Abstract
In the present paper, we study the new type of generating function of Apostol–Euler based poly Daehee polynomials. By making use of this generating function, we investigate some new and interesting identities for the Apostol–Euler based poly Daehee polynomials. We also derive some implicit summation formulae for Apostol–Euler based poly Daehee polynomials by applying the series manipulation techniques.
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Carlitz, L.: A note on Bernoulli and Euler polynomials of the second kind. Scr. Math. 25, 323–330 (1961)
Choi, J., Khan, N., Usman, T., Aman, M.: Certain unified polynomials. Integral Transforms Spec. Funct. 30(1), 28–40 (2019)
Dattoli, G., Lorenzutta, S., Cesarano, C.: Finite sums and generalized forms of Bernoulli polynomials. Rend. Math. 19, 385–391 (1999)
Jolany, H., Aliabadi, M., Corcino, R.B., Darafsheh, M.R.: A note on multi poly-Euler numbers and Bernoulli polynomials. arXiv preprint arXiv:1401.2645 (2014)
Kim, T.: On the multiple q-Genocchi and Euler numbers. Russ. J. Math. Phys. 15(4), 481–486 (2008)
Khan, W.A.: A new class of Hermite poly-Genocchi polynomials. J. Anal. Number Theory 4, 1–8 (2016)
Khan, W.A.: A note on Hermite-based poly-Euler and multi poly-Euler polynomials. Palest. J. Math 5, 17–26 (2016)
Khan, N.U., Usman, T.: A new class of Laguerre-based generalized Apostol polynomials. Faciculi Math. 57, 67–89 (2016)
Khan, N.U., Usman, T.: A new class of Laguerre poly-Bernoulli numbers and polynomial. Adv. Stud. Contemp. Math. 27, 229–241 (2017)
Khan, N.U., Usman, T., Aman, M.: Generating functions for Legendre-based poly-Bernoulli numbers and polynomials. Honam Math. J. 39, 217–231 (2017)
Khan, N.U., Usman, T.: A new class of Laguerre-based poly-Euler and multi poly-Euler polynomials. J. Anal. Number Theory 4, 113–120 (2016)
Khan, N., Usman, T., Choi, J.: A new class of generalized polynomials. Turk. J. Math. 42(3), 1366–1379 (2018)
Khan, S., Nahid, T.: Finding non-linear differential equations and certain identities for the Bernoulli–Euler and Bernoulli–Genocchi numbers. SN Appl. Sci. 1(3), 217 (2019)
Kim, D.S., Kim, T.: A study on the integral of the product of several Bernoulli polynomials. Rocky Mt. J. Math. 44(4), 1251–1263 (2014)
Kim, D.S., Kim, T.: Some identities involving Genocchi polynomials and numbers. Ars Comb. 121, 403–412 (2015)
Kim, D.S., Kim, T., Lee, S.H., Seo, J.-J.: Higher order Daehee numbers and polynomials. Int. J. Math. Anal. 8(6), 273–283 (2014)
Kim, T., Kwon, H.I., Lee, S.H., Seo, J.J.: A note on poly-Bernoulli numbers and polynomials of the second kind. Adv. Differ. Equ. 2014(1), 219 (2014)
Komatsu, T., Luca, F.: Some relationships between poly-Cauchy numbers and poly Bernoulli numbers. Ann. Math. Inform. 41, 99–105 (2013)
Luo, Q.M., Srivastava, H.M.: Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials. J. Math. Anal. Appl. 308(1), 290–302 (2005)
Luo, Q.M.: Apostol–Euler polynomials of higher order and the Gaussian hypergeometric function. Taiwan. J. Math. 10(4), 917–925 (2006)
Luo, Q.M.: Extension for the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 48, 291–309 (2011)
Rainville, E.D.: Special Functions. The MacMillan Comp., New York (1960)
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Saif, M., Nadeem, R. Evaluation of Apostol–Euler Based Poly Daehee Polynomials. Int. J. Appl. Comput. Math 6, 1 (2020). https://doi.org/10.1007/s40819-019-0748-2
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DOI: https://doi.org/10.1007/s40819-019-0748-2