Formulas for Poisson–Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach

  • Yilmaz Simsek
Original Paper


The main propose of this article is to investigate and modify Hermite type polynomials, Milne-Thomson type polynomials and Poisson–Charlier type polynomials by using generating functions and their functional equations. By using functional equations of the generating functions for these polynomials, we not only derive some identities and relations including the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Poisson–Charlier polynomials, the Milne-Thomson polynomials and the Hermite polynomials, but also study some fundamental properties of these functions and polynomials. Moreover, we survey orthogonality properties of these polynomials. Finally, by applying another method which is related to p-adic integrals, we derive some formulas and combinatorial sums associated with some well-known numbers and polynomials.


Generating function Functional equation Orthogonal polynomials Bernoulli numbers and polynomials Euler numbers and polynomials Stirling numbers Milne-Thomson polynomials Poisson–Charlier polynomials Hermite polynomials Special functions Special numbers and polynomials p-adic integral 

Mathematics Subject Classification

05A15 11B83 11B68 11S80 33C45 



The present paper was supported by Scientific Research Project Administration of Akdeniz University.


  1. 1.
    Bayad, A., Simsek, Y., Srivastava, H.M.: Some array type polynomials associated with special numbers and polynomials. Appl. Math. Compute 244, 149–157 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bretti, G., Ricci, P.E.: Multidimensional extensions of the Bernoulli and Appell polynomials. Taiwan. J. Math. 8, 415–428 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cakic, N.P., Milovanovic, G.V.: On generalized Stirling numbers and polynomials. Mathematica Balkanica 18, 241–248 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Campos, R.G., Marcellán, F.: Quadratures and integral transforms arising from generating functions. Appl. Math. Comput. 297, 8–18 (2017)MathSciNetGoogle Scholar
  5. 5.
    Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions (Translated from the French by J. W. Nienhuys). Reidel, Dordrecht and Boston (1974)Google Scholar
  6. 6.
    Degroot, M.H., Schervish, M.J.: Probability and Statistics, 4th edn. Addison-Wesley, Boston (2012)Google Scholar
  7. 7.
    Dere, R., Simsek, Y.: Hermite base Bernoulli type polynomials on the umbral algebra. Russ. J. Math. Phys. 22(1), 1–5 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Djordjevic, G.B., Milovanovic, G.V.: Special classes of polynomials. University of Nis, Faculty of Technology Leskovac (2014)Google Scholar
  9. 9.
    Gautschi, W.: Orthogonal polynomials: computation and approximation. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  10. 10.
    Jang, L.C., Kim, T.: A new approach to \(q\)-Euler numbers and polynomials. J. Concr. Appl. Math. 6, 159–168 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Khan, N., Usman, T., Choi, J.: A new class of generalized polynomials. Turk. J. Math. (to appear)
  12. 12.
    Kim, D., Stanton, D., Zeng, J.: The combinatorics of the Al-Salam-Chihara-Charlier polynomials (preprint)Google Scholar
  13. 13.
    Kim, D.S., Kim, T., Rim, S.-H., Lee, S.H.: Hermite polynomials and their applications associated with bernoulli and euler numbers. Discret. Dyn. Nat. Soc. 2012, 13 (2012).
  14. 14.
    Kim, D.S., Kim, T., Seo, J.: A note on Changhee numbers and polynomials. Adv. Stud. Theor. Phys. 7, 993–1003 (2013)CrossRefGoogle Scholar
  15. 15.
    Kim, D.S., Kim, T.: Daehee numbers and polynomials. Appl. Math. Sci. (Ruse) 7(120), 5969–5976 (2013)MathSciNetGoogle Scholar
  16. 16.
    Kim, T.: \(q\)-Euler numbers and polynomials associated with \(p\)-adic \(q\)-integral and basic \(q\)-zeta function. Trend Math. Inf. Center Math. Sci. 9, 7–12 (2006)Google Scholar
  17. 17.
