Startseite Representation by degenerate Frobenius–Euler polynomials
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Representation by degenerate Frobenius–Euler polynomials

  • Taekyun Kim EMAIL logo und Dae San Kim
Veröffentlicht/Copyright: 29. Juni 2022
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

The aim of this paper is to represent any polynomial in terms of degenerate Frobenius–Euler polynomials and, more generally, of higher-order degenerate Frobenius–Euler polynomials. Explicit formulas with the help of umbral calculus are derived and the obtained results are illustrated by some examples.

Funding statement: The authors would like to thank Jangjeon Institute for Mathematical Sciences for the support of this research.

Acknowledgements

The authors would like to thank the reviewer for the suggestions that helped improve the original manuscript in its present form.

References

[1] L. Carlitz, The product of two Eulerian polynomials, Math. Mag. 36 (1963), no. 1, 37–41. 10.1080/0025570X.1963.11975384Suche in Google Scholar

[2] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51–88. Suche in Google Scholar

[3] R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 433–438. Suche in Google Scholar

[4] G. V. Dunne and C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys. 7 (2013), no. 2, 225–249. 10.4310/CNTP.2013.v7.n2.a1Suche in Google Scholar

[5] C. Faber and R. Pandharipande, Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. 10.1007/s002229900028Suche in Google Scholar

[6] I. M. Gessel, On Miki’s identity for Bernoulli numbers, J. Number Theory 110 (2005), no. 1, 75–82. 10.1016/j.jnt.2003.08.010Suche in Google Scholar

[7] Y. He and S. J. Wang, New formulae of products of the Frobenius–Euler polynomials, J. Inequal. Appl. 2014 (2014), Article ID 261. 10.1186/1029-242X-2014-261Suche in Google Scholar

[8] D. S. Kim and T. Kim, Some identities of Frobenius–Euler polynomials arising from umbral calculus, Adv. Difference Equ. 2012 (2012), Article ID 196. 10.1186/1687-1847-2012-196Suche in Google Scholar

[9] D. S. Kim and T. Kim, Some new identities of Frobenius–Euler numbers and polynomials, J. Inequal. Appl. 2012 (2012), Article ID 307. 10.1186/1029-242X-2012-307Suche in Google Scholar

[10] D. S. Kim and T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct. 24 (2013), no. 9, 734–738. 10.1080/10652469.2012.754756Suche in Google Scholar

[11] D. S. Kim, T. Kim, D. V. Dolgy and S.-H. Rim, Higher-order Bernoulli, Euler and Hermite polynomials, Adv. Difference Equ. 2013 (2013), Article ID 103. 10.1186/1687-1847-2013-103Suche in Google Scholar

[12] D. S. Kim, T. Kim, S.-H. Lee and Y.-H. Kim, Some identities for the product of two Bernoulli and Euler polynomials, Adv. Difference Equ. 2012 (2012), Article ID 95. 10.1186/1687-1847-2012-95Suche in Google Scholar

[13] D. S. Kim, T. Kim and T. Mansour, Euler basis and the product of several Bernoulli and Euler polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 24 (2014), no. 4, 535–547. Suche in Google Scholar

[14] T. Kim and D. S. Kim, An identity of symmetry for the degenerate Frobenius–Euler polynomials, Math. Slovaca 68 (2018), no. 1, 239–243. 10.1515/ms-2017-0096Suche in Google Scholar

[15] T. Kim, D. S. Kim, D. V. Dolgy and S. H. Rim, Some identities on the Euler numbers arising from Euler basis polynomials, Ars Combin. 109 (2013), 433–446. 10.1186/1687-1847-2013-73Suche in Google Scholar

[16] T. Kim, D. S. Kim, G.-W. Jang and J. Kwon, Fourier series of sums of products of Euler functions, J. Comput. Anal. Appl. 275 (2019), no. 2, 345–360. Suche in Google Scholar

[17] H. Miki, A relation between Bernoulli numbers, J. Number Theory 10 (1978), no. 3, 297–302. 10.1016/0022-314X(78)90026-4Suche in Google Scholar

[18] N. Nielsen, Traité Élémentaire des Nombres de Bernoulli, Gauthier-Villars, Paris, 1923. Suche in Google Scholar

[19] J. Pan and F. Yang, Some convolution identities for Frobenius–Euler polynomials, Adv. Difference Equ. 2017 (2017), Paper No. 6. 10.1186/s13662-016-1054-5Suche in Google Scholar

[20] S. Roman, The Umbral Calculus, Pure Appl. Math. 111, Academic Press, New York, 1984. Suche in Google Scholar

[21] K. Shiratani and S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A 36 (1982), no. 1, 73–83. 10.2206/kyushumfs.36.73Suche in Google Scholar

[22] Y. Simsek, Special numbers and polynomials including their generating functions in umbral analysis methods, Axioms 7 (2018), no. 2, 1–12. 10.3390/axioms7020022Suche in Google Scholar

Received: 2021-10-01
Revised: 2022-02-02
Accepted: 2022-03-14
Published Online: 2022-06-29
Published in Print: 2022-10-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 28.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/gmj-2022-2167/html
Button zum nach oben scrollen