Genocchi Polynomials Research Articles

A note on (<i>p</i>, <i>q</i>)-analogue type of Fubini numbers and polynomials

In this paper, we introduce a new class of (p, q)-analogue type of Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we consider some relationships for (p, q)-Fubini polynomials associated with (p, q)-Bernoulli polynomials, (p, q)-Euler polynomials and (p, q)-Genocchi polynomials and (p, q)-Stirling numbers of the second kind.

Hermite-based hybrid polynomials and some related properties

In this paper, the two-variable one-parameter generalized Hermite-based hybrid polynomials are introduced by means of generating function, series definition and determinant definition. The recurrence relations, shift operators, differential, integro-differential and partial differential equations for these polynomials are established via factorization method. The two-variable one-parameter generalized Hermite-based Bernoulli, Euler and Genocchi polynomials are studied as the particular members and some examples are considered in terms of these polynomials to give the applications of main results. The graphical representation and interpretation is also shown for these polynomials.

Numerical Approach of Fractional Abel Differential Equation by Genocchi Polynomials

In the present paper, we study a computational technique for solving a fractional Abel differential equation. The fractional derivative is considered in the Caputo sense. Genocchi polynomials is applied in this technique. To this aim, the operational matrices of the fractional integration and product are applied. The primary motivation of this study is by aid of these matrices, the aforesaid equation is reduced a system of algebraic equations. To show the accuracy of the proposed method, some examples are provided.

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

In this paper, we present a novel discrete scheme based on Genocchi polynomials and fractional Laguerre functions to solve multiterm variable‐order time‐fractional partial differential equations (M‐V‐TFPDEs) in the large interval. In this purpose, the accurate modified operational matrices are constructed to reduce the problems into a system of algebraic equations. Also, the computational algorithm based on the method and modified operational matrices in the large interval is easily implemented. Furthermore, we discuss the error estimation of the proposed method. Ultimately, to confirm our theoretical analysis and accuracy of numerical approach, several examples are presented.

SOME COMPUTATIONS BETWEEN SUMS OF POWERS OF CONSECUTIVE INTEGERS AND ALTERNATING SUMS OF POWERS OF CONSECUTIVE INTEGERS

We study on the sums of powers of consecutive integers and alternating sums of power of consecutive integers. We derive many identities and correlations including Bernoulli, Euler and Genocchi polynomials and numbers.

A note on partially degenerate Legendre–Genocchi polynomials

A note on partially degenerate Legendre–Genocchi polynomials

Differential and integral equations for the Laguerre–Gould–Hopper-based Appell and related polynomials

In this article, recurrence relation, differential equations and integral equation for the Laguerre–Gould–Hopper-based Appell polynomials are derived using the factorization method. An analogous study of these results for the Laguerre–Gould–Hopper-based Bernoulli, Euler and Genocchi polynomials is also presented.

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.

Combinatorial Identities with Generalized Higher-order Genocchi Sequences

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established.Â

CAS wavelets analytic solution and Genocchi polynomials numerical solutions for the integral and integro-differential equations

In this paper, we proposed a new operational matrix of differentiation using CAS wavelets. By aid of these matrices, CAS wavelets method (CASWM) is developed for the Integro-differential equations. Properties of CAS wavelets are used to solve integral and integro-differential equations. Also, we obtain the numerical solutions for Integral equations using Genocchi polynomials with collocation method by reducing a system of algebraic equations. Illustrative numerical problems are considered to demonstrate the applicability of the proposed algorithms such as CASWM yields the exact solution but Genocchi polynomial collocation method (GPCM) yield approximation solution.

A Unified Family of Generalized $q$-Hermite Apostol Type Polynomials and its Applications

The intended objective of this paper is to introduce a new class of generalized $q$-Hermite based Apostol type polynomials by combining the $q$-Hermite polynomials and a unified family of $q$-Apostol-type polynomials. The generating function, series definition and several explicit representations for these polynomials are established. The $q$-Hermite-Apostol Bernoulli, $q$-Hermite-Apostol Euler and $q$-Hermite-Apostol Genocchi polynomials are studied as special members of this family and corresponding relations for these polynomials are obtained.

Finding non-linear differential equations and certain identities for the Bernoulli–Euler and Bernoulli–Genocchi numbers

In this research paper, the authors derived the non-linear differential equations for certain hybrid special polynomials related to the Bernoulli polynomials. The families of non-linear differential equations arising from the generating functions of the Bernoulli–Euler and Bernoulli–Genocchi polynomials are derived. Further, these non-linear differential equations are used to derive certain identities and formulas for the Bernoulli–Euler and Bernoulli–Genocchi numbers. However, to provided an exception, a linear differential equation is derived from the generating function of the Genocchi–Euler polynomials.

