Generalized Gould-Hopper Polynomials

Yıl 2021, Cilt: 9 Sayı: 1, 102 - 111, 28.04.2021

Nejla Özmen Mustafa Topaloğlu

Öz

In this paper, we derive generating functions for the generalized Gould-Hopper polynomials in terms of the generalized Lauricella function by using series rearrangement techniques. Further, we derive the summation formulae for that polynomials by using different analytical means on its generating function or by using certain operational techniques. Also, generating functions and summation formulae for the polynomials related to generalized Gould-Hopper polynomials are obtained as applications of main results. In addition, we derive a theorem giving certain families of bilateral generating functions for the generalized Gould-Hopper polynomials. The results obtained here include various families of bilinear and bilateral generating functions, miscellaneous properties and also some special cases for these polynomials.

Anahtar Kelimeler

Generalized Gould-Hopper polynomials, Lauricella functions, summation formulae, generating function

Kaynakça

• [1] P. Appell, J. Kampe´ de Feriet,´ Fonctions hypergeom´etriques´ Polynomesˆ d’Hermite, Gauthier-Villars, Paris, 1926.
• [2] G. Bretti, P.E. Ricci, Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwan. J. Math., Vol. 8 (2004), 415-428.
• [3] G. Dattoli, S. Lorenzutta, C. Cesarano, Finite sums and generalized forms of Bernoulli polynomials, Rend. Math. Appl., Vol. 19 (1999), 385-391.
• [4] M. A. Pathan, W. A. Khan, Some implicit summation formulas and symmetric identities for the generalized Hermite based-polynomials, Acta Univ. Apulensis Math. Inform., Vol. 39 (2014), 113-136.
• [5] E. Horozov, Generalized Gould Hoper Polynomials, arXiv:1609.06157v1[math.CA], 2016.
• [6] S. Khan, M.W.M. Al-Saad, Summation formulae for Gould–Hopper generalized Hermite polynomials, Comp. Math. Appl., Vol. 61, No.6 (2011), 1536-1541.
• [7] G. Dattoli, Generalized polynomials, operational identities and their applications, J. Comput. Appl. Math., Vol. 118 (2000), 111-123.
• [8] H.M. Srivastava, G. B. Djordjevic, Some generating functions and other properties associated with the generalized Hermite and related polynomials, Integral Transforms Spec. Func., Vol. 22, No. 12 (2011), 895-906.
• [9] G.B. Djordjevic,´ Polynomials related to generalized Chebyshev polynomials, Filomat, Vol. 23, No. 3 (2009), 279-290.
• [10] B. Yilmaz, M. A. Ozarslan, Differential equations for the extended 2D Bernoulli and Euler polynomials, Adv. Difference Equ., Vol. 2013 (2013), 16 pages.
• [11] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.
• [12] N. Ozmen and E. Erkus-Duman, Some results for a family of multivariable polynomials, AIP Conference Proceedings, Vol. 1558 (2013), 1124-1127.
• [13] N. Ozmen and E. Erkus-Duman, Some families of generating functions for the generalized Cesaro´ polynomials, J. Comput. Anal. Appl., Vol. 25 (2018), 670-683.
• [14] M. Lahiri, On a generalization of Hermite polynomials, Proc. Amer. Math. Soc. Vol. 27 (1971), 117–121.