Abstract
Eight basic identities of symmetry in three variables, which are related to degenerate Euler polynomials and alternating generalized falling factorial sums, are derived. These are the degenerate versions of the symmetric identities in three variables obtained in a previous paper. The derivations of identities are based on the p-adic integral expression of the generating function for the degenerate Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized falling factorial sums. Those eight basic identities and most of their corollaries are new, since there have been results only about identities of symmetry in two variables.
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Bayad A, Kim T, Choi J, Kim YH, Lee B (2011) On the symmetry properties of the generalized higher-order Euler polynomials. J Appl Math Inf 29(1–2):511–516
Carlitz L (1979) Degenerate Stirling, Bernoulli and Eulerian numbers. Util Math 15:51–88
Deeba E, Rodriguez D (1991) Stirling’s and Bernoulli numbers. Am Math Mon 98:423–426
Howard FT (1995) Applications of a recurrence for the Bernoulli numbers. J Number Theory 52:157–172
Kim T (2008) Symmetry \(p\)-adic invariant integral on \(\mathbb{Z}_{p}\) for Bernoulli and Euler polynomials. J Differ Equ Appl 14:1267–1277
Kim T (2009a) Symmetry of power sum polynomials and multivariate fermionic \(p\)-adic invariant integral on \({\mathbb{Z}}_p\). Russ J Math Phys 16:93–96
Kim T (2009b) Symmetry identities for the twisted generalized Euler polynomials. Adv Stud Cotemp Math (Kyungshang) 19:151–155
Kim T (2011) An identity of the symmetry for the Frobenius–Euler polynomials associated with the fermionic \(p\)-adic invariant \(q\)-integrals on \({\mathbb{Z}}_p\). Rocky Mt J Math 41:239–247
Kim DS, Kim T (2014) Three variable symmetric identities involving Carlitz-type q-Euler polynomials. Math Sci 8:147–152
Kim DS, Kim T (2015) Identities of symmetry for the generalized degenerate Euler polynomials. In: Singh VK, Srivastava HM, Venturino E, Resch M, Gupta V (eds) Modern Mathematical Methods and High Performance Computing in Science and Technology, M3HPCST. Springer Proceedings in Mathematics & Statistics, vol 171. Springer, Ghaziabad, India, p 33–43. doi:10.1007/978-981-10-1454-3
Kim DS, Kim T (2016) Symmetric identities of higher-order degenerate q-Euler polynomials. J Nonlinear Sci Appl 9:443–451
Kim DS, KIM T (2017) Symmetric identities of higher-order degenerate Euler polynomials. arXiv:1704.04025
Kim DS, Lee N, Na J, Park KH (2011) Identities of symmetry for higher-order Euler polynomials in three variables (II). J Math Anal Appl 379:388–400
Kim DS, Lee N, Na J, Park KH (2012) Identities of symmetry for higher-order Euler polynomials in three variables (I). Adv Stud Contemp Math (Kyungshang) 22(1):51–74
Kim DS, Lee N, Na J, Park KH (2013a) Abundant symmetry for higher-order Bernoulli polynomials (I). Adv Stud Contemp Math (Kyungshang) 23(3):461–482
Kim DS, Lee N, Na J, Park KH (2013b) Abundant symmetry for higher-order Bernoulli polynomials (II). Proc Jangjeon Math Soc 16(3):359–378
Kim T, Dolgy DV, Kim DS (2015) Some identities of \(q\)-Bernoulli polynomials under symmetry group \(S_3\). J Nonlinear Convex Anal 9:1869–1880
Kim T, Kim DS, Dolgy DV, Kwon HI, Seo JJ (2016) Identities of symmetry for degenerate Bernoulli polynomials and generalized falling factorial sums. Global J Pure Appl Math 12(5):4363–4383
Ozden H, Simsek Y (2008) A new extension of \(q\)-Euler numbers and polynomials related to their interpolation functions. Appl Math Lett 21(9):934–939
Ozden H, Cangul IN, Simsek Y (2008a) Multivariate interpolation functions of higher-order \(q\)-Euler numbers and their applications. Abstr Appl Anal Article ID 390857
Ozden H, Cangul IN, Simsek Y (2008b) Remarks on sum of products of \((h,q)\)-twisted Euler polynomials and numbers. J Inequal Appl Article ID 816129
Simsek Y (2010) Complete sum of products of \((h, q)\)-extension of Euler polynomials and numbers. J Differ Equ Appl 16(11):1331–1348
Tuenter H (2001) A symmetry of power sum polynomials and Bernoulli numbers. Am Math Mon 108:258–261
Wu M, Pan H (2014) Sums of products of the degenerate Euler numbers. Adv Differ Equ 2014:40
Yang S (2008) An identity of symmetry for the Bernoulli polynomials. Discrete Math 308:550–554
Acknowledgements
The first author is appointed as a chair professor at Tianjin Polytechnic University by Tianjin City in China from August 2015 to August 2019. The authors would like to express their sincere gratitude to referees for their valuable comments and information.
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Kim, T., Kim, D.S. Identities of Symmetry for Degenerate Euler Polynomials and Alternating Generalized Falling Factorial Sums. Iran J Sci Technol Trans Sci 41, 939–949 (2017). https://doi.org/10.1007/s40995-017-0326-6
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DOI: https://doi.org/10.1007/s40995-017-0326-6
Keywords
- Degenerate Euler polynomial
- Alternating generalized falling factorial sum
- Fermionic p-adic integral
- Identities of symmetry