    Kim, T.: \(q\)-Volkenborn integration. Russ. J. Math. Phys. 19, 288–299 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kim, T.: On the \(q\)-extension of Euler and Genocchi numbers. J. Math. Anal. Appl. 326(2), 1458–1465 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kim, T., Rim, S.-H., Simsek, Y., Kim, D.: On the analogs of Bernoulli and Euler numbers, related identities and zeta and \(l\)-functions. J. Korean Math. Soc. 45(2), 435–453 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric orthogonal polynomials and their \(q\)-analogues. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Lorentz, G.G.: Bernstein Polynomials. Chelsea Pub. Comp, New York (1986)zbMATHGoogle Scholar
  22. 22.
    Luo, Q.M., Srivastava, H.M.: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Compute 217, 5702–5728 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Milne-Thomson, L.M.: Two classes of generalized polynomials. Proc. Lond. Math. Soc. s2–35(1), 514–522 (1933)Google Scholar
  24. 24.
    Milovanovic, G.V.: Chapter 23: Computer algorithms and software packages. In: Brezinski, C., Sameh, A. (eds.) Walter Gautschi: Selected Works and Commentaries, vol. 3, pp. 9–10. Birkhuser, Basel (2014)CrossRefGoogle Scholar
  25. 25.
    Milovanovic, G.V.: Chapter 11: Orthogonal polynomials on the real line. In: Brezinski, C., Sameh, A. (eds.) Walter Gautschi: Selected Works and Commentaries, vol. 2, pp. 3–16. Birkhuser, Basel (2014)CrossRefGoogle Scholar
  26. 26.
    Ozden, H., Simsek, Y.: Modification and unification of the Apostol-type numbers and polynomials and their applications. Appl. Math. Compute 235, 338–351 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ozmen, N., Erkus-Duman, E.: On the Poisson-Charlier polynomials. Serdica Math. J. 41, 457–470 (2015)MathSciNetGoogle Scholar
  28. 28.
    Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)zbMATHGoogle Scholar
  29. 29.
    Riordan, J.: Introduction to Combinatorial Analysis. Princeton University Press, Princeton (1958)zbMATHGoogle Scholar
  30. 30.
    Roman, S.: The Umbral Calculus. Dover Publ. Inc., New York (2005)zbMATHGoogle Scholar
  31. 31.
    Schikhof, W.H.: Ultrametric Calculus: An Introduction to \(p\)-adic Analysis. Cambridge Studies in Advanced Mathematics 4. Cambridge University Press, Cambridge (1984)Google Scholar
  32. 32.
    Simsek, Y.: Special functions related to Dedekind-type DC-sums and their applications. Russ. J. Math. Phys. 17(4), 495–508 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Simsek, Y.: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 2013(87), 1–28 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Simsek, Y.: Special numbers on analytic functions. Appl. Math. 5, 1091–1098 (2014)CrossRefGoogle Scholar
  35. 35.
    Simsek, Y.: Generating functions for the Bernstein type polynomıals: a new approach to derving identities and applications for the polynomials. Hacet. J. Math. Stat. 43(1), 1–14 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Simsek, Y.: Complete sum of products of \((h;q)\)-extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 16, 1331–1348 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Simsek, Y.: Twisted \((h;q)\)-Bernoulli numbers and polynomials related to twisted \((h;q)\)-zeta function and L-function. J. Math. Anal. Appl. 324, 790–804 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Srivastava, H.M.: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Cambridge Philos. Soc. 129, 77–84 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Srivastava, H.M.: Some generalizations and basic (or \(q\)-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inform. Sci. 5, 390–444 (2011)MathSciNetGoogle Scholar
  40. 40.
    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Wiley, New York (1984)zbMATHGoogle Scholar
  41. 41.
    Srivastava, H.M., Choi, J.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier Science, Amsterdam, London and New York (2012)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsFaculty of Science University of AkdenizAntalyaTurkey

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