On the three families of extended Laguerre-based Apostol-type polynomials
Keywords:Gould-Hopper polynomials, Laguerre type exponential, Unified Apostol type polynomials, Extended Laguerre-based Apostol type polynomials, Fubini polynomials, Bell polynomials
In this paper, we introduce a new class of generalized extended Laguerre-based Apostol-type-Bernoulli, Apostol-type-Euler and Apostoltype-Genocchi polynomials. These Apostol type polynomials are used to connect Fubini-Hermite and Bell-Hermite polynomials and to find new representations. We derive some implicit summation formulae and symmetric identities for these families of special functions by applying the generating functions.
L. C. Andrews, Special functions for engineers and mathematicians. New York, NY: Macmillan, 1985.
E. T. Bell, “Exponential polynomials”, Annals of mathematical, vol. 35, no. 2, pp. 258-277, 1934, doi: 10.2307/1968431
B. C. Berndt, Ramanujan's notebooks, vol. 1. New York, NY: Springer, 1985, doi: 10.1007/978-1-4612-1088-7
A. E. Bernarini, G. K. Dattoli, and P. A. Ricci, “L-exponentials and higher order Laguerre polynomials”, in Proceeding of the Fourth Annual Conference of the Society for Special Functions and their Applications, Jaipur (India), March 4-6, 2003.
K. N. Boyadzhiev, “A series transformation formula and related polynomials”, International journal mathematical sciences, vol. 2005, Art ID. 792107, 2005, doi: 10.1155/IJMMS.2005.3849
K. N. Boyadzhiev, “Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials”, Advances and applications in discrete mathematics, vol. 1, no. 2, pp. 109-122, 2008. [On line]. Available: https://bit.ly/3rqe1Z7
G. Bretti, C. Cesarano, and P. Ricci, “Laguerre-type exponentials and generalized Appell polynomials”, Computers & mathematics with applications, vol. 48, no. 5-6, pp. 833–839, 2004, doi: 10.1016/j.camwa.2003.09.031
Y. B. Cheikh, “Some results on quasi-monomiality”, Applied mathematics and computation, vol. 141, no. 1, pp. 63–76, 2003, doi: 10.1016/s0096-3003(02)00321-1
C. Cesarano, B. Germano, and P. E. Ricci, “Laguerre-type Bessel functions”, Integral transforms and special functions, vol. 16, no. 4, pp. 315–322, 2005, doi: 10.1080/10652460412331270629
G. Dattoli and P. E. Ricci, “Multi-index polynomials and applications to statistical problems”, Nuovo Cimento della Societa Italiana di Fisica B, vol. 118, no. 6, pp. 625-633, 2003.
G. Dattoli, A. Arena, P. E. and Ricci, “Laguerrian eigenvalues problems and Wright functions”, Mathematical computer modeling, vol. 40, no. 7-8, pp. 877-881, 2004, doi: 10.1016/j.mcm.2004.10.017
G. Dattoli, S. Lorenzutta, and C. Cesarano, “Finite sums and generalized forms of Bernoulli polynomials”, Rendiconti di mathematica, vol. 19, pp. 385-391, 1999.
G. Dattoli, “Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle,” in Proceeding. of the Workshop on Advanced Special Functions and Applications, Melfi (PZ) Italia 9-12 May 1999, 2006, pp. 147–164, doi: 10.4399/97888799926578
G. Dattoli and P. E. Ricci, “Laguerre-type exponentials and the relevant L-circular and L-hyperbolic functions”, Georgian mathematical journal, vol. 10, no. 3, 2003, pp. 481-494, 2003, doi: 10.1515/GMJ.2003.481
H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite polynomials”, Duke mathematical journal, vol. 29, no. 1, 1962, pp. 51-62, doi: 10.1215/S0012-7094-62-02907-1
B. Kurt, “A further generalization of Bernoulli polynomials and on the 2D-Bernoulli polynomials B2n (x, y) ”, Applied mathematical sciences, vol. 4, no. 47, pp. 2315-2322, 2010. [On line]. Available: https://bit.ly/36HyKPZ
M. A. Özarslan, “Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials”, Advances differential equations, vol. 2013, Art ID. 116, 2013, doi: 10.1186/1687-1847-2013-116
M. A. Özarslan, “Unified Apostol-Bernoulli, Euler and Genocchi polynomials”, Computer mathematical applied, vol. 62, no. 1, pp. 2452-2462, 2011, doi: 10.1016/j.camwa.2011.07.031
H. Özden, “Unification of generating functions of the Bernoulli, Euler and Genocchi numbers and polynomials”, in ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, vol. 1281, T. E. Simos, G. Psihoyios , and C. Tsitouras, Eds. 2010, doi: 10.1063/1.3497848
H. Özden, “Generating function of the unified representation of the Bernoulli, Euler and Genocchi polynomials of higher order”, in Numerical analysis and applied mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, vol. 1389, T. E. Simos, G. Psihoyios, C. Tsitouras, and Z. Anastassi, Eds. 2011, doi: 10.1063/1.3636736
H. Özden, Y. Simsek, and H. Srivastava, “A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials”, Computers & mathematics with applications, vol. 60, no. 10, pp. 2779–2787, 2010, doi: 10.1016/j.camwa.2010.09.031
M. A. Pathan and W. Khan, “Some implicit summation formulas and symmetric identities for the generalized Hermite-Based polynomials”, Acta Universitatis Apulensis, vol. 39, pp. 113–136, 2014, doi: 10.17114/j.aua.2014.39.11
M. A. Pathan and W. A. Khan, “Some implicit summation formulas and symmetric identities for the generalized Hermite–Bernoulli polynomials”, Mediterranean journal of mathematics, vol. 12, no. 3, pp. 679–695, 2014, doi: 10.1007/s00009-014-0423-0
M. A. Pathan and W. A. Khan, “A new class of generalized polynomials associated with Hermite and Euler polynomials”, Mediterranean journal of mathematics, vol. 13, no. 3, pp. 913–928, 2015, doi: 10.1007/s00009-015-0551-1
M. A. Pathan, “A new class of generalized Hermite-Bernoulli polynomials”, Georgian mathematical journal, vol. 19, no. 3, pp. 559-573, 2012, doi: 10.1515/gmj-2012-0019
Y. Simsek, “Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications”, Fixed point theory and applications, vol. 2013, Art ID. 87, 2013, doi: 10.1186/1687-1812-2013-87
How to Cite
Copyright (c) 2021 M. A. Pathan, Waseem A. Khan
This work is licensed under a Creative Commons Attribution 4.0 International License.