A new class of generalized polynomials involving Laguerre and Euler polynomials

Year 2021, Volume: 50 Issue: 1, 1 - 13, 04.02.2021

Nabiullah Khan Talha Usman , Junesang Choi

https://doi.org/10.15672/hujms.555416

Cited By: 8

https://izlik.org/JA46RA65RW

Abstract

Motivated by their importance and potential for applications in a variety of research fields, recently, numerous polynomials and their extensions have been introduced and investigated. In this paper, we modify the known generating functions of polynomials, due to both Milne-Thomsons and Dere-Simsek, to introduce a new class of polynomials and present some involved properties. As obvious special cases of the newly introduced polynomials, we also introduce power sum-Laguerre-Hermite polynomials and generalized Laguerre and Euler polynomials and give certain involved identities and formulas. We point out that our main results, being very general, are specialised to yield a number of known and new identities involving relatively simple and familiar polynomials.

Keywords

Milne-Thomsons polynomials , Dere-Simsek polynomials , Laguerre polynomials , Hermite polynomials , Euler polynomials , generalized Laguerre-Euler polynomials , summation formulae , symmetric identities

References

• [1] L.C. Andrews, Special Functions for Engineer and Mathematician, Macmillan Company, New York, 1985.
• [2] E.T. Bell, Exponential polynomials, Ann. Math. 35 (2), 258–277, 1934.
• [3] G. Betti and P.E. Ricci, Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese J. Math. 8 (3), 415–428, 2004.
• [4] J. Choi, Notes on formal manipulations of double series, Commun. Korean Math. Soc. 18 (4), 781–789, 2003.
• [5] J. Choi, N.U. Khan and T. Usman, A note on Legendre-based multi poly-Euler polynomials, Bull. Iran. Math. Soc. 44, 707–717, 2018.
• [6] G. Dattoli, S. Lorenzutta and C. Cesarano, Finite sums and generalized forms of Bernoulli polynomials, Rend. Mat. 19, 385–391, 1999.
• [7] G. Dattoli and A. Torre, Theory and Applications of Generalized Bessel Function, Aracne, Rome, 1996.
• [8] G. Dattoli and A. Torre, Operational methods and two variable Laguerre polynomials, Atti Acad. Torino 132, 1–7, 1998.
• [9] G. Dattoli, A. Torre and A.M. Mancho, The generalized Laguerre polynomials, the associated Bessel functions and applications to propagation problems, Radiat. Phys. Chem. 59, 229–237, 2000.
• [10] R. Dere and Y. Simsek, Hermite base Bernoulli type polynomials on the umbral algebra, Russian J. Math. Phys. 22 (1), 1–5, 2015.
• [11] B.N. Guo and F. Qi, Generalization of Bernoulli polynomials, J. Math. Ed. Sci. Tech. 33 (3), 428–431, 2002.
• [12] N.U. Khan and T. Usman, A new class of Laguerre-based generalized Apostol polynomials, Fasciculli. Math. 57, 67–89, 2016.
• [13] N.U. Khan and T. Usman, A new class of Laguerre-based poly-Euler and multi poly- Euler polynomials, J. Anal. Num. Theor. 4 (2), 113–120, 2016.
• [14] N.U. Khan and T. Usman, A new class of Laguerre poly-Bernoulli numbers and polynomials, Adv. Stud. Contemporary Math. 27 (2), 229–241, 2017.
• [15] N.U. Khan, T. Usman and A. Aman, Generating functions for Legendre-Based poly- Bernoulli numbers and polynomials, Honam Math. J. 39 (2), 217–231, 2017.
• [16] N.U. Khan, T. Usman and J. Choi, Certain generating function of Hermite-Bernoulli- Laguerre polynomials, Far East J. Math. Sci. 101 (4), 893–908, 2017.
• [17] N.U. Khan, T. Usman and J. Choi, A new generalization of Apostol type Laguerre- Genocchi polynomials, C. R. Acad. Sci. Paris, Ser. I, 355, 607–617, 2017.
• [18] N.U. Khan, T. Usman and J. Choi, A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials, Turkish J. Math. 43, 486–497, 2019.
• [19] B. Kurt and Y. Simsek, Notes on generalization of the Bernoulli type polynomials, Appl. Math. Comput. 218, 906–911, 2011.
• [20] Q.-M. Luo, B.N. Guo, F. Qi and L. Debnath, Generalization of Bernoulli numbers and polynomials, Int. J. Math. Math. Sci. 59, 3769–3776, 2003.
• [21] Q.-M. Luo, F. Qi and L. Debnath, Generalization of Euler numbers and polynomials, Int. J. Math. Math. Sci. 61, 3893–3901, 2003.
• [22] L.M. Milne-Thomsons, Two classes of generalized polynomials, Proc. London Math. Soc. 35 (1), 514–522, 1933.
• [23] M.A. Pathan, A new class of generalized Hermite-Bernoulli polynomials, Georgian Math. J. 19, 559–573, 2012.
• [24] M.A. Pathan and W.A. Khan, A new class of generalized polynomials associated with Hermite and Euler polynomials, Mediterr. J. Math. 13 (3), 913–928, 2016.
• [25] F. Qi and B.N. Guo, Generalization of Bernoulli polynomials, RGMIA Res. Rep. Coll. 4 (4), Article 10, 691–695, 2001.
• [26] E.D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
• [27] H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
• [28] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
• [29] S. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308, 550–554, 2008.
• [30] Z. Zhang and H. Yang, Several identities for the generalized Apostol Bernoulli polynomials, Comput. Math. Appl. 56 (12), 2993–2999, 2008.