A family of complex Appell polynomial sets

In the present sequel to a recent work by Srivastava et al. (Rocky Mt J Math 49 (in press), 2019), the authors propose to show that the real and imaginary parts of a general set of complex Appell polynomials can be represented in terms of the Chebyshev polynomials of the first and second kind. Furthermore, by applying a general technique based upon the monomiality principle and quasi-monomial sets [see, for details, Ben Cheikh (Appl Math Comput 141:63–76, 2003), Dattoli (in: Cocolicchio, Dattoli, Srivastava (eds), Advanced special functions and applications (Proceedings of the Melfi School on advanced topics in mathematics and physics; Melfi, May 9–12), Aracne Editrice, Rome, 2000) and Steffensen (Acta Math 73:333–366, 1941)], the differential equations satisfied by the Bernoulli, Euler and Genocchi polynomials are derived.

Functional Expansions Based Multilayer Perceptron Neural Network for Classification Task

Artificial neural network has been proved among the best tools in data mining for classification tasks. Where, Multilayer Perceptron (MLP) is known as benchmarked technique for classification tasks due to common use and easy implementation. Meanwhile, it is fail to make high combination of inputs from lower feature space to higher feature space. In this paper, Shifted Genocchi polynomials and Chebyshev Wavelets functional expansions based Multilayer Perceptron techniques with Levenberg Marquardt back propagation learning are proposed to deal with high dimension problems in classification tasks. Five datasets from UCI repository and KEEL datasets were collected to evaluate the performance in terms of five evaluation measures. T-test was applied to check the significance of the proposed techniques. The comparison results show that the proposed models outperform in terms of accuracy, sensitivity, specificity, precision and f-measure.

On degenerate Apostol-type polynomials and applications

The main object of the current paper is to introduce and investigate a new unified class of the degenerate Apostol-type polynomials. These polynomials are studied by means of the generating function, series definition and are framed within the context of monomiality principle. Several important recurrence relations and explicit representations for the antecedent class of polynomials are derived. As the special cases, the degenerate Apostol–Bernoulli, Euler and Genocchi polynomials are obtained and corresponding results are also proved. A fascinating example is constructed in terms of truncated-exponential polynomials, which gives the applications of these polynomials to produce their hybridized forms.

Recurrence relations and determinant forms for general polynomial sequences. Application to Genocchi polynomials

ABSTRACTIn this paper the set of general polynomial sequences is considered. An elementary systematic approach is proposed. In fact a structure of group is given and for every element of this group recurrence relations and determinant forms are derived. Applications of the derived determinant forms are considered. In particular, the general linear interpolation and bounds of the zeros of each polynomial of the sequence are sketched. Finally, as an illustrative example, the shifted (with respect to the degree) Genocchi polynomial sequence is analyzed.

Applications of Fourier Series and Zeta Functions to Genocchi Polynomials

Applications of Fourier Series and Zeta Functions to Genocchi Polynomials

An Extended Generalized q-Extensions for the Apostol Type Polynomials

Through a modification on the parameters associated with generating function of the q-extensions for the Apostol type polynomials of order α and level m, we obtain some new results related to a unified presentation of the q-analog of the generalized Apostol type polynomials of order α and level m. In addition, we introduce some algebraic and differential properties for the q-analog of the generalized Apostol type polynomials of order α and level m and the relation of these with the q-Stirling numbers of the second kind, the generalized q-Bernoulli polynomials of level m, the generalized q-Apostol type Bernoulli polynomials, the generalized q-Apostol type Euler polynomials, the generalized q-Apostol type Genocchi polynomials of order α and level m, and the q-Bernstein polynomials.

On some classes of differential equations and associated integral equations for the Laguerre–Appell polynomials

Abstract The article aims to explore some new classes of differential equations and associated integral equations for some hybrid families of Laguerre polynomials. The recurrence relations and differential, integro-differential and partial differential equations for the hybrid Laguerre–Appell polynomials are derived via the factorization method. An analogous study of these results for the hybrid Laguerre–Bernoulli, Euler and Genocchi polynomials is presented. Further, the Volterra integral equations for the hybrid Laguerre–Appell polynomials and for their corresponding members are also explored.

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

In this paper, two-dimensional Genocchi polynomials and the Ritz–Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein–Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples.