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

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Published: 3 June 2025

Last updated: 31 July 2025

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1

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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2

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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3

Komatsu, Takao, and Florian Luca. "Generalized incomplete poly-Bernoulli polynomials and generalized incomplete poly-Cauchy polynomials." International Journal of Number Theory 13, no. 02 (2017): 371–91. http://dx.doi.org/10.1142/s1793042117500221.

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By using the restricted and associated Stirling numbers of the first kind, we define the generalized restricted and associated poly-Cauchy polynomials. By using the restricted and associated Stirling numbers of the second kind, we define the generalized restricted and associated poly-Bernoulli polynomials. These polynomials are generalizations of original poly-Cauchy polynomials and original poly-Bernoulli polynomials, respectively. We also study their characteristic and combinatorial properties.

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4

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

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5

Komatsu, Takao, and Genki Shibukawa. "Poly-Cauchy polynomials and generalized Bernoulli polynomials." Acta Scientiarum Mathematicarum 80, no. 34 (2014): 373–88. http://dx.doi.org/10.14232/actasm-013-761-9.

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6

Kim, Taekyun, and Dae Kim. "A note on degenerate multi-poly-Bernoulli numbers and polynomials." Applicable Analysis and Discrete Mathematics, no. 00 (2022): 5. http://dx.doi.org/10.2298/aadm200510005k.

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In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of the multiple polylogarithms and degenerate versions of the multi-poly-Bernoulli numbers and polynomials. We investigate some properties for those numbers and polynomials. In addition, we give some identities and relations for the degenerate multi-poly- Bernoulli numbers and polynomials.

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7

Muhiuddin, G., W. A. Khan, U. Duran, and D. Al-Kadi. "Some Identities of the Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable." Journal of Function Spaces 2021 (June 1, 2021): 1–8. http://dx.doi.org/10.1155/2021/7172054.

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In this paper, we introduce degenerate multi-poly-Bernoulli polynomials and derive some identities of these polynomials. We give some relationship between degenerate multi-poly-Bernoulli polynomials degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate multi-poly-Bernoulli polynomials of complex variables, and then, we derive several properties and relations.

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8

Kargin, Levent, Mehmet Cenkci, Ayhan Dil, and Mumun Can. "Generalized harmonic numbers via poly-Bernoulli polynomials." Publicationes Mathematicae Debrecen 100, no. 3-4 (2022): 365–86. http://dx.doi.org/10.5486/pmd.2022.9074.

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9

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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10

Bayad, Abdelmejid, and Yoshinori Hamahata. "Multiple polylogarithms and multi-poly-Bernoulli polynomials." Functiones et Approximatio Commentarii Mathematici 46, no. 1 (2012): 45–61. http://dx.doi.org/10.7169/facm/2012.46.1.4.

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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 Ber

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12

Bayad, Abdelmejid, and Yoshinori Hamahata. "Arakawa–Kaneko L-functions and generalized poly-Bernoulli polynomials." Journal of Number Theory 131, no. 6 (2011): 1020–36. http://dx.doi.org/10.1016/j.jnt.2010.11.005.

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13

Khan, Waseem Ahmad, Divesh Srivastava, and Kottakkaran Sooppy Nisar. "A new class of generalized polynomials associated with Milne-Thomson-based poly-Bernoulli polynomials." Miskolc Mathematical Notes 25, no. 2 (2024): 793. https://doi.org/10.18514/mmn.2024.2820.

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Motivated by their importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis, and other field of applied mathematics, a variety of polynomials and numbers with their variants and extensions have recently been introduced and investigated. In this sequel, we modify the known generating functions of polynomials, due to both Milne-Thomson and Dere and Simsek, to introduce a new class of generalized polynomials and present some of their involved properties. As obvious special cases of the newly introduced polynomials, we also cal

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14

Kargın, Levent. "Poly-p-Bernoulli polynomials and generalized Arakawa–Kaneko zeta function." Lithuanian Mathematical Journal 60, no. 1 (2019): 29–50. http://dx.doi.org/10.1007/s10986-019-09448-7.

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15

Pathan, M. A., and Waseem A. Khan. "A new class of generalized polynomials associated with Hermite and poly-Bernoulli polynomials." Miskolc Mathematical Notes 22, no. 1 (2021): 317. http://dx.doi.org/10.18514/mmn.2021.1684.

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16

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

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17

Kim, T., and D. S. Kim. "Explicit Formulas for Probabilistic Multi-Poly-Bernoulli Polynomials and Numbers." Russian Journal of Mathematical Physics 31, no. 3 (2024): 450–60. http://dx.doi.org/10.1134/s1061920824030087.

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18

Komatsu, Takao, José L. Ramírez, and Víctor F. Sirvent. "Multi-poly-Bernoulli numbers and polynomials with a q parameter." Lithuanian Mathematical Journal 57, no. 4 (2017): 490–505. http://dx.doi.org/10.1007/s10986-017-9371-2.

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19

Acala, Nestor G., and Edward Rowe M. Aleluya. "On Generalized Arakawa–Kaneko Zeta Functions with Parameters a,b,c." International Journal of Mathematics and Mathematical Sciences 2020 (July 27, 2020): 1–6. http://dx.doi.org/10.1155/2020/2041262.

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For k∈ℤ, the generalized Arakawa–Kaneko zeta functions with a, b, c parameters are given by the Laplace-Mellin integral ξks,x;a,b,c=1/Γs∫0∞Lik1−ab−t/bt−a−tc−xtts−1dt, where ℜs>0 and x>0 if k≥1, and ℜs>0 and x>k+1 if k≤0. In this paper, an interpolation formula between these generalized zeta functions and the poly-Bernoulli polynomials with a,b,c parameters is obtained. Moreover, explicit, difference, and Raabe’s formulas for ξks,x;a,b,c are derived.

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20

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,

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21

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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22

Pita-Ruiz, Claudio. "On bi-variate poly-Bernoulli polynomials." Communications in Mathematics Volume 31 (2023), Issue 1 (November 22, 2022). http://dx.doi.org/10.46298/cm.10327.

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We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on standard Bernoulli polynomials, as the addition formula and the binomial formula. We also prove a result that allows us to obtain poly-Bernoulli polynomial identities from polynomial identities, and we use this result to obtain several identities involving products of poly-Bernoulli and/or standard Bernoulli polynomials. We prove two generalized recurrences

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23

Ohno, Yasuo, and Yoshitaka Sasaki. "Recursion formulas for poly-Bernoulli numbers and their applications." International Journal of Number Theory, October 12, 2020, 1–15. http://dx.doi.org/10.1142/s1793042121500081.

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Recurrence formulas for generalized poly-Bernoulli polynomials are given. The formula gives a positive answer to a question raised by Kaneko. Further, as applications, annihilation formulas for Arakawa-Kaneko type zeta-functions and a counting formula for lonesum matrices of a certain type are also discussed.

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24

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. T

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25

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 attachme

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