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http://dx.doi.org/10.4134/CKMS.c200275

HIGHER ORDER APOSTOL-TYPE POLY-GENOCCHI POLYNOMIALS WITH PARAMETERS a, b AND c  

Corcino, Cristina B. (Research Institute for Computational Mathematics and Physics Mathematics Department Cebu Normal University)
Corcino, Roberto B. (Research Institute for Computational Mathematics and Physics Mathematics Department Cebu Normal University)
Publication Information
Communications of the Korean Mathematical Society / v.36, no.3, 2021 , pp. 423-445 More about this Journal
Abstract
In this paper, a new form of poly-Genocchi polynomials is defined by means of polylogarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which are used to express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials that possesses a double generating function is introduced. Finally, the type 2 Apostolpoly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.
Keywords
Genocchi polynomials; Bernoulli polynomials; Frobenius polynomials; Appell polynomials; polylogarithm; polyexponential function; Apostol-type polynomials;
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  • Reference
1 L. Comtet, Advanced Combinatorics, revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
2 B. Kurt, Some identities for the generalized poly-Genocchi polynomials with the parameters a, b and c, J. Math. Anal. 8 (2017), no. 1, 156-163.
3 T. Nahid, P. Alam, and J. Choi, Truncated-exponential-based Appell-type Changhee polynomials, Symmetry 12 (2020), 1588. https://doi.org/10.3390/sym12101588   DOI
4 T. Usman, M. Aman, O. Khan, K. S. Nisar, and S. Araci, Construction of partially degenerate Laguerre-Genocchi polynomials with their applications, AIMS Math. 5 (2020), no. 5, 4399-4411. https://doi.org/10.3934/math.2020280   DOI
5 N. Khan, T. Usman, and J. Choi, A new generalization of Apostol-type Laguerre-Genocchi polynomials, C. R. Math. Acad. Sci. Paris 355 (2017), no. 6, 607-617. https://doi.org/10.1016/j.crma.2017.04.010   DOI
6 Y. He, Some new results on products of the Apostol-Genocchi polynomials, J. Comput. Anal. Appl. 22 (2017), no. 4, 591-600.
7 M. Kaneko, Poly-Bernoulli numbers, J. Theor. Nombres Bordeaux 9 (1997), no. 1, 221-228.   DOI
8 J. Choi, N. Khan, T. Usman, and M. Aman, Certain unified polynomials, Integral Transforms Spec. Funct. 30 (2019), no. 1, 28-40. https://doi.org/10.1080/10652469.2018.1534   DOI
9 Y. He, S. Araci, H. M. Srivastava, and M. Acikgoz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials, Appl. Math. Comput. 262 (2015), 31-41. https://doi.org/10.1016/j.amc.2015.03.132   DOI
10 H. Jolany, M. R. Darafsheh, and R. E. Alikelaye, Generalizations on poly-Bernoulli numbers and polynomials, Int. J. Math. Combin. 2 (2010), 7-14.
11 N. U. Khan, T. Usman, and M. Aman, Generating functions for Legendre-based poly-Bernoulli numbers and polynomials, Honam Math. J. 39 (2017), no. 2, 217-231.   DOI
12 N. Khan, T. Usman, and J. Choi, Certain generating function of Hermite-BernoulliLaguerre polynomials, Far East J. Math. Sci. 101 (2017), no. 4, 893-908.
13 N. Khan, T. Usman, and J. Choi, A new class of generalized polynomials associated with Laguerre and Bernoulli polynomials, Turkish J. Math. 43 (2019), no. 1, 486-497. https://doi.org/10.3906/mat-1811-56   DOI
14 N. Khan, T. Usman, and K. S. Nisar, A study of generalized Laguerre poly-Genocchi polynomials, Mathematics 7 (2019), 219. doi.org/10.3390/math7030219.   DOI
15 T. Kim, Y. S. Jang, and J. J. Seo, A note on poly-Genocchi numbers and polynomials, Appl. Math. Sci. 8 (2014), 4775-4781. http://dx.doi.org/10.12988/ams.2014.46465   DOI
16 Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transforms Spec. Funct. 20 (2009), no. 5-6, 377-391. https://doi.org/10.1080/10652460802564324   DOI
17 D. S. Kim, D. V. Dolgy, T. Kim, and S. Rim, Some formulae for the product of two Bernoulli and Euler polynomials, Abstr. Appl. Anal. 2012 (2012), Art. ID 784307, 15 pp. https://doi.org/10.1155/2012/784307   DOI
18 T. Kim, S. Rim, D. V. Dolgy, and S. Lee, Some identities of Genocchi polynomials arising from Genocchi basis, J. Inequal. Appl. 2013 (2013), 43, 6 pp. https://doi.org/10.1186/1029-242X-2013-43   DOI
19 D. W. Lee, On multiple Appell polynomials, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2133-2141. https://doi.org/10.1090/S0002-9939-2010-10648-2   DOI
20 L. Toscano, Polinomi ortogonali o reciproci di ortogonali nella classe di Appell, Matematiche (Catania) 11 (1956), 168-174.
21 D. S. Kim and T. Kim, A note on polyexponential and unipoly functions, Russ. J. Math. Phys. 26 (2019), no. 1, 40-49. https://doi.org/10.1134/S1061920819010047   DOI
22 A. Bayad and Y. Hamahata, Arakawa-Kaneko L-functions and generalized poly-Bernoulli polynomials, J. Number Theory 131 (2011), no. 6, 1020-1036. https://doi.org/10.1016/j.jnt.2010.11.005   DOI
23 J. Shohat, The Relation of the Classical Orthogonal Polynomials to the Polynomials of Appell, Amer. J. Math. 58 (1936), no. 3, 453-464. https://doi.org/10.2307/2370962   DOI
24 S. Araci, W. A. Khan, M. Acikgoz, C. Ozel, and P. Kumam, A new generaliztion of Apostol type Hermite-Genocchi polynomials and its applications, Springerplus 5 (2016), Art. ID 860.
25 S. Araci, E. S,en, and M. Acikgoz, Theorems on Genocchi polynomials of higher order arising from Genocchi basis, Taiwanese J. Math. 18 (2014), no. 2, 473-482. https://doi.org/10.11650/tjm.18.2014.3006   DOI
26 H. Jolany and R. B. Corcino, Explicit formula for generalization of poly-Bernoulli numbers and polynomials with a, b, c parameters, J. Class. Anal. 6 (2015), no. 2, 119-135. https://doi.org/10.7153/jca-06-10   DOI
27 N. U. Khan, T. Usman, and M. Aman, Certain generating funtion of generalized Apostol type Legendre-based polynomials, Note Mat. 37 (2017), no. 2, 21-43. https://doi.org/10.1285/i15900932v37n2p21   DOI
28 T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 23-28.
29 T. Kim, Y. S. Jang, and J. J. Seo, Poly-Bernoulli polynomials and their applications, Int. J. Math. Anal. 8 (2014), no. 30, 1495-1503.   DOI