Apostol-Genocchi Polynomials Research Articles

APPROXIMATION BY A COMPOSITION OF APOSTOL-GENOCCHI AND PALTANEA-DURRMEYER OPERATORS

The present paper deals with the Durrmeyer construction of operators based on a class of orthogonal polynomials called Apostol-Genocchi polynomials. For the proposed operators, we first establish a global approximation result followed by its convergence estimate in terms of usual, r-th and weighted modulus of continuity. We further study the asymptotic type results such as the Voronovskaya theorem and quantitative Voronovskaya theorem. Moreover, we estimate the rate of pointwise convergence of the proposed operators for functions of bounded variation defined on the interval (0,∞). Finally, the results are validated through graphical representations and an absolute error table.

Degenerate Apostol-Frobenius Type Poly-Genocchi Polynomials of Higher Order with Parameters a and b

This paper introduces another variation of poly-Genocchi polynomials by mixing the concept of modified degenerate polyexponential function, Apostol-Genocchi polynomials and Frobenius polynomials. These polynomials are called the degenerate Apostol-Frobenius-type poly-Genocchi polynomials with parameters a and b. Several identities and formulas are derived including recurrence relations, explicit formulas and certain differential identity. Moreover, some relations are established connecting these polynomials to degenerate Stirling numbers of the first and second kind, higher order degenerate Bernoulli polynomials, and higher order degenerate Frobenius-Euler polynomials.

Asymptotic approximations for Generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials in terms of Hyperbolic Functions

Asymptotic approximation formulas for polynomials of the type Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi with integer order and real parameters are obtained via hyperbolic functions. The derivation of the formulas is done using the principle of saddle point and expansion of appropriate hyperbolic function about a saddle point.

Integral modification of Beta-Apostol-Genocchi operators

<p style='text-indent:20px;'>We propose certain Durrmeyer-type operators for Apostol-Genocchi polynomials in this research. We explore these operators' approximation attributes and measure the rate of convergence. In addition, we present a direct approximation theorem based on first and second-order modulus of continuity, local approximation findings for Lipschitz class functions and a direct theorem based on the typical modulus of continuity. Finally, we showed a graph illustrating the convergence of the suggested operators and an error table.</p>

A new family of Apostol–Genocchi polynomials associated with their certain identities

ABSTRACT In this paper, we provide a generating function for mix type Apostol–Genocchi polynomials of order η associated with Bell polynomials. We also derive certain important identities of Apostol Genocchi polynomials of order η based on Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some correlation with Stirling numbers and their special instances. Moreover, we discover some symmetric identities and their related known results.

Some new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order $\alpha$ 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.

New Results for Degenerated Generalized Apostol–bernoulli, Apostol–euler and Apostol–genocchi Polynomials

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.

Some Generalized Properties of Poly-Daehee Numbers and Polynomials Based on Apostol–Genocchi Polynomials

Numerous polynomial variations and their extensions have been explored extensively and found applications in a variety of research fields. The purpose of this research is to establish a unified class of Apostol–Genocchi polynomials based on poly-Daehee polynomials and to explore some of their features and identities. We investigate these polynomials via generating functions and deduce various identities, summation formulae, differential and integral formulas, implicit summation formulae, and several characterized generating functions for new numbers and polynomials. Finally, by using an operational version of Apostol–Genocchi polynomials, we derive some results in terms of new special polynomials. Due to the generic nature of the findings described here, they are used to reduce and generate certain known or novel formulae and identities for relatively simple polynomials and numbers.

A Family of Generalized Legendre-Based Apostol-Type Polynomials

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

Approximation by Sz醩z Type Operators Involving Apostol-Genocchi Polynomials

The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of order α. Applying the Korovkin theorem, we arrive at the convergence of the operator with the aid of moments and central moments. We determine the rate of convergence of the operator using several tools such as -functional, modulus of continuity, second modulus of continuity. We also give a type of Voronovskaya theorem for estimating error. Moreover, we investigate some results about convergence properties of the operator in a weighted space. Finally, we give numerical examples to support our theorems by using the Maple.

Formulas for special numbers and polynomials derived from functional equations of their generating functions

The main purpose of this paper is to investigate various formulas, identities and relations involving Apostol type numbers and parametric type polynomials. By using generating functions and their functional equations, we give many relations among the certain family of combinatorial numbers, the Vieta polynomials, the two-parametric types of the Apostol-Euler polynomials, the Apostol-Bernoulli polynomials, the Apostol-Genocchi polynomials, the Fibonacci and Lucas numbers, the Chebyshev polynomials, and other special numbers and polynomials. Moreover, we give some formulas related to trigonometric functions, special numbers and special polynomials. Finally, some remarks and observations on the results of this paper are given.

Approximation by a new sequence of operators involving Apostol-Genocchi polynomials

Abstract The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.

Fourier expansion and integral representation generalized Apostol-type Frobenius\u2013Euler polynomials

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.

Generalized Apostol Type Polynomials Based on Twin-Basic Numbers

In this work, we consider a class of new generating function for \((p,q)\)-analog of Apostol type polynomials of order \(\alpha\) including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order \(\alpha\). By making use of their generating function, we derive some useful identities. We also introduce the generating functions of \((p,q)\)-analogues of the Stirling numbers of second kind of order \(\tau\) and the Bernstein polynomials by which we construct diverse correlations including aforementioned polynomials and the \((p,q)\)-gamma function.

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.

A parametric kind of the Fubini-type polynomials

By introducing two specific generating functions, we define a kind of parametric Fubini-type polynomials. We investigate some fundamental properties of these parametric-kind Fubini-type polynomials. In particular, we derive the relations between the parametric-kind Fubini-type polynomials and other polynomials including (for example) the parametric-kind Apostol–Bernoulli, the Apostol–Euler and the Apostol–Genocchi polynomials of order $$\alpha $$. Finally, we derive several families of multilinear and multilateral generating functions for the parametric-kind Fubini-type polynomials as well as other polynomials which are considered in this paper.

Some new generalizations and applications of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

By means of six specific generating functions, we introduce a type of generalized parametric Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials and systematically study their basic properties. As an application of the new polynomials, we use them in computing a new series of Taylor-type.

Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by \[ \Big ( \frac{2}{\lambda e^t+1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{E}^{(\alpha )}_{n}(x;\lambda ) \frac{t^n}{n!}\,, \qquad \lambda \in \mathbb{C}\setminus \lbrace -1\rbrace \,, \] and as an “exceptional family” \[ \Big ( \frac{t}{e^t-1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{B}^{(\alpha )}_{n}(x) \frac{t^n}{n!}\,, \] both of these for $\alpha \in \mathbb{C}$.

Higher-Order Convolutions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas.

Identities and relations involving the modified degenerate hermite-based Apostol–Bernoulli and Apostol–Euler polynomials

In recent years, many researchers (see, for example, Araci et al. in Springer Plus5(1), Article ID 860. https://doi.org/10.1186/s40064-016-2357-4 , 2016 to Zhang and Yang in Comput Math Appl 56:2993–2999, 2008) worked on the Apostol–Bernoulli type polynomials and numbers. They introduced and investigated some properties of these types of polynomials and numbers including several identities and symmetric relations for them. Carlitz (Script Math 25:323–330, 1961, Utilitas Math 15:51–88, 1979) introduced the degenerate Bernoulli numbers. Dolgy et al. (Adv Stud Contemp Math 26:203–209, 2016) and Kwon et al. (Filomat 26:1–9, 2016) introduced and investigated the modified degenerate Bernoulli polynomials and the modified degenerate Euler polynomials, respectively. They gave some relations for these polynomials. Ozarslan (Comput Math Appl 62:2452–2462, 2011) and Khan et al. (J Math Anal Appl 351:756–764, 2009) considered the Hermite-based unified Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Khan et al. (J Nonlinear Sci Appl 10:5072–5081, 2017) introduced the partially degenerate Hermite–Genocchi polynomials. In this article, we define the modified degenerate Hermite-based Apostol–Bernoulli, the modified degenerate Hermite-based Apostol–Euler and the modified Hermite-based Apostol–Genocchi polynomials. We prove two theorems and several symmetry relations for each of these families of polynomials. We also derive finite summation formulas for the modified degenerate unified Hermite-based Apostol type polynomials.