Abstract
In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials \({\Phi_{n}^{(\alpha)}(x,\nu)}\) of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E n , Euler polynomials E n (x), generalized Euler numbers E n (a, b), generalized Euler polynomials E n (x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials \({{_HE}_n(x,y)}\) of Dattoli et al. and \({{_HE}_n^{(\alpha)} (x,y)}\) of Pathan are generalized to the one \({ {_HE}_n^{(\alpha)}(x,y,a,b,c)}\) which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E n , E n (x), E n (a, b), E n (x; a, b, c) and \({{}_HE_n^{(\alpha)}(x,y;a,b,c)}\) are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.
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Pathan, M.A., Khan, W.A. A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials. Mediterr. J. Math. 13, 913–928 (2016). https://doi.org/10.1007/s00009-015-0551-1
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DOI: https://doi.org/10.1007/s00009-015-0551-1