Article ID: MTJPAM-D-24-00056

Title: Generating relations associated with the extended bivariate Srivastava type polynomials


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-24-00056; Volume 6 / Issue 3 / Year 2024, Pages 208-216

Document Type: Research Paper

Author(s): ‪Talha Usman a , Nabiullah Khan b , Mohd Aman c , Junesang Choi d

aPreparatory Studies Center, Mathematics and Computing Unit, University of Technology and Applied Sciences, Sur-411, Sultanate of Oman

bDepartment of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India

cDepartment of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India

dDepartment of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea

Received: 20 April 2024, Accepted: 14 May 2024, Published: 9 June 2024

Corresponding Author: Junesang Choi (Email address: junesang@dongguk.ac.kr)

Full Text: PDF

Abstract

Since their introduction in 1972, the Srivastava polynomials and their extensions have undergone extensive investigation and diverse applications. This paper aims to establish various families of bilateral and mixed multilateral generating relations concerning the extended bivariate Srivastava type polynomials and arbitrary sequences of polynomials or functions. The outcomes presented herein, characterized by their generality, are poised to yield numerous generating relations encompassing established polynomials and sequences of functions, supplemented by several explicit demonstrations.

Keywords: Bilateral generating relations, multilateral generating relations, extended Srivastava type polynomials

References:
  1. A. Altin, E. Erkuş and M. A. Özarslan, Families of linear generating function of polynomials in two variables, Integral Transforms Spec. Funct. 17 (5), 315–320, 2006; https://doi.org/10.1080/10652460600568162.
  2. A. A. Al Gonah, Some generating relations involving 2-variable Laguerre polynomials and extended Srivastava polynomials, Konuralp J. Math. 2 (3), 131–139, 2015.
  3. P. Appell and J. Kampé de Fériet, Fonctions hypergétriques et hypersphriques, polyn$\hat{o}$mes d’Hermite, Gauthier-villars, Paris 1926.
  4. A. Bayad and Y. Hamahata, Arakawa–Kaneko L-functions and generalized poly-Bernoulli polynomials, J. Number Theory 131, 1020–1036, 2011; https://doi.org/10.1016/j.jnt.2010.11.005.
  5. J. Choi, A generalization of Gottlieb polynomials in several variables, Appl. Math. Lett. 25, 43–46, 2012.
  6. J. Choi, Notes on formal manipulations of double series, Commun. Korean Math. Soc. 18 (4), 781–789, 2003.
  7. G. Dattoli, Generalized polynomials, operational identities and their applications, J. Comput. Appl. Math. 118 (1-2), 111–123, 2000.
  8. G. Dattoli, Hermite Bessel and Laguerre Bessel function: A by product of the monomiality principle, In: Advanced Special Function and Applications, Melfi, 1999; In: Proc. Melfi Sch. Adv. Top. Math. Phys. 1, 147–164, 2000.
  9. G. Dattoli, S. Lorenzutta, A. M. Mancho and A. Torre, Generalized polynomials and associated operational identities, J. Comput. Appl. Math. 108 (1-2), 209–218, 1999.
  10. G. Dattoli, P. E. Ricci and C. Cesarano, The Lagrange polynomials, the associated generalizations and the umbral calculus, Integral Transforms Spec. Funct. 14 (2), 181–186, 2003; https://doi.org/10.1080/1065246031000098186.
  11. G. Dattoli, P. E. Ricci and C. Cesarano, A note on Legendre polynomials, Int. J. Nonlinear Sci. Numer. Simul. 2 (4), 365–370, 2001.
  12. G. Dattoli, A. Torre and S. Lorenzutta, Operational identities and properties of ordinary and generalized special function, J. Math. Anal. Appl. 236 (2), 399–414, 1999.
  13. B. González, J. Matera and H. M. Srivastava, Some q-generating function and associated generalized hypergeometric polynomials, Math. Comput. Model. 34 (1-2), 133–175, 2001.
  14. N. U. Khan, M. Aman, T. Usman and J. Choi, Legendre-Gould Hopper-based Sheffer polynomials and operational methods, Symmetry 12 (12), 2020; Article ID: 2051, https://doi.org/10.3390/sym12122051.
  15. N. U. Khan and T. Usman, Certain generating functions of Hermite-Bernoulli-Legendre polynomials, Ufa Math. J. 10 (2), 118–126, 2018.
  16. N. U. Khan and T. Usman, A new class of Laguerre-based generalized Apostol polynomials, Fasc. Math. 57, 67–89, 2016.
  17. N. U. Khan and T. Usman, A new class of Laguerre-based poly-Euler and multi poly-Euler polynomials, J. Anal. Number Theory 4 (2), 113–120, 2016.
  18. N. U. Khan and T. Usman, A new class of Laguerre poly-Bernoulli numbers and polynomials, Adv. Stud. Contemp. Math. 27 (2), 229–241, 2017.
  19. N. U. Khan, T. Usman and M. Aman, Generating functions for Legendre-based poly-Bernoulli numbers and polynomials, Honam Math. J. 39 (2), 217–231, 2017; https://doi.org/10.5831/HMJ.2017.39.2.217.
  20. N. U. Khan, T. Usman and J. Choi, Certain generating function of Hermite-Bernoulli-Laguerre polynomials, Far East J. Math. Sci. 101 (4), 893–908, 2017.
  21. N. U. Khan, T. Usman and J. Choi, A New generalization of Apostol type Laguerre-Genocchi polynomials, C. R. Acad. Sci. Paris, Ser. I 355, 607– 617, 2017.
  22. N. U. Khan, T. Usman and J. Choi, A new class of generalized polynomials involving Laguerre and Euler polynomials, Hacet. J. Math. Stat. 50 (1), 1–13, 2021; https://doi.org/10.15672/hujms.555416.
  23. B. Kurt, A note on the Apostol type q-Frobenius-Euler polynomials and generalizations of the Srivastava-Pinter addition theorems, Filomat 30 (1), 65–72, 2016; DOI: 10.2298/FIL1601065K.
  24. B. Kurt and Y. Simsek, Notes on generalization of the Bernoulli type polynomials, Appl. Math. Comput. 218, 906–911, 2011.
  25. Q.-M. Luo, B. N. Guo, F. Qi and L. Debnath, Generalization of Bernoulli numbers and polynomials, Int. J. Math. Math. Sci. 59, 3769-–3776, 2003.
  26. T. Nahid, P. Alam and J. Choi, Truncated-exponential-based Appell-type Changhee polynomials, Symmetry 12 (10), 2020; Article ID: 1588; https://doi.org/10.3390/sym12101588.
  27. M. A. Özarslan, Some families of generating function for the extended Srivastava polynomials, Appl. Math. Comput. 218 (3), 959–964, 2011.
  28. H. Ozden and Y. Simsek, Unified representation of the family of L-functions, J. Inequal. Appl. 2013, 2013; Article ID: 64, https://doi.org/10. 1186/1029-242X-2013-64.
  29. E. Özergin, M. A. Özarslan and H. M. Srivastava, Some families of generating functions for a class of bivariate polynomials, Math. Comput. Model. 50, 1113–1120, 2009.
  30. H. M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math. 14, 1–6, 1972.
  31. H. M. Srivastava and J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  32. H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.

Cite this article

How to cite this article: T. Usman, N. Khan, M. Aman and J. Choi, Generating relations associated with the extended bivariate Srivastava type polynomials, Montes Taurus J. Pure Appl. Math. 6 (3), 208-216, 2024; Article ID: MTJPAM-D-24-00056.