On Poly-Euler Polynomials and Arakawa-Kaneko Type Zeta Functions of Parameters \(a,b,c\)

Authors

  • Nestor G. Acala Department of Mathematics, Mindanao State University, Marawi City
  • Roberto B. Corcino Research Institute for Computational Mathematics and Physics (RICMP), Cebu Normal University, Cebu City

DOI:

https://doi.org/10.26713/cma.v12i3.1514

Keywords:

Euler numbers and polynomials, Bernoulli numbers and polynomials, Riemann zeta functions, Arakawa-Kaneko zeta functions, Poly-Euler numbers and polynomials, Poly-Bernoulli numbers and polynomials, Generalized poly-Euler numbers and polynomials

Abstract

In this paper, we investigate a class of generalized poly-Euler polynomials with \(a,b,c\) parameters, a generalization of the classical Euler numbers and polynomials. Various properties of these generalized polynomials are established. We also introduce the Arakawa-Kaneko type zeta functions for the poly-Euler polynomials with \(a,b,c\) parameters and obtain an interpolation formula for the generalization of poly-Euler numbers and polynomials with \(a,b,c\) parameters. Furthermore, we establish the relationship between the Arakawa-Kaneko type zeta functions for generalized poly-Euler polynomials and the Arakawa-Kaneko zeta functions for generalized poly-Bernoulli polynomials defined in [1].

Downloads

Download data is not yet available.

References

N. G. Acala and E. R. M. Aleluya, On generalized Arakawa-Kaneko zeta functions with parameters a,b, c, International Journal of Mathematics and Mathematical Sciences 2020 (2020), Article ID 2041262, 6 pages, DOI: 10.1155/2020/2041262.

T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers and related zeta functions, Nagoya Mathematical Journal 153 (1999), 189 – 209, https://www2.math.kyushuu.ac.jp/~mkaneko/papers/18MZV_pB_Nagoya.pdf.

T. Arakawa and M. Kaneko, On poly-Bernoulli numbers, Commentarii Mathematici Universitatis Sancti Pauli. Rikkyo University (St. Paul's University) 48 (1999), 159 – 167, https://www2.math.kyushu-u.ac.jp/~mkaneko/papers/pb2.pdf.

A. Bayad and Y. Hamahata, Polylogarithms and poly-Bernoulli polynomials, Kyushu Journal of Mathematics 65 (2011), 15 – 24, DOI: 10.2206/kyushujm.65.15.

L. Carlitz, Degenerate Stirling, Bernoulli, and Eulerian numbers, Utilitas Mathematica 15 (1979), 51 – 58.

L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company (1974).

M.-A. Coppo and B. Candelpergher, The Arakawa-Kaneko zeta function, The Ramanujan Journal 22(2010), 153 – 162, DOI: 10.1007/s11139-009-9205-x.

R. B. Corcino, C. B. Corcino, J. M. Ontolan and H. Jolany, Some identities on the generalized poly-Euler and poly-Bernoulli polynomials, Matimyas Matimatika 39 (2) (2016), 43 – 58, URL: http://mathsociety.ph/matimyas/images/vol39/Corcino.pdf.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, Reading, MA (1994), https://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf.

Y. Hamahata, Poly-Euler polynomials and Arakawa-Kaneko type zeta functions, Functiones et Approximatio Commentarii Mathematici 51(1) (2014), 7 – 22, DOI: 10.7169/facm/2014.51.1.1.

H. Jolany, M. R. Darafsheh and R. E. Alikelaye, Generalizations of poly-Bernoulli numbers and polynomials, International Journal of Mathematical Combinatorics 2 (2010), 7 – 14, http://fs.unm.edu/IJMC/GeneralizationsOfPolyBernoulliNumbers.pdf.

H. Jolany and R. Corcino, Explicit formula for the generalization of poly-Bernoulli numbers and polynomials with a,b, c parameters, Journal of Classical Analysis 6 (2015), 119 – 135, DOI: 10.7153/jca-06-10.

M. Kaneko, Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bordeaux 9 (1997), 221 – 228, https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_221_0/.

Q.-M. Luo, B.-N. Guo, F. Qi and L. Debnath, Generalizations of Bernoulli numbers and polynomials, International Journal of Mathematics and Mathematical Sciences 59 (2003), 3769 – 3776, DOI: 10.1155/S0161171203112070.

Q.-M. Luo, F. Qi and L. Debnath, Generalizations of Euler numbers and polynomials, International Journal of Mathematics and Mathematical Sciences 61 (2003), 3893 – 3901, DOI: 10.1155/S016117120321108X.

Downloads

Published

30-09-2021
CITATION

How to Cite

Acala, N. G., & Corcino, R. B. (2021). On Poly-Euler Polynomials and Arakawa-Kaneko Type Zeta Functions of Parameters \(a,b,c\). Communications in Mathematics and Applications, 12(3), 401–415. https://doi.org/10.26713/cma.v12i3.1514

Issue

Vol. 12 No. 3 (2021)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.