A note on Hermite-based truncated Euler polynomials
DOI:
https://doi.org/10.5269/bspm.51960Abstract
In this paper, we introduce a new class of truncated Hermite-Euler polynomials and numbers as a generalization of Hermite-Euler polynomials. Furthermore, the discussion is on properties and relations with the hypergeometric Bernoulli polynomials, Frobenius-Euler polynomials and Stirling numbers of the second kind. As a result, we derive some implicit summation formulas of this polynomials.
References
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2. Bell, E. T, Exponential polynomials, Ann. of Math., 35, 258-277, (1934). https://doi.org/10.2307/1968431
3. Choi, J, Kim, D. S, Kim, T and Kim, Y. H, A note on some identities of Frobenius-Euler numbers and polynomials, Inter. J. Math. Math. Sci., 2012, 1-9, (2012). https://doi.org/10.1155/2012/861797
4. Dattoli, G, Lorenzutta, S and Cesarano, C., Finite sums and generalized forms of Bernoulli polynomials, Rendiconti di Mathematica, 19, 385-391, (1999).
5. Duran, U, Acikgoz, M and Araci, S, Hermite-based poly-Bernoulli polynomials with a q-paramenter, Adv. Stud. Contemp. Math. (Kyungshang), 28(2), 285-296, (2018). https://doi.org/10.20944/preprints201802.0145.v1
6. Duran, U and Acikgoz, M, Truncated Fubini Polynomials, Mathematics, 7, 431, (2019). https://doi.org/10.3390/math7050431
7. Duran, U and Acikgoz, M, On Degenerate Truncated special polynomials, Mathematics, 8, 144, (2020). https://doi.org/10.3390/math8010144
8. Hassen, A and Nguyen, H. D, Hypergeometric Bernoulli polynomials and Appell sequences, Int. J. Number Theory, 4, 767-774, (2008). https://doi.org/10.1142/S1793042108001754
9. Hassen, A and Nguyen, H. D, Hypergeometric zeta functions, Int. J. Number Theory, 6, 99-126, (2010). https://doi.org/10.1142/S179304211000282X
10. Jang, L. C, Kim, D. S, Jang, G. -W and Kwon, J, Some identities for q-Bernoulli numbers and polynomials arising form q-Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 28(4), 659-667, (2018).
11. Jang, G. W and Kim, T, A note on type 2 degenerate Euler and Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 29(1), 147-159, (2019).
12. Kamano, K, Sums of products of hypergeometric Bernoulli numbers, J. Number Theory, 130, 2259-2271, (2010). https://doi.org/10.1016/j.jnt.2010.04.005
13. Komatsu, K and Ruiz C. P, Truncated Euler polynomials, Mathematica Slovaca, 68(3), 527-536, (2018). https://doi.org/10.1515/ms-2017-0122
14. Kim, D. S and Kim, T, Some identities of Frobenius-Euler polynomials arising from umbral calculus, Adv. Difference Equ., 2012, Article number: 196(2012). https://doi.org/10.1186/1687-1847-2012-196
15. Kim, D. S, Kim, T, Rim, S. H and Dolgy, D. V, Barne's multiple BErnoulli and Hermite mixed-type polynomials, Proc. Jangjeon Math. Soc., 18(1), 7-19, (2015).
16. Khan, W. A, Some properties of the generalized Apostol type Hermite-Based polynomials, Kyungpook Math. J., 55, 597-614, (2015). https://doi.org/10.5666/KMJ.2015.55.3.597
17. Khan, W. A and Ahmad, M, Partially degenerate poly-Bernoulli polynomials associated with Hermite polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 28(3), 487-496, (2018).
18. Khan, W. A, A new class of heigher-order hypergeometric Bernoulli polynomials associated with Hermite polynomials, Bol. Soc. Paran. Mat., Accepted (2020).
19. Kim, D. S and Kim, T, Some new identities of Frobenius-Euler numbers and polynomials, J. Ineq. Appl., 2012, 1-10, (2012). https://doi.org/10.1155/2012/619197
20. Kim, T, Kwon, H. -I and Jang, G. W, Symmetric identities of heigher-order of degenerate q-Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 27(1), 31-41, (2017).
21. Kumam, W, Srivastava, H. M and Kumam, P, Truncated-exponential-based Frobenius-Euler polynomials, Advan. Diff. Eq., 2019:530, (2019). https://doi.org/10.1186/s13662-019-2462-0
22. Kurt, B and Simsek, Y, On the generalized Apostol type Frobenius Euler polynomials, Advances in Differences equations, (2013), 1-9, (2013). https://doi.org/10.1186/1687-1847-2013-1
23. Pathan, M. A and Khan, W. A, Some implicit summation formulas and symmetric identities for the generalized Hermite-Euler polynomials, East-West J. Maths., 16(1), 92-109, (2014).
24. Srivastava, H. M, Araci, S, Khan, W. A and Acikgoz, M, A note on the truncated-exponential based Apostol-type polynomials, Symmetry 11, Article ID 538,doi:10.3390/sym11040538, (2019). https://doi.org/10.3390/sym11040538
25. Srivastava, H. M and Acikgoz, M, A new approach to Legendre-truncated-exponential-based Sheffer sequences via Riordan arrays, Applied Mathematics and Computation 369, 124683, (2020). https://doi.org/10.1016/j.amc.2019.124683
26. Srivastava, H. M and Manocha, H. L, A treatise on generating functions, Ellis Horwood Limited, New York, 1984.
27. Simsek, Y, Generating functions for generalized Striling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Th. Appl., DOI: 1186/1687-1812-2013-87, (2013). https://doi.org/10.1186/1687-1812-2013-87
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2022-12-21
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