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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References
Bell E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)
Betti G., Ricci P.E.: Multidimensional extensions of the Bernoulli and Appell polynomials. Taiwan. J. Math. 8(3), 415–428 (2004)
Dattoli G., Lorenzutta S., Cesarano C.: Finite sums and generalized forms of Bernoulli polynomials. Rendiconti di Mathematica 19, 385–391 (1999)
Dere R., Simsek R.: : Bernoulli type polynomials on Umbral algebra. Russian J. Math. Phys. 23(1), 1–6 (2015)
Khan S., Pathan M.A, Hassan N.A.M., Yasmin G.: Implicit summation formula for Hermite and related polynomials. J. Math. Anal. Appl. 344, 408–416 (2008)
Kurt B., Simsek Y.: Notes on generalization of the Bernoulli type polynomials. Appl. Math. Comput. 218, 906–911 (2011)
Luo Q-M., Qi F., Debnath L.: Generalization of Euler numbers and polynomials. Int. J. Math. Math. Sci. 61, 3893–3901 (2003)
Luo Q.-M., Guo B.N., Qi F., Debnath L.: Generalization of Bernoulli numbers and polynomials. Int. J. Math. Math. Sci. 59, 3769–3776 (2003)
Milne Thomsons L.M.: Two classes of generalized polynomials. Proc. Lond. Math. Soc. 35(1), 514–522 (1933)
Pathan M.A.: A new class of generalized Hermite–Bernoulli polynomials. Georg. Math. J. 19, 559–573 (2012)
Pathan, M.A., Khan, W.A.: Some implicit summation formulas and symmetric identities for the generalized Hermite–Bernoulli polynomials. Mediterr. J. Math. (2014). doi:10.1007/s00009-014-0423-0
Pathan M.A., Khan W.A.: Some implicit summation formulas and symmetric identities for the generalized Hermite–Euler polynomials. East–West J. Math. 16(1), 92–109 (2014)
Pathan, M.A., Khan, W.A.: A new class of generalized polynomials associated with Hermite and Bernoulli polynomials. LE MATEMATICHE. 70(Fasc.1), 1–19 (To appear, 2015)
Qui F., Guo B.N.: : Generalization of Bernoulli polynomials. RGMIA Res. Rep. Coll. 4(4), 691–695 (2001) (Article 10)
Simsek Y:: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 2013, 87 (2013)
Simsek Y.: Complete sum of product of (h,q) extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 16(11), 1331–1348 (2010)
Srivastava H.M., Manocha H.L.: A Treatise on Generating Functions. Ellis Horwood Limited. Co., New York (1984)
Srivastava H.M., Kurt B., Simsek Y.: Some families of Genocchi type polynomials and their interpolation functions. Integr. Trans. Spec. Funct. 23(12), 919–938 (2012)
Tuenter H.J.H.: A symmetry power sum of polynomials and Bernoulli numbers. Am. Math. Mon. 108, 258–261 (2001)
Yang S.L., Qiao Z.K.: Some symmetry identities for the Euler polynomials. J. Math. Res. Expos. 30(3), 457–464 (2010)
Zhang Z., Yang H.: Several identities for the generalized Apostol Bernoulli polynomials. Comput. Math. Appl. 56, 2993–2999 (2008)
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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