On unified Gould-Hopper based Apostol-type polynomials

Volume 24, Issue 4, pp 287--298 http://dx.doi.org/10.22436/jmcs.024.04.01

Publication Date: March 25, 2021 Submission Date: January 08, 2021 Revision Date: January 28, 2021 Accteptance Date: February 27, 2021

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Authors

Waseem Ahmad Khan - Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P. O. Box 1664, Al Khobar 31952, Saudi Arabia. Kottakkaran Sooppy Nisar - Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia. Mehmet Acikgoz - Department of Mathematics, Faculty of Arts and Sciences, University of Gaziantep, TR-27310 Gaziantep, Turkey. Ugur Duran - Department of the Basic Concepts of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkey. Abdallah Hassan Abusufian - Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia.

Abstract

In this paper, we consider unified Gould-Hopper based Apostol-type polynomials and investigate some of their formulas including several implicit summation formulae and some symmetric identities by the series manipulation method. Moreover, we acquire several new results for unified Gould-Hopper based Apostol-type polynomials using appropriate operational rules.

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Keywords

• Gould-Hopper polynomials
• monomiality principle
• unified Apostol-type polynomials
• summation formula
• symmetric identity

MSC

•  11B68
•  33C45
•  05A10

References

• [1] L. C. Andrews, Special functions for engineers and mathematicians, Macmillan Co., New York (1985)
  • MathSciNet
  • Google Scholar


• [2] T. M. Apostol, On the Lerch zeta function, Pacific J. Math., 1 (1951), 161--167
  • View Article
  • MathSciNet


• [3] P. Appell, Sur une classe de polynomes, Ann. Sci. Ecole Norm. Sup., 9 (1880), 119--144
  • View Article
  • MathSciNet


• [4] G. Bretti, C. Cesarno, P. E. Ricci, Laguerre-type exponentials and generalized Appell polynomials, Comput. Math. Appl., 48 (2004), 833--839
  • View Article
  • MathSciNet


• [5] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, Proc. Melfi Sch. Adv. Top. Math. Phys., 1 (2000), 147--164
  • MathSciNet
  • Google Scholar


• [6] H. W. Gould, A. T. Hopper, Operational formulas connected with two generalization of Hermite polynomials, Duke Math. J., 29 (1952), 51--63
  • View Article
  • MathSciNet


• [7] W. A. Khan, Some properties of the generalized Apostol type Hermite-based polynomials, Kyungpook Math. J., 55 (2015), 597--614
  • View Article
  • MathSciNet


• [8] S. Khan, M. W. Al-Saad, R. Khan, Laguerre-based Appell polynomials: properties and applications, Math. Comput. Modelling, 52 (2010), 247--259
  • View Article
  • MathSciNet


• [9] S. Khan, N. Raza, General-Appell polynomials within the context of monomiality principle, Int. J. Anal., 2013 (2013), 11 pages
  • View Article
  • MathSciNet


• [10] S. Khan, G. Yasmin, R. Khan, N. A. M. Hassan, Hermite-based Appell polynomials, properties and applications, J. Math. Anal. Appl, 351 (2009), 756--764
  • View Article
  • MathSciNet


• [11] V. Kurt, Some symmetry identities for the unified Apostol-type polynomials and multiple power sums, Filomat, 30 (2016), 583--592
  • View Article
  • MathSciNet


• [12] V. Kurt, On the unified family of generalized Apostol-type polinomials of higher order and multiple power sums, Filomat, 30 (2016), 929--935
  • View Article
  • Google Scholar


• [13] B. Kurt, Notes on unified q-Apostol-type polynomials, Filomat, 30 (2016), 921--927
  • View Article
  • MathSciNet


• [14] B. Kurt, S. Bilgic, Symmetry identities for the 2-variable unified Apostol-type polynomials, J. Inequal. Spec. Funct., 8 (2017), 96--103
  • View Article
  • MathSciNet


• [15] Q.-M. Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials, Math. Comp., 78 (2009), 2193--2208
  • View Article
  • MathSciNet


• [16] Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transforms Spec. Funct., 20 (2009), 377--391
  • View Article
  • MathSciNet


• [17] Q.-M. Luo, Some formulas for Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments, Appl. Anal. Discrete Math., 3 (2009), 336--346
  • View Article
  • MathSciNet


• [18] Q.-M. Luo, Fourier expansions and integral representations for the Genocchi polynomials, J. Integer Seq., 12 (2009), 1--9
  • View Article
  • Google Scholar


• [19] Q.-M. Luo, q-extensions for the Apostol-Genocchi polynomials, Gen. Math., 17 (2009), 113--125
  • View Article
  • MathSciNet


• [20] Q.-M. Luo, Extensions of the Genocchi polynomials and their Fourier expansions and integral representations, Osaka J. Math., 48 (2011), 291--309
  • View Article
  • MathSciNet


• [21] Q.-M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308 (2005), 290--302
  • View Article
  • MathSciNet


• [22] Q.-M. Luo, H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl., 51 (2006), 631--642
  • View Article
  • MathSciNet


• [23] Q.-M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput., 217 (2011), 5702--5728
  • View Article
  • MathSciNet


• [24] M. A. Ozarslan, Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials, Adv. Difference Equ., 2013 (2013), 13 pages
  • View Article
  • MathSciNet


• [25] M. A. Ozarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl., 62 (2011), 2452--2462
  • View Article
  • MathSciNet


• [26] H. Ozden, Generating function of the unified representation of the Bernoulli, Euler and Genocchi polynomials of higher order, AIP Conf. Proc., 1389 (2011), 349--352
  • View Article
  • Google Scholar


• [27] H. Ozden, Y. Simsek, H. M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl., 60 (2010), 2779--2287
  • View Article
  • MathSciNet


• [28] M. A. Pathan, W. A. Khan, Some implicit summation formulas and symmetric identities for the generalized Hermite-based polynomials, Acta Univ. Apulensis Math. Inform., 39 (2014), 113--136
  • View Article
  • MathSciNet


• [29] M. A. Pathan, W. A. Khan, Some implicit summation formulas and symmetric identities for the generalized Hermite-Bernoulli polynomials, Mediterr. J. Math., 12 (2015), 679--695
  • View Article
  • MathSciNet
  • Google Scholar


• [30] M. A. Pathan, W. A. Khan, A new class of generalized polynomials associated with Hermite and Euler polynomials, Mediterr. J. Math., 13 (2016), 913--928
  • View Article
  • MathSciNet


• [31] J. Sandor, B. Crstici, Handbook of number theory. II, Kluwer Academic Publishers, Dordrecht (2004)
  • View Article
  • MathSciNet


• [32] Y. Smirnov, A. Turbiner, , Errata: "Lie algebraic discretization of differential equations'', Modern Phys. Lett. A, 10 (1995), 1795--1802
  • View Article
  • MathSciNet


• [33] H. M. Srivastava, S. Araci, W. A. Khan, M. Acikgoz, A Note on the Truncated-Exponential Based Apostol-Type Polynomials, Symmetry, 11 (2019), 1--20
  • View Article
  • Google Scholar


• [34] H. M. Srivastava, H. L. Manocha, A treatise on generating functions, Ellis Horwood Limited. Co., New York (1984)
  • MathSciNet
  • Google Scholar


• [35] J. F. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math., 73 (1941), 333--366
  • View Article
  • MathSciNet