Representation by degenerate Frobenius–Euler polynomials

References

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

[2] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51–88. Search 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. Search 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.a1Search 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/s002229900028Search 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.2003.08.010Search 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-2014-261Search 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. 10.1186/1687-1847-2012-196Search 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-2012-307Search 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. 10.1080/10652469.2012.754756Search 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. 10.1186/1687-1847-2013-103Search 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. 10.1186/1687-1847-2012-95Search 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. Search 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-2017-0096Search 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. 10.1186/1687-1847-2013-73Search 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. Search 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)90026-4Search in Google Scholar

[18] N. Nielsen, Traité Élémentaire des Nombres de Bernoulli, Gauthier-Villars, Paris, 1923. Search 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/s13662-016-1054-5Search in Google Scholar

[20] S. Roman, The Umbral Calculus, Pure Appl. Math. 111, Academic Press, New York, 1984. Search 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.36.73Search 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/axioms7020022Search in Google Scholar