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Journal articles on the topic 'Generalized Multi poly-Euler polynomials'

Author: Grafiati

Published: 3 June 2025

Last updated: 6 July 2025

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1

Corcino, Roberto B., Hassan Jolany, Cristina B. Corcino, and Takao Komatsu. "On Generalized Multi Poly-Euler Polynomials." Fibonacci Quarterly 55, no. 1 (2017): 41–53. http://dx.doi.org/10.1080/00150517.2017.12427790.

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2

Kim, Taekyun, Dae San Kim, Jin-Woo Park, and Jongkyum Kwon. "A Note on Multi-Euler–Genocchi and Degenerate Multi-Euler–Genocchi Polynomials." Journal of Mathematics 2023 (August 26, 2023): 1–7. http://dx.doi.org/10.1155/2023/3810046.

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Recently, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials are introduced. The aim of this note is to study the multi-Euler–Genocchi and degenerate multi-Euler–Genocchi polynomials which are defined by means of the multiple logarithm and generalize, respectively, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials. Especially, we express the former by the generalized Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind, and the latter by the generalized degenerate Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind.

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3

El-Desouky, Beih, Rabab Gomaa, and Alia Magar. "The multi-variable unified family of generalized Apostol-type polynomials." Applicable Analysis and Discrete Mathematics, no. 00 (2020): 15. http://dx.doi.org/10.2298/aadm190405015e.

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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.

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4

Sweilam, N. H., S. M. AL-Mekhlafi, and D. Baleanu. "Efficient numerical treatments for a fractional optimal control nonlinear Tuberculosis model." International Journal of Biomathematics 11, no. 08 (2018): 1850115. http://dx.doi.org/10.1142/s1793524518501152.

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In this paper, the general nonlinear multi-strain Tuberculosis model is controlled using the merits of Jacobi spectral collocation method. In order to have a variety of accurate results to simulate the reality, a fractional order model of multi-strain Tuberculosis with its control is introduced, where the derivatives are adopted from Caputo’s definition. The shifted Jacobi polynomials are used to approximate the optimality system. Subsequently, Newton’s iterative method will be used to solve the resultant nonlinear algebraic equations. A comparative study of the values of the objective functional, between both the generalized Euler method and the proposed technique is presented. We can claim that the proposed technique reveals better results when compared to the generalized Euler method.

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5

Araci, Serkan, Mumtaz Riyasat, Shahid Wani, and Subuhi Khan. "A New Class of Hermite-Apostol Type Frobenius-Euler Polynomials and Its Applications." Symmetry 10, no. 11 (2018): 652. http://dx.doi.org/10.3390/sym10110652.

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The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations.

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6

Bilgic, Secil, and Veli Kurt. "On generalized q-poly-Bernoulli numbers and polynomials." Filomat 34, no. 2 (2020): 515–20. http://dx.doi.org/10.2298/fil2002515b.

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Many mathematicians in ([1],[2],[5],[14],[20]) introduced and investigated the generalized q-Bernoulli numbers and polynomials and the generalized q-Euler numbers and polynomials and the generalized q-Gennochi numbers and polynomials. Mahmudov ([15],[16]) considered and investigated the q-Bernoulli polynomials B(?)n,q(x,y) in x,y of order ? and the q-Euler polynomials E(?) n,q (x,y)in x,y of order ?. In this work, we define generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ?. We give new relations between the generalized q-poly-Bernoulli polynomials B[k,?] n,q (x,y) in x,y of order ? and the generalized q-poly-Euler polynomials ?[k,?] n,q (x,y) in x,y of order ? and the q-Stirling numbers of the second kind S2,q(n,k).

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7

Kurt, Veli. "On the generalized q-poly-Euler polynomials of the second kind." Filomat 34, no. 2 (2020): 475–82. http://dx.doi.org/10.2298/fil2002475k.

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In this work, we define the generalized q-poly-Euler numbers of the second kind of order ? and the generalized q-poly-Euler polynomials of the second kind of order ?. We investigate some basic properties for these polynomials and numbers. In addition, we obtain many identities, relations including the Roger-Sz?go polynomials, the Al-Salam Carlitz polynomials, q-analogue Stirling numbers of the second kind and two variable Bernoulli polynomials.

