Generating Functions for Special Polynomials and Numbers Including Apostol-Type and Humbert-Type Polynomials

  • Gulsah Ozdemir
  • Yilmaz Simsek
  • Gradimir V. MilovanovićEmail author


The aim of this paper is to give generating functions and to prove various properties for some new families of special polynomials and numbers. Several interesting properties of such families and their connections with other polynomials and numbers of the Bernoulli, Euler, Apostol–Bernoulli, Apostol–Euler, Genocchi and Fibonacci type are presented. Furthermore, the Fibonacci-type polynomials of higher order in two variables and a new family of special polynomials \((x,y)\mapsto \mathbb {G}_{d}(x,y;k,m,n)\), including several particular cases, are introduced and studied. Finally, a class of polynomials and corresponding numbers, obtained by a modification of the generating function of Humbert’s polynomials, is also considered.


Generating function Fibonacci polynomials Humbert polynomials Bernoulli polynomials and numbers Euler polynomials and numbers Apostol–Bernoulli polynomials and numbers Apostol–Euler polynomials and numbers Genocchi polynomials Stirling numbers 

Mathematics Subject Classification

05A15 11B39 11B68 11B73 11B83 


  1. 1.
    Apostol, T.M.: On the Lerch Zeta function. Pacific J. Math. 1(2), 161–167 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayad, A., Simsek, Y., Srivastava, H.M.: Some array type polynomials associated with special numbers and polynomials. Appl. Math. Comput. 244, 149–157 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cakić, N.P., Milovanović, G.V.: On generalized Stirling numbers and polynomials. Math. Balkanica 18, 241–248 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht and Boston (1974)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dere, R., Simsek, Y., Srivastava, H.M.: A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra. J. Number Theory 133, 3245–3263 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Djordjević, G.B.: Polynomials related to generalized Chebyshev polynomials. Filomat 23(3), 279–290 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Djordjević, G.B., Milovanović, G.V.: Special Classes of Polynomials. University of Niš, Leskovac (2014)Google Scholar
  8. 8.
    Gegenbauer, L.: Zur Theorie der Functionen \(C_{n}^{v}(x)\). Osterreichische Akademie der Wissenschaften Mathematisch Naturwissen Schaftliche Klasse Denkscriften 48, 293–316 (1884)zbMATHGoogle Scholar
  9. 9.
    Gould, H.W.: Inverse series relations and other expansions involving Humbert polynomials. Duke Math. J. 32(4), 697–712 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Henry, W., He, T.: Characterization of \((c)\)-Riordan arrays, Gegenbauer–Humbert-type polynomial sequences, and \((c)\)-Bell polynomials. J. Math. Res. Appl. 33(5), 505–527 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Horadam, A.F.: Genocchi polynomials. In: Applications of Fibonacci Numbers, vol. 4 (Winston-Salem, NC, 1990), pp. 145–166. Kluwer Acad. Publ., Dordrecht (1991)Google Scholar
  12. 12.
    Humbert, P.: Some extensions of Pincherle’s polynomials. Proc. Edinburgh Math. Soc. 1(39), 21–24 (1921)Google Scholar
  13. 13.
    Kim, T.: On the \(q\)-Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 17, 9–15 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lu, D.-Q., Srivastava, H.M.: Some series identities involving the generalized Apostol type and related polynomials. Comput. Math. Appl. 62, 3591–2602 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Luo, Q.-M.: On the Apostol–Bernoulli polynomials. Cent. Eur. J. Math. 2(4), 509–515 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Luo, Q.-M., Srivastava, H.M.: Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 217, 5702–5728 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Luo, Q.-M., Srivastava, H.M.: Some relationships between the Apostol–Bernoulli and Apostol–Euler polynomials. Comput. Math. Appl. 51, 631–642 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Milovanović, G.V., Djordević, G.P.: On some properties of Humbert’s polynomials. Fibonacci Q. 25, 356–360 (1987)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Milovanović, G.V., Djordević, G.P.: On some properties of Humbert’s polynomials. II. Facta Univ. Ser. Math. Inform. 6, 23–30 (1991)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Milovanović, G.V., Mitrinović, D.S., Rassias, ThM: Topics in Polynomials: Extremal Problems, Inequalities. Zeros. World Scientific Publ. Co., Singapore (1994)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ozdemir, G., Simsek, Y.: Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers. Filomat 30(4), 969–975 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ozden, H., Simsek, Y., Srivastava, H.M.: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 60, 2779–2787 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pincherle, S.: Una nuova extensione delle funzione spherich. Mem. R. Accad. Bologna 5, 337–362 (1890)Google Scholar
  24. 24.
    Simsek, Y.: Generating functions for generalized Stirling type numbers. Array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 87, 28 (2013). doi: 10.1186/1687-1812-2013-87
  25. 25.
    Simsek, Y.: Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers. Math. Meth. Appl. Sci. (2016). doi: 10.1002/mma.4143 zbMATHGoogle Scholar
  26. 26.
    Srivastava, H.M.: Some generalizations and basic \((\)or \(q\)-\()\) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5(3), 390–444 (2011)MathSciNetGoogle Scholar
  27. 27.
    Srivastava, H.M., Choi, J.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier Inc, Amsterdam (2012)zbMATHGoogle Scholar
  28. 28.
    Srivastava, H.M., Manocha, H.L.: A treatise on generating functions. In: Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press John Wiley & Sons, Inc., New York (1984)Google Scholar
  29. 29.
    Srivastava, H.M., Kurt, B., Simsek, Y.: Some families of Genocchi type polynomials and their interpolation functions. Integral Transforms Spec. Funct. 23, 919–938 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Gulsah Ozdemir
    • 1
  • Yilmaz Simsek
    • 1
  • Gradimir V. Milovanović
    • 2
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of ScienceAkdeniz UniversityAntalyaTurkey
  2. 2.Serbian Academy of Sciences and ArtsBeogradSerbia
  3. 3.Faculty of Science and MathematicsUniversity of NišNišSerbia

Personalised recommendations