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New degenerate Bernoulli, Euler, and Genocchi polynomials


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[1] M. Acikgoz, S. Araci and U. Duran, New extensions of some known special polynomials under the theory of multiple q-calculus, Turkish J. Anal. Num. Theory, 3 (2015) 128–139.10.12691/tjant-3-5-4Search in Google Scholar

[2] M. Acikgoz and Y. Simsek, On multiple interpolation functions of the Nörlund type q-Euler polynomials, Abstr. Appl. Anal., 2009 (2009) Article ID 382574, 14pp.10.1155/2009/382574Search in Google Scholar

[3] S. Araci, M. Acikgoz and E. Şen, Some new formulae for Genocchi numbers and polynomials involving Bernoulli and Euler polynomials, Int. J. Math. Math. Sci., 2014 (2014) Article ID 760613, 7 pp.10.1155/2014/760613Search in Google Scholar

[4] E. P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Phys. A., 340 (2004) 95–101.10.1016/j.physa.2004.03.082Search in Google Scholar

[5] E. P. Borges, On a q-generalization of circular and hyperbolic functions, J. Phys. A, 31 (1998) 5281–5288.10.1088/0305-4470/31/23/011Search in Google Scholar

[6] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel), 7 (1956) 28–33.10.1007/BF01900520Search in Google Scholar

[7] R. Dere, Y. Simsek and H. M. Srivastava, A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, J. Number Theory, 133 (2013) 3245–3263.10.1016/j.jnt.2013.03.004Search in Google Scholar

[8] K. Dilcher and C. Vignat, General convolution identities for Bernoulli and Euler polynomials, J. Math. Anal. Appl., 435 (2016) 1478–1498.10.1016/j.jmaa.2015.11.006Search in Google Scholar

[9] Y. He, S. Araci, H. M. Srivastava and M. Acikgoz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials, Appl. Math. Comput., 262 (2015) 31–41.10.1016/j.amc.2015.03.132Search in Google Scholar

[10] D. S. Kim, T. Kim and J.-J. Seo, Umbral calculus associated with new degenerate Bernoulli polynomials, J. Comput. Anal. Appl., 22 (2017) 831–840.Search in Google Scholar

[11] D. Lim, Some identities of degenerate Genocchi polynomials, Bull. Korean Math. Soc., 53 (2016) 569–579.Search in Google Scholar

[12] H. Ozden, Y. Simsek and H. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl., 60 (2010) 2779–2787.Search in Google Scholar

[13] S. Roman, The theory of the umbral calculus. I, J. Math. Anal. Appl., 87 (1982) 58–115.Search in Google Scholar

[14] S. Roman, The umbral calculus, Pure and Applied Mathematics, 111. Academic Press, Inc., New York, 1984.Search in Google Scholar

[15] Y. Simsek, Complete sum of products of (h, q)-extension of Euler polynomials and numbers, J. Di erence Equ. Appl., 16 (2010) 1331–1348.Search in Google Scholar

[16] H. M. Srivastava, B. Kurt and V. Kurt, Identities and relations involving the modified degenerate Hermite-based Apostol-Bernoulli and Apostol-Euler polynomials, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113 (2019) 1299–1313.Search in Google Scholar

[17] J. Szmidt, J. Urbanowicz and D. Zagier, Congruences among generalized Bernoulli numbers, Acta Arith., 71 (1995) 273–278.Search in Google Scholar

[18] C. Tsallis, Possible generalizations of Boltzmann-Gibbs statistics, J. Stat. Phys., 52 (1988) 479–487.Search in Google Scholar

[19] C. Tsallis, What are the numbers that experiments provide?, Quimica Nova, 17 (1994) 468–471.Search in Google Scholar

[20] P. T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, J. Number Theory, 128 (2008) 738–758.Search in Google Scholar

[21] D. Zagier, A modified Bernoulli number, Nieuw Arch. Wiskd. (5), 16 (1998) 63–72.Search in Google Scholar