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8

Khan, Waseem Ahmad, Mehmet Acikgoz, and Ugur Duran. "Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials." Symmetry 12, no. 10 (2020): 1691. http://dx.doi.org/10.3390/sym12101691.

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Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

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9

Jolany, Hassan, Roberto B. Corcino, and Takao Komatsu. "More properties on multi-poly-Euler polynomials." Boletín de la Sociedad Matemática Mexicana 21, no. 2 (2015): 149–62. http://dx.doi.org/10.1007/s40590-015-0061-y.

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10

Corcino, Roberto, Cristina Corcino, and Waseem Khan. "On Type 2 Degenerate Poly-Frobenius-Euler Polynomials." Recoletos Multidisciplinary Research Journal 13, no. 1 (2025): 13–31. https://doi.org/10.32871/rmrj2513.01.02.

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Background: This paper introduces a class of special polynomials called Type 2 degenerate poly-Frobenius-Euler polynomials, defined using the polyexponential function. Motivated by the expanding theory of degenerate versions of classical polynomials, the paper seeks to enrich the mathematical landscape by constructing generalized structures with deeper combinatorial and analytic properties. Methods: The study employs the method of generating functions combined with Cauchy's rule for the product of two series to derive explicit formulas and identities, enabling systematic manipulation of series expansions. From an analytic perspective, the authors utilized the comparison test and principles of uniform convergence to establish that certain integral representations correspond to holomorphic functions. Results: The researchers successfully derived explicit formulas and identities for the Type 2 degenerate poly-Frobenius-Euler polynomials. They established meaningful connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, they introduced the Type 2 degenerate unipoly-poly-Frobenius-Euler polynomials, defined via the unipoly function, and thoroughly investigated their various properties, including behaviors under differentiation and integration. Conclusion: The study significantly advances the theory of degenerate polynomials by constructing new polynomial families, derivation of explicit identities, and establishing analytic properties. It opens new avenues for future research by bridging classical and generalized combinatorial sequences within a robust analytic framework.

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11

Corcino, Roberto Bagsarsa, and Cristina Corcino. "Generalized Laguerre-Apostol-Frobenius-Type Poly-Genocchi Polynomials of Higher Order with Parameters a, b and c." European Journal of Pure and Applied Mathematics 15, no. 4 (2022): 1549–65. http://dx.doi.org/10.29020/nybg.ejpam.v15i4.4505.

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    In this paper, the generalized Laguerre-Apostol-Frobenius-type poly-Genocchi polyno- mials of higher order with parameters a, b and c are defined using the concept of polylogarithm, Laguerre, Apostol and Frobenius polynomials. These polynomials possess numerous properties including recurrence relations, explicit formulas and certain differential identity. Moreover, some connections of these higher order generalized Laguerre-Apostol-Frobenius-type poly-Genocchi poly- nomials to Stirling numbers of the second kind and different variations of higher order Euler and Bernoulli-type polynomials are obtained.   

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12

Khan, N. U., and T. Usman. "A New Class of Laguerre-Based Poly-Euler and Multi Poly-Euler Polynomials." Journal of Analysis & Number Theory 4, no. 2 (2016): 113–20. http://dx.doi.org/10.18576/jant/040205.

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13

Corcino, Roberto Bagsarsa, Mark Laurente, and Mary Ann Ritzell Vega. "On Multi Poly-Genocchi Polynomials with Parameters a, b and c." European Journal of Pure and Applied Mathematics 13, no. 3 (2020): 444–58. http://dx.doi.org/10.29020/nybg.ejpam.v13i3.3676.

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Most identities of Genocchi numbers and polynomials are related to the well-knownBenoulli and Euler polynomials. In this paper, multi poly-Genocchi polynomials withparameters a, b and c are dened by means of multiple parameters polylogarithm. Several properties of these polynomials are established including some recurrence relations and explicit formulas.

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14

CAMPILLO, A., F. DELGADO, and S. M. GUSEIN-ZADE. "THE ALEXANDER POLYNOMIAL OF A PLANE CURVE SINGULARITY AND INTEGRALS WITH RESPECT TO THE EULER CHARACTERISTIC." International Journal of Mathematics 14, no. 01 (2003): 47–54. http://dx.doi.org/10.1142/s0129167x03001703.

