Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME IDENTITIES ASSOCIATED WITH 2-VARIABLE TRUNCATED EXPONENTIAL BASED SHEFFER POLYNOMIAL SEQUENCES

  • Choi, Junesang (Department of Mathematics Dongguk University) ;
  • Jabee, Saima (Department of Applied Sciences and Humanities Faculty of Engineering and Technology Jamia Millia Islamia (A Central University)) ;
  • Shadab, Mohd (Department of Applied Sciences and Humanities Faculty of Engineering and Technology Jamia Millia Islamia (A Central University))
  • Received : 2019.03.19
  • Accepted : 2019.10.14
  • Published : 2020.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

Since Sheffer introduced the so-called Sheffer polynomials in 1939, the polynomials have been extensively investigated, applied and classified. In this paper, by using matrix algebra, specifically, some properties of Pascal and Wronskian matrices, we aim to present certain interesting identities involving the 2-variable truncated exponential based Sheffer polynomial sequences. Also, we use the main results to give some interesting identities involving so-called 2-variable truncated exponential based Miller-Lee type polynomials. Further, we remark that a number of different identities involving the above polynomial sequences can be derived by applying the method here to other combined generating functions.

Keywords

References

  1. L. Aceto and I. Cacao, A matrix approach to Sheffer polynomials, J. Math. Anal. Appl. 446 (2017), no. 1, 87-100. https://doi.org/10.1016/j.jmaa.2016.08.038
  2. G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999. https://doi.org/10.1017/CBO9781107325937
  3. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Co., New York, 1985.
  4. S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput. 233 (2014), 599-607. https://doi.org/10.1016/j.amc.2014.01.013
  5. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach Science Publishers, New York, 1978.
  6. G. Dattoli, C. Cesarano, and D. Sacchetti, A note on truncated polynomials, Appl. Math. Comput. 134 (2003), no. 2-3, 595-605. https://doi.org/10.1016/S0096-3003(01)00310-1
  7. G. Dattoli, S. Lorenzutta, and D. Sacchetti, Integral representations of new families of polynomials, Ital. J. Pure Appl. Math. No. 15 (2004), 19-28.
  8. G. Dattoli, M. Migliorati, and H. M. Srivastava, A class of Bessel summation formulas and associated operational methods, Fract. Calc. Appl. Anal. 7 (2004), no. 2, 169-176.
  9. G. Dattoli, M. Migliorati, and H. M. Srivastava, Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials, Math. Comput. Modelling 45 (2007), no. 9-10, 1033-1041. https://doi.org/10.1016/j.mcm.2006.08.010
  10. A. Di Bucchianico and D. Loeb, A selected survey of umbral calculus, Electron. J. Combin. 2 (1995), Dynamic Survey 3, 28 pp.
  11. L. G. Garza, L. E. Garza, F. Marcellan, and N. C. Pinzon-Cortes, A matrix approach for the semiclassical and coherent orthogonal polynomials, Appl. Math. Comput. 256 (2015), 459-471. https://doi.org/10.1016/j.amc.2015.01.071
  12. 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
  13. M. X. He and P. E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002), no. 2, 231-237. https://doi.org/10.1016/S0377-0427(01)00423-X
  14. S. Khan, G. Yasmin, and N. Ahmad, On a new family related to truncated exponential and Sheffer polynomials, J. Math. Anal. Appl. 418 (2014), no. 2, 921-937. https://doi.org/10.1016/j.jmaa.2014.04.028
  15. S. Khan, G. Yasmin, R. Khan, and N. A. M. Hassan, Hermite-based Appell polynomials: properties and applications, J. Math. Anal. Appl. 351 (2009), no. 2, 756-764. https://doi.org/10.1016/j.jmaa.2008.11.002
  16. D. S. Kim and T. Kim, A matrix approach to some identities involving Sheffer polynomial sequences, Appl. Math. Comput. 253 (2015), 83-101. https://doi.org/10.1016/j.amc.2014.12.048
  17. D. H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly 95 (1988), no. 10, 905-911. https://doi.org/10.2307/2322383
  18. E. D. Rainville, Special Functions, The Macmillan Co., New York, 1960.
  19. S. Roman, The theory of the umbral calculus. I, J. Math. Anal. Appl. 87 (1982), no. 1, 58-115. https://doi.org/10.1016/0022-247X(82)90154-8
  20. S. Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, Inc., New York, 1984.
  21. S. M. Roman and G.-C. Rota, The umbral calculus, Advances in Math. 27 (1978), no. 2, 95-188. https://doi.org/10.1016/0001-8708(78)90087-7
  22. K. Rosen, Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boca Raton, Florida, 2000.
  23. I. M. Sheffer, A differential equation for Appell polynomials, Bull. Amer. Math. Soc. 41 (1935), no. 12, 914-923. https://doi.org/10.1090/S0002-9904-1935-06218-4
  24. I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590-622. http://projecteuclid.org/euclid.dmj/1077491414 https://doi.org/10.1215/S0012-7094-39-00549-1
  25. Y. Yang and C. Micek, Generalized Pascal functional matrix and its applications, Linear Algebra Appl. 423 (2007), no. 2-3, 230-245. https://doi.org/10.1016/j.laa.2006.12.014
  26. Y. Yang and H. Youn, Appell polynomial sequences: a linear algebra approach, JP J. Algebra Number Theory Appl. 13 (2009), no. 1, 65-98.
  27. H.-Y. Youn and Y.-Z. Yang, Differential equation and recursive formulas of Sheffer polynomial sequences, Discrete Math. 2011 (2011), Article ID 476462, 1-16.