and the generalized Apostol-type Frobenius-Euler polynomials H
2020
The main purpose of this paper is to investigate the Fourier series representation of the general... more The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius-Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius-Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz-Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius-Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius-Genocchi polynomials.
This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E. Ta... more This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of E. We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices.
This paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials B [... more This paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials B [m−1] n (x; q) of level m, as well as some others identities which connect this polynomial class with the q-generalized Bernoulli polynomials of level m, as well as the q-gamma function, and the q-Stirling numbers of the second kind and the q-Bernstein polynomials.
The aim of this paper is to introduce the generalized Apostol-type polynomial matrix W [m−1,α] (x... more The aim of this paper is to introduce the generalized Apostol-type polynomial matrix W [m−1,α] (x; c, a; λ; µ; ν) and the generalized Apos-tol-type matrix W [m−1,α] (c, a; λ; µ; ν). Using some properties of the generalized Apostol-type polynomials and numbers, we deduce a product formula for W [m−1,α] (x; c, a; λ; µ; ν) and provide some factorizations of the Apostol-type polynomial matrix W [m−1] (x; c, a; λ; µ; ν), involving the generalized Pascal matrix, Fibonacci and Lucas matrices, respectively.
Through a modification on the parameters associated with generating function of the í µí±-extens... more Through a modification on the parameters associated with generating function of the í µí±-extensions for the Apostol type polynomials of order í µí»¼ and level í µí±, we obtain some new results related to a unified presentation of the í µí±-analog of the generalized Apostol type polynomials of order í µí»¼ and level í µí±. In addition, we introduce some algebraic and differential properties for the í µí±-analog of the generalized Apostol type polynomials of order í µí»¼ and level í µí± and the relation of these with the í µí±-Stirling numbers of the second kind, the generalized í µí±-Bernoulli polynomials of level í µí±, the generalized í µí±-Apostol type Bernoulli polynomials, the generalized í µí±-Apostol type Euler polynomials, the generalized í µí±-Apostol type Genocchi polynomials of order í µí»¼ and level í µí±, and the í µí±-Bernstein polynomials.
An operational matrix method based on generalized Bernoulli polynomials of level m is introduced ... more An operational matrix method based on generalized Bernoulli polynomials of level m is introduced and analyzed in order to obtain numerical solutions of initial value problems. The novelty of our method comes essentially from the incorporation of the generalized Bernoulli polynomials of level m, which generalize the classical Bernoulli polynomials. Also, a comparison between the numerical solutions of initial value problems associated to dierent levels is done. Keywords. Bernoulli polynomials Generalized Bernoulli polynomials of level m Operational matrix Galerkin method.
CONSTRUCTIVE MATHEMATICAL ANALYSIS, 2024
The primary objective of this paper is to introduce and examine the new class of discrete orthogo... more The primary objective of this paper is to introduce and examine the new class of discrete orthogonal
polynomials called U–Bernoulli Korobov-type polynomials. Furthermore, we derive essential recurrence relations and
explicit representations for this polynomial class. Most of the results are proven through the utilization of generating
function methods. Lastly, we place particular emphasis on investigating the orthogonality relation associated with
these polynomials.
CarpathianMath. Publ., 2024
This article investigates the properties and monomiality principle within Bell-based Apostol- Ber... more This article investigates the properties and monomiality principle within Bell-based Apostol- Bernoulli-type polynomials. Beginning with the establishment of a generating function, the study proceeds to derive explicit expressions for these polynomials, providing insight into their structural characteristics. Summation formulae are then derived, facilitating efficient computation and manipulation. Implicit formulae are also examined, revealing underlying patterns and relationships.
Through the lens of themonomiality principle, connections between various polynomial aspects are elucidated, uncovering hidden symmetries and algebraic properties. Moreover, connection formulae are derived, enabling seamless transitions between different polynomial representations. This analysis contributes to a comprehensive understanding of Bell-based Apostol-Bernoulli-type polynomials,
offering valuable insights into their mathematical nature and applications.
Communications in Applied and Industrial Mathematics Communications in Applied and Industrial Mathematics , 2024
In this article, we introduce a new class of polynomials, known as Apostol Hermite Bernoulli-type... more 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.
Communications in Applied and Industrial Mathematics, 2024
This article explores some properties of degenerate hypergeometric Bernoulli polynomials, which a... more This article explores some properties of degenerate hypergeometric Bernoulli polynomials, which are defined through the following generating function t m e x λ (t) e λ (t) − m−1 l=0 (1) l,λ t l l!
REPORTS ON MATHEMATICAL PHYSICS, 2024
In the realm of specialized functions, the allure of 𝑞-calculus beckons to many scholars, captiva... more In the realm of specialized functions, the allure of 𝑞-calculus beckons to many scholars, captivating them with its prowess in shaping models of quantum computing, noncommutative probability, combinatorics, functional analysis, mathematical physics, approximation theory, and
beyond. This study explores a new idea called the multidimensional 𝑞-Hermite polynomials, using different 𝑞-calculus techniques. Numerous properties and novel findings regarding these polynomials are derived, encompassing their generating function, series representations, recurrence relations, 𝑞-differential formula, and operational principles. Further, we proved that these polynomials are quasi-monomial in 𝑞-aspect. As the applications, these findings are subsequently employed to address connection between the multidimensional 𝑞-Hermite polynomials and the
two-variable 𝑞-Legendre polynomials for the first time. Various characterizations are examined, as well the graphical representations of the two-variable 𝑞-Legendre polynomials are provided by the surface plots and graphs of distribution of zeros for the 𝑞-Legendre polynomials with
some specific set of parameters are shown using Mathematica. Our investigations shed light on the intricate nature of these polynomials, elucidating their behaviour and facilitating deeper
understanding within the realm of 𝑞-calculus.
AIMS Mathematics, 2024
In this research, we leverage various q-calculus identities to introduce the notion of q-Hermite-... more In this research, we leverage various q-calculus identities to introduce the notion of q-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these q-Hermite-Appell polynomials, encompassing their generating function, series representation, summation equations, recurrence relations, q-differential formula, and operational principles. Our investigation sheds light on the intricate nature of these polynomials, elucidating their behavior and facilitating deeper understanding within the realm of q-calculus.
The main objective of this work is to investigate a novel class of polynomials, called the ∆ h-Sh... more The main objective of this work is to investigate a novel class of polynomials, called the ∆ h-Sheffer polynomials and to explore their various properties. The generating function, explicit representations, quasi-monomiality, and certain novel identities involving ∆ h-Sheffer polynomials are obtained. Also, the ∆ h-Sheffer polynomials are explored via determinant representation. Further, the ∆ h Gould-Hopper-Sheffer polynomials are introduced with the help of ∆ h-Sheffer and ∆ h Gould-Hopper polynomials. Certain fascinating results, such as the generating function, determinant form, multiplicative, and derivative operators and many more results for these hybrid form of the ∆ h-Sheffer polynomials are also obtained. Certain examples are considered as the special cases of ∆ h Gould-Hopper-Sheffer polynomials.
We know that the matrices provide a flexible framework to study combinatorial structures. In fact... more We know that the matrices provide a flexible framework to study combinatorial structures. In fact, the generalized Fibonacci matrices allow us to develop the applications to coding theory. In the beginning of this work, a new family of generalized Bernoulli-Fibonacci polynomials of order m is introduced followed by investigating various properties associated with this polynomial class, as well as its relationships with other polynomial families and numbers. These include explicit relations, difference equations, summation formulae, linear and differential recurrence relations. Furthermore, we focus on matrix approach associated with this family by providing the generalized Fibo-Bernoulli polynomials matrix, Fibo-Pascal polynomial matrix and other important matrices. Some product and inverse formulae for the generalized Fibo-Bernoulli polynomials matrix involving other matrices are also derived at the end.
AIMS mathematics, 2024
This study explores the evolution and application of integral transformations, initially rooted i... more This study explores the evolution and application of integral transformations, initially rooted in mathematical physics but now widely employed across diverse mathematical disciplines. Integral transformations offer a comprehensive framework comprising recurrence relations, generating expressions, operational formalism, and special functions, enabling the construction and analysis of specialized polynomials. Specifically, the research investigates a novel extended family of Frobenius-Genocchi polynomials of the Hermite-Apostol-type, incorporating multivariable variables defined through fractional operators. It introduces an operational rule for this generalized family, establishes a generating connection, and derives recurring relations. Moreover, the study highlights the practical applications of this generalized family, demonstrating its potential to provide solutions for specific scenarios.
In this paper, we present a new family of generalized Bernoulli-type polynomials, as well as its ... more In this paper, we present a new family of generalized Bernoulli-type polynomials, as well as its numbers. In addition, we obtain some results such as algebraic and differential properties for this new family of Bernoulli-type polynomials. Likewise, the generalized Bernoulli-type polynomials matrix R (α) (x) is introduced. We deduce some product formulae for R (α) (x) and also, the inverse of the Bernoullitype matrix R is determined. Furthermore, we establish some explicit expressions for the Bernoullitype polynomial matrix R(x), which involve the generalized Pascal matrix and finally we study the summation formula of Euler-Maclaurin type and the Riemann zeta function applied to these Bernoullitype polynomials.
Originally developed within the realm of mathematical physics, integral transformations have tran... more Originally developed within the realm of mathematical physics, integral transformations have transcended their origins and now find wide application across various mathematical domains. Among these applications, the construction and analysis of special polynomials benefit significantly from the elucidation of generating expressions, operational principles, and other distinctive properties. This study delves into a pioneering exploration of an extended lineage of Frobenius-Euler polynomials belonging to the Hermite-Apostol type, incorporating multivariable variables through fractional operators. Motivated by the exigencies of contemporary engineering challenges, the research endeavors to uncover the operational rules and establishing connections inherent within these extended polynomials. In doing so, it seeks to chart a course towards harnessing these mathematical constructs within diverse engineering contexts, where their unique attributes hold the potential for yielding profound insights. The study deduces operational rules for this generalized family, facilitating the establishment of generating connections and the identification of recurrence relations. Furthermore, it showcases compelling applications, demonstrating how these derived polynomials may offer meaningful solutions within specific engineering scenarios.
Journal of Mathematics and Computer Science, 2024
The article introduces a novel class of polynomials, H Q [∆ h ] m (q 1 , q 2 , q 3 , q 4 , q 5 ; ... more The article introduces a novel class of polynomials, H Q [∆ h ] m (q 1 , q 2 , q 3 , q 4 , q 5 ; h), termed ∆ h Hermite-based Appell polynomials, utilizing the monomiality principle. These polynomials exhibit close connections with ∆ h Hermite-based Bernoulli, Euler, and Genocchi polynomials, elucidating their specific properties and explicit forms. Moreover, the research establishes generating relations for these polynomials, facilitating profound insights applicable across diverse domains such as mathematics, physics, and engineering sciences.
Dolomites Research Notes on Approximation, 2023
The main objective of this paper is to introduce and explore two novel classes of degenerate bipa... more The main objective of this paper is to introduce and explore two novel classes of degenerate biparametric Apostol-type polynomials, which are based on a definition of degenerate Apostol-type polynomials provided by Subuhi Khan et al. We derive various algebraic and differential properties associated with these polynomials. Additionally, we provide a series of illustrative examples for these newly introduced polynomial families along with their corresponding graphs. The majority of the results are proven utilizing well-established generating functions and identities.
This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–... more This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function methods for Apostol–Euler and Apostol–Genocchi Hermite polynomials of level m.
Carpathian Mathematical Publications
The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apost... more The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order alpha\alphaalpha and level mmm in the variable xxx. Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.
WSEAS TRANSACTIONS ON MATHEMATICS
The main objective of this work is to deduce some interesting algebraic relationships that connec... more The main objective of this work is to deduce some interesting algebraic relationships that connect the degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol– Genocchi polynomials and other families of polynomials such as the generalized Bernoulli polynomials of level m and the Genocchi polynomials. Futher, find new recurrence formulas for these three families of polynomials to study.
Axioms
In this paper, we introduce a class of new classes of degenerate unified polynomials and we show ... more In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some examples.
A new class of degenerate Apostol-type Hermite polynomials and applications, 2022
In this article, a new class of the degenerate Apostol-type Hermite polynomials is introduced. Ce... more In this article, a new class of the degenerate Apostol-type Hermite polynomials is introduced. Certain algebraic and differential properties of there polynomials are derived. Most of the results are proved by using generating function methods.
WSEAS transactions on mathematics, Aug 5, 2022
We introduce two biparametric families of Apostol-Frobenius-Euler polynomials of level m. We give... more 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 λ-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.