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It was shown that the Alexander polynomial (of several variables) of a (reducible) plane curve singularity coincides with the (generalized) Poincaré polynomial of the multi-indexed filtration defined by the curve on the ring [Formula: see text] of germs of functions of two variables. The initial proof of the result was rather complicated (it used analytical, topological and combinatorial arguments). Here we give a new proof based on the notion of the integral with respect to the Euler characteristic over the projectivization of the space [Formula: see text] — the notion similar to (and inspired by) the notion of the motivic integration.

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15

Pan, Linyu, and Yuanpeng Zhu. "Grey Model Prediction Enhancement via Bernoulli Equation with Dynamic Polynomial Terms." Symmetry 17, no. 5 (2025): 713. https://doi.org/10.3390/sym17050713.

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The grey prediction model is designed to characterize systems comprising both partially known information (referred to as white) and partially unknown dynamics (referred to as black). However, traditional GM(1,1) models are based on linear differential equations, which limits their capacity to capture nonlinear and non-stationary behaviors. To address this issue, this paper develops a generalized grey differential prediction approach based on the Bernoulli equation framework. We incorporate the Bernoulli mechanism with a nonlinear exponent n and a dynamic polynomial-driven term. In this work, we propose a new model designated as BPGM(1,1). Another key innovation of this work is the adoption of a nonlinear least squares direct parameter identification strategy to calculate the exponent and polynomial parameters in the Bernoulli equation, which achieves a higher degree of freedom in parameter selection and effectively circumvents the model distortion issues caused by traditional background value estimation. Furthermore, the Euler discretization method is utilized for numerical solving, reducing the reliance on traditional analytical solutions for linear structures. Numerical experiments indicate that BPGM(1,1) surpasses GM(1,1), NFBM(1,1), and their improved versions. By leveraging the synergistic mechanism between Bernoulli-type nonlinear regulation and polynomial-driven external excitation, this framework significantly enhances prediction accuracy for systems characterized by non-stationary behaviors and multi-scale trends.

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16

Alekseev, Aleksander. "CONTROL OF A COMPLEX OBJECTS, STATES OF WHICH ARE DESCRIBING BY THE MATRIX RATING MECHANISM." Applied Mathematics and Control Sciences, no. 1 (March 27, 2020): 114–39. http://dx.doi.org/10.15593/2499-9873/2020.1.08.

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The control problem of a multi-criteria object is considered. Controlled object that has several criteria that are significant for a decision maker. Each criterion characterizes a control object in terms of a particular result of activity or an efficiency indicator. To evaluate the effectiveness of the functioning of the managed facility as a whole, the rating matrix mechanism is used, taking into account all the criteria in the complex. The optimal control problem is formulated as a search for the values of aggregated criteria that provide a given value of a complex indicator with minimal costs for providing values of particular criteria. The generalized cost function was reduced to an equation with one variable. The analytical equation of the level line of the indicator aggregated as a result of the convolution of two criteria is obtained. The line equation is found for an arbitrary binary convolution matrix, including the elements of which are given continuous values. It is shown that the objective function is reduced to a fourth-order polynomial, which can be analytically solved using the Ferrari or Descartes-Euler methods. It is shown that the task of searching for the values of two particular criteria describing the state of the control object for which the complex indicator calculated using the additive-multiplicative approach to complex assessment is equal to the given value and the costs for their provision are minimal, has a solution in general form for arbitrary nondecreasing convolution matrix of two criteria. Particular solutions to the control problem are found using costly functions, which are the inverse function of the Cobb-Douglas production function. It was shown that the cost function of the aggregate indicator has additional terms and is described by an algebraic equation with nonzero coefficients for variables and an additional constant. Based on what it was concluded that the cost functions, which are the inverse function e of the Cobb-Douglas production function, can be applied to control objects that have only two criteria. A similar formulation of the control problem for an arbitrary non-decreasing convolution matrix of two criteria is considered when using the additive-multiplicative approach to aggregation and when using cost functions described by a second-order algebraic equation in general form. As a result of the study, it is shown that the form of the cost function for the aggregated indicator is preserved. Thus, using cost functions in the form of second-order equations, the control problem has a solution in the general form for any number of criteria.

