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ON p-ADIC INTEGRAL FOR GENERALIZED DEGENERATE HERMITE-BERNOULLI POLYNOMIALS ATTACHED TO χ OF HIGHER ORDER

Khan, Waseem Ahmad;Haroon, Hiba

  • Received : 2018.06.10
  • Accepted : 2019.01.14
  • Published : 2019.03.25

Abstract

In the current investigation, we obtain the generating function for Hermite-based degenerate Bernoulli polynomials attached to ${\chi}$ of higher order using p-adic methods over the ring of integers. Useful identities, formulae and relations with well known families of polynomials and numbers including the Bernoulli numbers, Daehee numbers and the Stirling numbers are established. We also give identities of symmetry and additive property for Hermite-based generalized degenerate Bernoulli polynomials attached to ${\chi}$ of higher order. Results are supported by remarks and corollaries.

Keywords

Multivariate p-adic integral;Generalized degenerate Bernoulli polynomials;Hermite polynomials;Daehee numbers;Stirling numbers

References

  1. L. Carlitz, q-Bernoulli numbers and polynomials, Duke. Math. J., 15(4) (1948), 987-1000 . https://doi.org/10.1215/S0012-7094-48-01588-9
  2. L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math. 15 (1979), 51-88.
  3. H. Haroon, W. A. Khan, Degenerate Bernoulli numbers and polynomials associated with degenerate Hermite polynomials, Commun. Korean. Math. Soc. 33(2) (2018), 651-669.
  4. G. W. Jang, D. S. Kim, T. Kim, Degenerate Changhee numbers and polynomials of the second kind, Adv. Stud. Contemp. Math. (Kyungshang) 27 (2017), 609-624.
  5. G. W. Jang, J. Kwon, J. G. Lee, Some identities of degenerate Daehee numbers arising from nonlinear differential equation, Adv. Differ. Equ., 2017:206, (2017). https://doi.org/10.1186/s13662-017-1265-4
  6. W. A. Khan, Degenerate Hermite-Bernoulli numbers and polynomials of the second kind, Prespacetime J. 7 (2016), 1297-1305.
  7. T. Kim, Degenerate Cauchy numbers and polynomials of the second kind, Adv. Stud. Contemp. Math. (Kyungshang) 27 (2017), 441-449.
  8. T. Kim, On degenerate q-Bernoulli polynomials, Bull. Korean Math. Soc. 53 (2016), 1149-1156.
  9. T. Kim, On the degenerate Cauchy numbers and polynomials, Proc. Jangjeon Math. Soc. 18 (2015), 307-312.
  10. T. Kim, D. V. Dolgy, On the identities of Symmetric for degenerate Bernoulli polynomials of order r, Adv. Stud. Contemp. Math. (Kyungshang) 25 (2015), 457-462.
  11. T. Kim, D. V. Dolgy, D. S. Kim, Symmetric identities for degenerate generalized Bernoulli polynomials, J. Nonlinear. Sci. Appl. 9 (2016), 677-683. https://doi.org/10.22436/jnsa.009.02.30
  12. T. Kim, L. C. Jang, Y. H. Kim, K. W. Hwang, On the identities of symmetry for the generalized Bernoulli polynomials attached to  of higher order, J. Inequalities Appl. (2009), 7-pages.
  13. T. Kim, S. H. Rim, J. J. Seo, Degenerate Bernoulli numbers and polynomials of the second kind, Int. J. Math. Anal. 9 (2015), 1269-1278. https://doi.org/10.12988/ijma.2015.5260
  14. D. S. Kim, T. Kim, A note on Boole polynomials, Integral transforms Spec. Funct. 25 (2014), no. 8, 627-633. https://doi.org/10.1080/10652469.2014.891586
  15. D. S. Kim, T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. 7 (2013), 5969-5976.
  16. D. S. Kim, T. Kim, On degenerate Bell numbers and polynomials, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales 111 (2017), 435-446.
  17. D. S. Kim, T. Kim, Some identities of degenerate special polynomials, De Gruyter Open 13 (2015), 380-389.
  18. D. S. Kim, T. Kim, Some identities of Korobov-type polynomials associated with p-adic integral on Zp, Adv. Difference Eqs. 2015:282 (2015). https://doi.org/10.1186/s13662-015-0602-8
  19. D. S. Kim, T. Kim, D. V. Dolgy, A note on degenerate Bernoulli numbers and polynomials associated with p-adic invariant integral on Zp, Appl. Math. Comp. 259 (2015), 198-204. https://doi.org/10.1016/j.amc.2015.02.068
  20. D. S. Kim, T. Kim, D. V. Dolgy, T .Komatsu, Barne's type Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 25(1) (2015), 121-146.
  21. D. S. Kim, T. Kim, S. H. Lee, J. J. Seo, Higher order Daehee numbers and polynomials, Int. J. Math. Anal. 8 (2014), 273-283. https://doi.org/10.12988/ijma.2014.4118
  22. D. S. Kim, T. Kim, Higher order degenerate Euler polynomials , Appl. Math. Sci. 9 (2015), 57-73. https://doi.org/10.12785/amis/090108
  23. D. Lim, Some identities of degenerate Genocchi polynomials, Bull. Korean Math. Soc. 53 (2016), 569-579. https://doi.org/10.4134/BKMS.2016.53.2.569
  24. S. S. Pyo, Some identities of degenerate Fubini polynomials arising from differential equations, J. Nonlinear Sci. Appl. 11 (2018), 383-393. https://doi.org/10.22436/jnsa.011.03.07
  25. S. Roman, The umbral calculus, Pure App. Math., Academic Press, Inc., New York 111 (1984).
  26. A. Volkenborn, Ein P-adisches Integral und seine Anwendungen. I, Manuscripta Math. 7 (1972), 341-373. https://doi.org/10.1007/BF01644073
  27. A. Volkenborn, Ein P-adisches Integral und seine Anwendungen. II, Manuscripta Math. 12 (1974), 17-46. https://doi.org/10.1007/BF01166232