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17

Hassan, Jolany, Mohebbi Hossein, and Alikelaye R.Eizadi. "Some Results on Generalized Multi Poly-Bernoulli and Euler Polynomials." May 30, 2011. https://doi.org/10.5281/zenodo.821407.

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The Arakawa-Kaneko zeta function has been introduced ten years ago by T. Arakawa and M. Kaneko in [22]. In [22], Arakawa and Kaneko have expressed the special values of this function at negative integers with the help of generalized Bernoulli numbers B(k) called poly-Bernoulli numbers. Kim-Kim [4] introduced Multi poly- Bernoulli numbers and proved that special values of certain zeta functions at non-positive integers can be described in terms of these numbers.

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18

Aguilar-Porro, Cristina, Mario L. Ruz, and Francisco J. Blanco-Rodríguez. "A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments." Journal of Vibration and Control, June 13, 2023. http://dx.doi.org/10.1177/10775463231177335.

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Free vibration characteristics in uniform beams with several lumped attachments are an important problem in engineering applications that have to deal with mounting different equipment (e.g. motors, oscillators or engines) on a structural beam. In order to solve the lack of a generalized automatic procedure, this investigation presents a simple solving approach based on analytical means applied to a secular frequency equation for obtaining the natural frequencies of an arbitrarily supported single-span, or multi-span Euler–Bernoulli beam carrying any combination of miscellaneous attachments. The approach is obtained by solving a characteristic polynomial equation using a classical method for computing the roots of a polynomial. Interestingly, if the number of elements is greater than one, a pole-zero cancellation is needed, but it does not require manual interventions such as initial values and iteration. The mathematical approach is validated with bibliographic references and evaluated for accuracy and computational effectiveness. A good agreement is observed with relative error values practically negligible mostly ranging between 10−3 and 10−9 in the first five natural frequencies, which confirms the validity of the presented approach in this paper. The MatLab code that has been developed with the solving approach is freely available as a supplementary material to this paper. Additionally, a MatLab graphical user interface has also been developed in this work which allows to obtain the eigenvalues of a simply supported Euler–Bernoulli beam carrying an undetermined number of lumped elements. The graphical user interface is also available for download, along with help facilities to be run in a Windows operating system and detailed instructions to reproduce the case studies presented here. The proposed scheme (and also the MatLab graphical user interface) is very easy to code, and can be slightly modified to accommodate beams with arbitrary supports.

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19

Aguilar, Cristina, Mario L. Ruz, and Francisco J. Blanco-Rodríguez. "A modified polynomial-based approach to obtaining the eigenvalues of a uniform Euler–Bernoulli beam carrying any number of attachments." Journal of vibration and control 30, no. 11-12 (2023). https://doi.org/10.1177/10775463231177335.

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Free vibration characteristics in uniform beams with several lumped attachments are an important problem in engineering applications that have to deal with mounting different equipment (e.g. motors, oscillators or engines) on a structural beam. In order to solve the lack of a generalized automatic procedure, this investigation presents a simple solving approach based on analytical means applied to a secular frequency equation for obtaining the natural frequencies of an arbitrarily supported single-span, or multi-span Euler–Bernoulli beam carrying any combination of miscellaneous attachments. The approach is obtained by solving a characteristic polynomial equation using a classical method for computing the roots of a polynomial. Interestingly, if the number of elements is greater than one, a pole-zero cancellation is needed, but it does not require manual interventions such as initial values and iteration. The mathematical approach is validated with bibliographic references and evaluated for accuracy and computational effectiveness. A good agreement is observed with relative error values practically negligible mostly ranging between 10<sup>−3</sup> and 10<sup>−9</sup> in the first five natural frequencies, which confirms the validity of the presented approach in this paper. The MatLab code that has been developed with the solving approach is freely available as a supplementary material to this paper. Additionally, a MatLab graphical user interface has also been developed in this work which allows to obtain the eigenvalues of a simply supported Euler–Bernoulli beam carrying an undetermined number of lumped elements. The graphical user interface is also available for download, along with help facilities to be run in a Windows operating system and detailed instructions to reproduce the case studies presented here. The proposed scheme (and also the MatLab graphical user interface) is very easy to code, and can be slightly modified to accommodate beams with arbitrary supports.

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